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is denned to within a phase factor, it is enough to show that |0(0)| = 1. To this end, we make use of the X2-normalization of and (41). First, compute for fixed N,
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Ph. Tdmmitchian
which is by definition equal to 27r|<jXO)| 2 . Letting N —> oo, leads to
which implies |(0)| = 1. In conclusion, we have
which is essentially the Fourier-side version of the dilation equation. We are now ready to construct the wavelet t/>. Step 3: Analysis of WQ. By definition, V\ = VQ 0 WQ. Consider a function f ( x ) € WQ and its Fourier transform /(w). Then there exists p(u), 27r-periodic, such that
Every function in VQ has a Fourier transform which can be put in the form
where m(u>) is any 2?r-periodic function. Consequently, the orthogonality between / and V® takes the form
or
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129
for every m(cj) 6 L2(T). Thus, we necessarily have
Let mi(u;) = e~zc"'mo('^ + f): the vector
is orthogonal to
and has unit norm. Therefore there exists a 27r-periodic function q((jj] such that
for every u. This implies that
where
and H/ll = 27r||g|| L 2( T ). These two relations show that the i]}(x — k) form an orthonormal basis of WQ. Finally, i(>(x) takes the form
in physical space. If the a^'s are the coefficients of the dilation equation, f/-' is then rapidly decreasing, along with its first r derivatives. This concludes the proof of Theorem 7. The wavelet ^'(x) has some additional cancellation properties, which justifies its name of "wavelet." For example, because
the integral of tf> is vanishes. A much more accurate characterization of the above remarks is given by the following result.
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Ph. Tchamitchian
Proposition 1. With the notation defined above, we have: i) All the polynomials of degree less or equal to r belong to the "closure" of V0, in the sense that if 0 < n < r, there exists coefficients van^ such that
Proof. Of course, ii) is a simple consequence of i). But i) is not trivial, except for n — 0. In that case, we have
because (0) = 1 and (j)(2irk) = 0 if k ^ 0. Without going into the details, let us draw a sketch of the proof for other values of n. If E j ( x , y ) denotes the kernel of the projection onto Vj, the inclusion Vo C Vj implies the identity
But when j —+ +00, the localization of z i—v E j ( x , z ) forces z to lie in the neighborhood of x. The idea is then to expand Eo(z,y) around EQ(X, y), as a function of z. This leads to a sum of terms of the general form
Next, define
Because EQ(X + 1,^+1) = EQ(X,Z), Pn is 1-periodic. We also have, by a scaling argument,
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131
Finally, the expansion of EQ(Z, y) up to order r — I gives
One deduces that, for almost every u and ,T,
(the sequence {u} — 23x} being equidistributed modulo 1); then again by scaling, we have
and finally
This proves i) of the theorem for every value of n < r — 1. When n — r, more care is required. Of course, the integral formula, for the last term in the Taylor expansion of E0(z, y) must be used. Taking into account the previous results, one is left with
and then with
If x is a Lebesgue point of -j^Eo(t, y), one can reduce this to
and argue as before to get the desired conclusion. Proposition 1 is thus proved. One can symbolize the relation
by the simple picture shown in Figure 9. (Notice that in most examples ij) is centered around |.) To simplify the notation, we associate the point Ik + |J 2"^ with the symbol A. Thus we shall sometimes use tl>\ instead of tyjk- Each V'?-fc = '0,\ is represented by the point ( A , 2 ~ J ) in the position-scale plane shown in Figure 10. A scale is the inverse of a frequency, so the small scales, i.e., the high frequencies, are near the A-axis.
Ph. Tchamitchian
132
Figure 10: Dyadic grid.fghjfgfkhkghfkghklfdgkfklfdghkkglhfklkg
Figure 10: Dyadic grid. 5.3
Examples
Let us now conclude by considering several examples in some detail. Example. Spline wavelets. The MRA of our first example leads to orthonormal bases of spline wavelets, independently discovered by G. Battle and P. G. Lemarie. The corresponding ip have an exponential decay at infinity, with a rate depending on the regularity of the wavelet. This will be our "geometric" example, in the sense that the definition of the Fj's begins with geometric considerations. Example. Dyadic interpolation. This type of interpolation has been introduced for fast interpolation procedures. Actually, these techniques were already known to researchers studying fractals [15] and computer graphics [11]. Indeed, the extension of the definition to several dimensions leads to an algorithm to very quickly generate surfaces. It seems actually that this kind of idea goes back to de Rhani. Here is the precise definition of VQ which depends on a, sequence
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of numbers {cfc}fce^, finitely supported in practice, otherwise in Z 1 , and such that J3fc ck = 1Definition 7. T/«e function f ( x ) € VQ when i) f is square-integrable and continuous; ii) f is determined by its sampling on TL through the formulae
The study of the MRA generated by this definition is instructive, and is left to the interested reader. Our previous analysis of general MRA suggests another approach, in which one constructs an arbitrary function mo to obtain bases of wavelets with desirable properties, without the use of geometric considerations. This line of research was developed by Y. Meyer, I. Daubechies, and others. Here, we only mention two of the MRA they obtained. Example. Y. Meyer's bases. Let 6 ( t ) be a C°° function such that 0(t) = 0(-t), 8(t) = 1 on [0, f], 0 ( t ) = O i f * > M ? and 0(jr-1) = l-B(t) on [f,^]. Then, if m 0 (w) = sin(^#(o;)) and c/) is denned through (43),
J
O
J
^
L
t
i
'
;
'
j
L
d
d
J
This is the smoothed version of our second example, leading to bandlimited wavelets. Example. Daubechies' compactly supported wavelets. I. Daubechies gave an algorithm for constructing bases of wavelets such that tj} has finite support. The regularity of ip can be arbitrarily high; the size of its support growing linearly with the regularity. These wavelets are especially useful in numerical analysis. We will also use them in the following sections since their compact support allows us to avoid some technicalities. Example. Biorthogonal bases. Finally, we note that orthonormality is often not necessary, and may
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Ph. Tchamitchian
be replaced by biorthogonality. In Cohen et al. (1990), the notion of biorthogonal MRA is developed. The concept of biorthogonality will be crucial in later sections. However, we temporarily postpone the question of its usefulness.
6
Operators
In the second part of this course, we present a more difficult set of results, theorems and proof outlines, describing some of the links between wavelets, elliptic differential operators and singular integral operators of Calderon-Zygmund type. The central idea is the following: there are large classes of operators for which the wavelets (or eventually suitably modified wavelets) axe nearly their eigenvectors. By this, we mean the following. If the wavelets were the given operator's true eigenvectors, its associated matrix would be diagonal. Instead, we will associate to our operators matrices with decreasing elements off the diagonal, in such a way that, up to a given precision, those matrices can be thought of as being band-limited. Of course, although the idea is simple, its concrete realization demands the solution of some specific problem areas, most of them dealing with the class of Calderon-Zygmund operators, which turns out to be of central importance in the subject. Section 7 introduces this class. However, before studying it in full generality, we first describe a subclass crucial to the sequel. We then characterize some functional spaces through the use of the wavelet bases. Sections 8 and 9 are devoted to the study of operators. In Section 8, we explain how to represent an operator in a given wavelet basis, and then construct its associated matrix. This is then used to complete our study of Calderon-Zygmund operators. We also consider the invertibility problem., which, may be presented as follows. Let M be a matrix associated with an operator T with a suitable property of concentration near the diagonal. Assume that T is invertible (on L2 for example). Then the question to answer is whether the inverse matrix, M~], has the same concentration property. If the wavelets used to construct M are reasonably close to eigenvectors of T, this should be true. But we will see that this is not always the case, and will describe the counterexamples and the
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135
results which are known so far. In Section 9, we turn to operators for which the concepts developed in the previous sections do not apply directly. We show that it suffices to adapt the wavelet bases to these operators in a suitable way, the key point being the need for a nonstandard cancellation property for the wavelets. Once this is done, we quickly cover the properties of these operators, which somehow parallel those discussed in Section 8.
7
Calderon-Zygmund Operators and Characterization of Functional Spaces
In this section, we describe the relationships between bases of wavelets and operator theory. Our first aim is to understand why wavelets provide unconditional bases for most classical functional spaces. The relationship between unconditional bases and operator theory can be seen as follows. First we recall the definition of an unconditional basis. Definition 8. An unconditional basis {e,}i^^ of a given Banach space E is a Schauder basis, which means that every f e E is the limit of finite sums £^=i c,-ej, where the coefficients c,- are unique, with the additional property that the series £^ c8-e; is unconditionally convergent. In other words, the convergence of J^; c;e; does not depend on the order of summation. It is well-known that this is equivalent to the existence of a constant C independent of /, and such that for every integer N and every sequence {cj}, where e; 6 [-1,4-1], one has
Intuitively, this means that inclusion in the space E is a property of the modulus of the coefficients c t -, independent of their phases. In operator language, for every sequence {e;} = e, the operator Pf, defined by P^e{) = e;e,-, is continuous on E, and is bounded independent of e. This is why, returning to a basis of wavelets V'jfc? we are interested in operators Pf such that
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Ph. Tchamitchian
These operators are obviously isometries of i 2 , but what happens in X p , Hl, BMO (bounded mean oscillatory) and other spaces? 7.1
A Class of Calderon-Zygmund
Operators
Although the operators Pc are particularly simple, they allow the introduction of a class of operators which will be of central importance 7 . It is the class of operators T denned by
such that the TJ^'S satisfies standard estimates. Definition 9. A family of functions r^, j, k G TL, is said to satisfy standard estimates if there exist two constants C, a > 0 such that
and
The intuitive meaning of these estimates is that all the functions Tjk have almost the same shape, obtained from an initial pattern by dilations and translations. It is an extension of the model case, in which Tjk(x) = 2 J / 2 r(2- ? ,T - k), for a localized and regular function T. By construction, the kernel of such an operator T is given by
in which case K(x,y) satisfies the classical Calderon-Zygmund estimates, also called standard estimates. Definition 10. K(x,y) satisfy standard estimates if it is a continuous function on IR x lR\{(,r, y ) , x = y}, such that there exists two constants C, 6 > 0, with
To facilitate further references, we denote this class of operators by C
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It is not hard to see why our K(x,y~) fulfills these conditions. The size estimate on A" reduces to the evaluation of
The summation over k leads to
But it is easy to see that
uniformly. This is a discrete version of the integral
hence we obtain the size estimate on the kernel K(x,y). For the regularity estimates, we only have to choose 6 < a. Then, proceeding as before, we are led to
However, these estimates do not imply that T is a CalderonZygmund operator (CZO), unless we know it is continuous on L2. Indeed, let us recall this last definition. Definition 11. A Calderon-Zygmund operator T is an operator whose kernel satisfies standard estimates, which is continuous on L2, and for which
when f and g are two test functions with, disjoint supports.
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Ph. Tchamitchian
(The condition on the supports of / and g allow us to include the identity, and more generally multiplication operators — by a bounded function —, in the class of Calderon-Zygmund operators.) For example, the previous operators Pf are CZO. But, if we want T, defined by (44), to be continuous on L 2 , then we must impose some additional properties on T. In the case where Tjk(x) = 2 J / 2 e~l 2J ' r ~ A ' : l , then T is not continuous on L2. In fact, when Tjk has the special form Tjk(x) — 2 j '/ 2 T(2 J ;c — k), T is continuous only if J r = 0. This latter fact shows that cancellation properties play a role. At present, we will be content with the following: Proposition 2. // / Tjk = 0 for every j,k, then T is continuous on L2. The proof is elementary, but relies on a construction which will be of constant use: the transposition of the problem in terms of matrices. Because {V-'jfc} is a basis of L 2 , the continuity of T is equivalent to
and in turn implies that
defines a continuous operator on I2. We now define the main class of matrices we will constantly refer to. (The set A, introduced below denotes the set of all indices A.) Definition 12. We call A4 the space of matrices M — {m\^\i}, A, A' G A, such that there exists two constants (7, 7 > 0 with
Perhaps the best way to understand the meaning of these estimates is to prove the following Lemma. Lemma 2. If {r\} is a set of functions satisfying standard estimates, and such that J T\ = 0 for every A, then the matrix (< T\,T\I >) belongs to j\A.
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In particular, if r\(x) = 2-7'/2r(2-'a; — fc), i.e. {rA} is a family of wavelets, with r a nice function of vanishing integral, then the scalar products < TA,T^ > satisfy the estimates (45). Now suppose that 3 > J''iT\' is at a coarser scale than r,\, and rA being of vanishing integral, the product < T\,TX> > is not a mean value of T\< around the point A, but a mean oscillation (see Section 2.3). In fact, we have
Using the estimates of Definition 9, we immediately deduce that
which is equivalent to (45) with 7 = f • (We present this type of estimate in its simplest form. More sophisticated versions can be obtained by decoupling the index of regularity and the index in the shape functions wa in Definitions 9 and 12. For example, if the functions T\ are decreasing rapidly enough at infinity, one has (45) with 7 = a, the index of regularity of the r\'s.) The continuity properties of the class Ai are described in the following simple and important result, due to P. G. Lemarie and Y. Meyer. Theorem 8. M. is an algebra, and every matrix in M defines a continuous operator on l^(A) and lfiw(A\ where l^(A) is the space of sequences {c\} such that J^\ ICA 2~J//2 < +00, and by duality, {c\} G 1™/W(A} means supx \cx\2J/2 < +00. The theorem has two consequences: The first is that, by interpolation, if M = {m\t\>} € M, then M is continuous on 12(A). Using Lemma 2, this proves Proposition 2. The second is that, if 7 is defined by (45), the matrices whose elements are and
are also continuous on l ^ ( A ) , lf/w(A'), l'2(A) etc., as long as \6\ < 7. The proof of Theorem 8 is not very much longer than its statement. We only show the continuity on /^, and leave the other parts to the reader.
140
Ph. Tchamitchian This continuity is equivalent to proving that
for every A'.
by a straightforward application of the Lemma 1, which in turn implies the desired inequality. We now turn our attention to functional analysis, and the characterization of spaces by the wavelets bases. All these characterizations follow the same pattern. Hence, we will not treat the problem completely, and consider only typical situations.
7.2
Characterizations of Hl, BMO, and Lp, (1 < p < oo)
The operators Pe, defined by P6(ijj\) = CA^A; where e\ — ±1, are CZO, with constants (in Definition 11) not depending on e. Hence, they are uniformly continuous from Hl to Ll:
for every / G H* and every sequence €. If dfjL denotes the probability measure defined on { —1, + 1}"4, we recall the celebrated consequence of Khintchine's inequality,
where ~ means that the two quantities above define equivalent norms on / 2 . Thus, integrating (8) with respect to e, we get
with CA =< /, V'A >• Now define a square function A ( f ) by
Wavelets, Functions, and Operators
141
By translating t/> by an integer, it is not difficult to see that A(f) € Lp if and only if
with equivalence of the norms. This implies that ||A(/)||i < C |/||#i. In fact, the converse inequality
is also true. Indeed, if H denotes the Hilbert transform, one has HPc(fa) = 6j f c(ff^)A, where
Consequently, the family (Htp)\ satisfy standard estimates, and with vanishing integrals. Proposition 2 applies, showing that HPt is continuous from -H1 to Ll. Because PfPf — /, we then have the continuity and the invertibility of Pf on H1, which gives us the existence of C such that
Again integrating with respect to e, we obtain
Theorem 9. / € H1 if and only if A ( f ) 6 Ll, and ||A(/)||i is equivalent to \\f\\fji.
Along the same lines, but even simpler, one can prove that
if 1 < p < +00. However, BMO is not characterized by requiring that A(f) 6 L°°: this condition is strictly stronger than membership in BMO. The characterization of BMO will be of Carleson's measure type. Before stating it, let us define, for each A, the interval I\ as the interval [k2-j,(k + l ) 2 ~ j [ , if A = (fc + |) 2~ J .
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Ph. Tchamitchian
Theorem 10. / — ^\c\ip\ belongs to BMO if and only if there exists a constant C such that, for every A,
Moreover, the best constant C defines an equivalent norm on BMO. To see why this condition is necessary, consider the function / = !C,\ c.vV'A in BMO, where we assume first that only a finite number of coefficients c\ are nonvanishing. For each A, let
By construction, A(a\) is supported in /\, which implies
Hence, a\ is in // ! , and
But this means precisely that
This can be extended in a straightforward mariner to the case where the c^'s are not finitely supported. The converse part of the proof may follow a standard pattern, and we will not develop it. Instead, we prefer to push a little further the investigation of H1. Maybe the reader has recognized in the previous functions a\ a kind of (nonnormalized) atom of If1. What is the relation between atomic decompositions of Hl and wavelets? To bypass some potential technical issues, assume now that the wavelet ^, from which we deduce each if>jki '1S compactly supported. Recall that an 7_Aatom of Hl is a function a, supported in an interval /, and such that f a = 0, while \\a\\p < l\l/p~l.
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Definition 13. An atomic decomposition of f G Hl is an expansion
where the a^s are Lp-atoms, for a given p G [l,+oo], and {at} is a sequence in I 1 , whose norm may be chosen equivalent to ||/||/fi, independently of f. Thus, each 2 J//2 ^>A is an i p -atom of H1, for every p G [1, +00]. However, the series
is not an atomic decomposition of /. If it were, we would have
but it can be shown that this relation is equivalent to / G B®'1, which is a space strictly embedded in H1. So, if we want to recover an atomic decomposition of / from its wavelet expansion, we must gather the wavelets, following an algorithm we now describe. The key is given by the square function A ( f ) . For each A, let us define
Then, we obviously have
If t > 0 is a given threshold, we define
If x G fi, there exists A such that L\ > t, and hence I\ C O. This allows us to write
where each Jn is a maximal dyadic interval in fL In particular, we have
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Ph. Tchamitchian
For each t = 2k, k 6 /X, we consider the corresponding fi^, and decompose it into
We say that A <E Ak,n when I\ C Jk,n, but I\ <£ Jk+i,P for any p (notice that each Jk+i,p is contained in one Jk,n)- We then define h,n by
By construction, / = Y^k,n^k,n- We now show that each bk,n is a iionnormalized atom. The central estimate is the following:
where Xj denotes the characteristic function of ./. The proof uses the simple geometry of dyadic intervals. A(bk,n) being supported in Jk,n, we pick up x in Jk,n- If A(f)(x) < 2 fc+1 , there is nothing to prove, so we assume x G fi^+i- Then, x 6 /(x), a dyadic maximal interval in fifc+i. If A 6 Afc, n and x 6 /A, either JA C /(x) or /(x) C /A, where I ( x ) is the "father" of /(,?:), i.e., the unique dyadic interval containing /(x) and twice its size. The former being forbidden by the definition of Ak,n and i(x), the latter is necessarily true. Consequently, A(6fc >ri )(x) < Z/ M < 2fc+1, where /i is such that /(x) = / M . A first corollary of this estimate is that
Also, if we define a/c jri by
then afc ire has finite support, in the image of ,/^.,n under a dilation by a factor depending only on the wa,velet if} (and centered at the middle of Jk,n)- We have
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which shows that if 1 < p < + 00. Of course, J o,k,n = 0. Each a^n is an iyp-atom of H1, for I
and 7
This means that ^ ak,n a-k.n is an atomic decomposition of /. Finally, the above-constructed atoms are not in i°°, a priori. We are faced with the impossibility of characterizing i00, by any kind of unconditional bases, for the weak-* topology. 7.3
Characterization of Sobolev Spaces
Characterizing functional spaces can be exploited much further. Because it will be useful later on, we show how to characterize the usual Sobolev spaces, denoted by B^'m, or Hs if no confusion with the real Hardy spa.ce is possible. The question is how can we measure ||-DS/||2 where D = }£? We begin by writing formally
and then by noticing that (with obvious notation)
Of course, we assume that 't/> is regular enough if s > 0, oscillating enough if s < 0. We have factorized the operator Ds, according to the formula where A is the diagonal operator defined by
146
Ph. Tchamitchian
and Ts is the operator which, to i/>\, associates (Dsijj)\ vmwhich is none other than the wavelet version of the well-known Calderon's factorization of Ds. The characterization of B'^'2 follows from Theorem 11. (i) {i'\} is an unconditional basis of B^'2 if and only if
(ii) {tf>\} is an unconditional basis of B'^' if and only if the abovedefined operator Ts is continuous and invertible on L2. (Hi) If {V'AJ is constructed from a MRA of regularity r as in Theorem 7 and Proposition 2, and if\s\ < r, then Ts is continuous and invertible on L2. Proof. Point (i) is again a consequence of Khintchine's theorem (46), and the point (ii) is an obvious corollary. Point (iii) uses Proposition 2, since under the hypothesis, both sets of functions (Ds^)x and (D~sif.')\ satisfy standard estimates, and their integrals vanish. Hence Ts is continuous on L2. It is also invertible, because the system of (D~sif})x is biorthogonalto (DsiJj)\. Consequently, the operator whose kernel is written
is formally the inverse of Ts. This operator is continuous on L2 thanks to Proposition 2.
8
Calderon-Zygmund Operators, Matrices and Functional Calculus
Here we examine more closely the relationship between wavelets, MRA and CZO, showing how one may analyze these operators, and what can be said about their functional calculus. For this purpose.
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we single out an algebra of CZO, and study issues related to the invertibility of operators. In what follows, we consider a MRA which generates a wavelet ij) compactly supported in [—M, M]. The MRA has a regularity r > 1, and hence (r + 1) vanishing moments. Denote by PJ and Qj the orthogonal projections onto Vj and Wj, and, depending on the context, by t/>jk or -(/>>, with A = (k + |j 2~ J , the wavelet 2j/2i{>(2jx - k). How are wavelets used to analyze operators? There are two approaches. Following Beylkin, Coifman and Rokhlin (1991), one approach consists in associating the matrix < T^^Vv > ^° *ne °P~ erator T. This matrix is called the standard representation of the operator T. The second representation, the non-standard one, begins with the telescopic series
8.1
The Non-Standard Expansion
This last expression involves three types of scalar products:
The first one can be understood as resulting from the interaction between wavelets at the same scale 2~ J through the action of T, while for the two other ones, the coarser scales are globally represented by the functions (j>jk, in their interaction with wavelets at scale 2~ J . Suppose that T is an operator whose kernel is regular enough to satisfy the estimates
148
Ph. Tchamitchian
where C depends only on T and the MRA. To see this, consider
and similar expressions for /3Jkt, 7^. This is possible because
The inequality follows from a Taylor expansion of K(x, y ) , up to order r, around the point (&2~ J , y) or the point (x,£2~^). The principal part of the expansion does not contribute because of t/Vs vanishing moments, while the remaining terms lead in a straightforward manner to the announced estimates. What happens when \k — i\ < 2M? The example of K(x, y) — \x — y\~l shows that, if we wish to understand the representation of Z 2 -continuous operators, we must make an additional assumption:
This property is referred to as the weak boundedness property. It is a much weaker condition than the i 2 -continuity of T. For example, if the operator T has an antisymmetric kernel, then it fulfills this condition. We now compute T f ( x ) , when / £ L2. Let djk =< /, i/>jk > 5 and assume first that only a finite number of these coefficients are nonvanishing. We have
Thanks to the previous estimates, we can define three sets of functions, {Tjk},{(Tjk} and {pjk}, by
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(Notice the different orderings of k and I.) We can then rewrite Tf as
It is easy to see that the families {rjk}, {&jk} and {pjk} satisfy standard estimates. This means that we have decomposed T into the sum of three operators related to the class C defined in Section 2.4. The first and the third belong to C: Ti(^jk) = Tjk and Ts^jk) = pjk while the second is the adjoint of an element of C: T£(i(>jk) = OjkWhether T is continuous or not on L2 then reduces to deciding when an operator of the class C is continuous. We gave a partial answer to this question in the previous chapter. Now we are in a position to consider the complete answer. Proposition 3. Let {0\} be a family of functions which satisfy standard estimates, and let U be the operator defined by U(^\) = 0\. Then U is continuous on L2 if and only if the numbers f 6\ satisfy a Carleson's condition, i.e.
Proof. The proof begins with the decomposition
if this family of functions satisfy standard estimates, and j 0\ = 0. By Proposition 2, the operator U such that U('^x) = @\ is continuous.
150
Ph. Tchamitchian There remains an operator V which acts on / according to
It is well-known that Carleson's condition implies
where, by definition,
But G' is a maximal function, and ||G||2 < C||^||2. This completes part one of the proof. The converse is straightforward since
and U*(l) belongs to BMO if U is continuous on i 2 , by the classical theory of CZO. Moreover, if (3 = ?7*(1), the operator V* is the socalled paraproduct
In fact, we have almost proved the celebrated Tl-theorem. Theorem 12. Tl-theorem (G. David, J. L. Journe). Let T be an operator whose kernel satisfies standard estimates. Then, T is continuous on L2 if and only if T has the weak boundedness property, T(l) e BMO, and T * ( l ) 6 BMO.
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When T is not known to be continuous, but has a CalderonZygmund kernel and satisfies the weak boundedness property (48), we decompose T as in (53), and define T(l) and T*(l) by
They are automatically elements of the Besov space B^°°. The Tltheorem now becomes a consequence of the previous proposition. The non-standard form, when applied in a numerical context, gives rise to non-standard matrices that are sparse, and therefore can be rapidly evaluated. The calculation of these matrices has been elaborated on and applied by G. Beylkin, R. Coifman and V. Rokhlin. It only is efficient for operators with a smooth nonoscillatory kernel (CZO's, pseudo-differential operators, fractional integration, etc.). We refer to Beylkin's chapter in this volume for numerical considerations. 8.2
The Standard Form
Though more natural, the standard matrix
is not always convenient for the analysis of T. Suppose for example that, for a given A, we have Tip\ =
well-defined by assumption. Then, the three following properties are equivalent:
152
Ph. Tchamitchian
ii) T't/'A = T\- where the functions T\ satisfy the standard estimates, and / T\ = 0 for every X.
Proof. That i) and ii) are equivalent is proved similarly to Lemma 2 and Theorem 8. The equivalence of i) and ii) to iii) is a consequence of the decomposition (53). Hence, by Theorem 8, the space
is an algebra of operators, the so-called Lemarie algebra which provides a, convenient framework in which to consider problems such as inyertibility.
8.3
Inversion of Calderon-Zygmund Operators
8.3.1
Results
The abstract setting of the problem is the following. We are given an algebra A of bounded operators acting on a Hilbert space H, which is not closed for the operator norm. The problem we address is to know whether the inverse of an invertible operator in A still belongs to A. If A were a von-Neumann algebra, i.e., closed for the operator norm, the answer would be in the affirmative. Indeed, if T is invertible, so is T*T, and because T~l = (T*T)~1T*, we may restrict the problem to the case where T is self-adjoint and positive 8 . Then, we have A < T < B for two constants A,B > 0, from which we deduce that \\T - ^±^|| < 1~A. This allows us to write
where \\U\\ < ^rj < 1. But it is now obvious that (I— U)~l belongs to A, and so does T"1. (Notice that the reduction to a perturbation of the identity is still valid in our own context.) 8
We will always assume that A is stable by taking the adjoint.
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Our main example will be the algebra £. We will also consider the following simpler algebras: £, the algebra of matrices {mk,e}k,e. € 2Z such that there exists two constants C, £ > 0 such that
C, the algebra of matrices {mk,e}k,t£j, snch that, for every n G 2Z, there exists Cn for which
and J, the algebra of matrices (mk,()k,ee^ such that there exists two constants (7, e > 0 for which
The answers to our problem for £, £ and I are affirmative, though the answer for C is not. Theorem 13. If M, element of £, C or I, is invertible on I2(rffj), then M~l belongs to £, C or J. This theorem is due to S. Jaffard and Y. Meyer9. However, on the algebra £, we proved that the situation is completely different. Theorem 14. For every on L2 but not on Lp if
there exists
invertible
There also exists T e £, invertible on L 2 and on Lp for 1 < p < +00, on Hl but not on BMO. It follows that T"1 and T"1 cannot be CZO. However, we proved also a positive result, corresponding to the limit case of the counterexamples. Theorem 15. // T G C is invertible on Hl and on BMO, then
T~l e C. 9
Except for the case of J, due to S. Jaffard and J. L. Journe.
154 8.3.2
Ph. Tchamitchian The counterexamples
The simplest way to prove Theorem 14 is given by a set of counterexamples designed by P.G. Lemarie. (For a different and more complete treatment, the interested reader can refer to Meyer (1991) and Cohen et al. (1990), the latter giving the more conceptual formalism.) Proof of Theorem 14. Let U be the operator defined by
U is a partial isometry of _L 2 , whose adjoint is given by
Consequently, for every a 6 [0,1[, / — all is an invertible CZO in the algebra £. The image of the basis {V'jfc} is the set of functions
which forms an unconditional basis of L2 associated with the biorthogonal system The operator (/ - all) is invertible on Lp or BMO if and only if {&jk} is an unconditional basis of Lp or Hl, and this is equivalent to asserting that {TJ^} is an unconditional basis of Lq, - + - = 1, or Hl. The point is that i_ i belongs to Lq if and only if a < 22 «. This can be seen by computing explicitly the square function A(T^O). Hence, / — all is not 1_!
_i
invertible on Lp as soon as a > If 2 , which is possible if a > 2~5. The end-point case a = 2~z is also interesting, for then TO,O ^ Hl, showing that T — I — 2~?U is not invertible on BMO. However, it is invertible on Hl. The operator T is obviously continuous on II1. To prove its invertibility, we show that *T~ 1 is continuous on BMO, by explicitly computing the image of any function / = Yljkcjk'i']kA direct calculation gives
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Let us return to the notation if)\, c,\, . -., if I\ is the interval [&2~- 7 , (fc+ l)2~jI we denote by s(/\) the interval [k2~j,(2k+ l)2~ j - 1 [. Then, we can rewrite *T~l f as
Suppose / € BMO. We have to estimate
By Cauchy-Schwarz inequality, it is dominated by
But, if /y C /A, for each value of n G IN, there is at most one /M C /\ such that s n ( I ^ ) — I\i. Consequently, the last expression is itself dominated by
b
proving immediately that f'T~lf 6 BMO. Notice that the factor 2~2" plays a weak role, in the sense that we only needed it to define a summable sequence. For example, the same proof works equally well for each (/—at/*)" 1 , a < 1, although they are far from being CZO. Finally, the operators Te announced in Theorem 14 may be defined as (J - otU)(I - of/*), for a = 2F~. All these counterexamples show how different two biorthogonal systems can be. Theorem 15, which will be proved later, gives the conditions under which two such systems are close to the case of wavelets. We now turn our attention to positive results. The difficulties we face are of two kinds: the first comes from the absence of commutativity. the second, which is specific to CZO, is related to the interaction between different scales. Let us see now what happens with algebras of matrices {mk,e}k,(e%i where only the first difficulty is present.
156 8.3.3
Ph. Tchamitchian Inversion of nearly diagonal matrices
We begin with the algebra £, which we recall is defined by
We need only consider the case of a perturbation of the identity M = I - V, \\U\\ < 1. Expanding M~l = ^n=o Un, we estimate the (TV + 1) first terms by
where A is a constant depending on e, and the remaining terms by
Then, the optimization of N leads to
where
It is remarkable that this proof works in a more general setting. The only ingredients of the proof are the estimates
where the distance function d(k,f.) satisfies
for every e' < e. With the algebra £, d is the usual Euclidean distance, which leads to nice estimates for the matrix elements. But with the algebra Z, the distance is log(l + \k — l\). Thus, this proof gives the estimate
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if M £ I, is invertible. This result is clearly too weak (notice that e' < 1 in any case). Proof of Theorem 13. Let us consider the case of the algebra C, intermediate between £ and T. The proof of Theorem 13 in this case proceeds differently, and follows an approach introduced by R. Beals, in connection with zero-order pseudo-differential operators. It consists in characterizing the algebra by the properties of some commutators, and then using the relations between commutators with M and commutators with M~l. To characterize the algebra £, define an unbounded operator X on £ 2 (Z2) by
The matrix associated with [X, M], if M — {?n^}, is then ([k — i\mkt). More generally, the n-th commutator[X[X • • • [X, M] • • •}] has a matrix with elements [k — (]nm/,e. It is then straightforward to prove that a matrix M belongs to £ if and only if all the commutators [X[X • • • [X, M} • • •}] define bounded operators on f2. Now, if M is invertible, we have
and similarly, the n-th commutator
is a finite sum of products between M~l and commutators with M, of lower orders. Hence, it is bounded on I2 for every n, and M""1 6 (. Finally, we consider the algebra J, for which we present a proof slightly different from Jaffard's original one. extensively using the calculus on commutators. Let M, an element of I, be invertible on t2, and such that (with a suitable renormalization)
for a given a > 0. If M were invertible on I1 and l°°, the proof would be almost complete. We notice that [X, M] maps I1 to t°°. Consequently, maps tl to l°°. Therefore, there exists a constant C such that
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where mj^,1 denotes the matrix element of M"1. To obtain the additional rate of decrea.se of the matrix elements off the diagonal, we will need the following lemma, whose proof is left to the reader.
(This clever trick is borrowed from Jaffard's proof.) If /? < 1, we have and
Consider the first sum. It is the matrix element of the product of three matrices, A, B, C defined by
The two other sums being treated similarly, we obtain
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for every (3 < a (if a < 1). Then, by a bootstrapping argument, we use this estimate and (56) to prove that in fact
We are finally left with the task of showing why M is invertible on I1 and £°°. The idea is to approximate M by a truncated version, denoted by M. Then, M is invertible on I1 and £°°, because it is a band-matrix. By carefully controlling the different constants, we will prove, inductively that M is invertible on tp for every p > 1. To begin, suppose M is invertible on tp and tp , where - + ^j = 1, and P < P'• If N is an integer, to be chosen later, we define M by
Hence, if TV is sufficiently large, M has an inverse on tp and lp , uniformly in N. Because M £ £ (or £), -^ is also invertible on ^1 and £°°. The objective is to estimate its norm. We will show that
This is derived from a commutator technique. Let s = 2 — | + ot- By construction, we have
and [X, M] is continuous from lp to lp' by Lemma 3 if s < 1 or by p < p' if s > 1. The same property is then true for [X, M""1], which implies that We also have the rough estimate
which shows that [X, M] and [X, [X, M}] are continuous on I2 with norms less than N3. We deduce that [X, [X, M~1}} is continuous on £ 2 , with norm less than C(M)N6. Hence,
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By optimizing e, this implies the desired estimates on HM" 1 ]^^ and HM-^IOO.OO- Now let pi € [l,p] and / e [0,1] such that
Then,
Knowing that M is invertible on each / ? , we write M = M(I + M~1R). Because
we see that M is invertible on 1P1 and lp'i as soon as [2 — -J t < a, nr J_ _ 1 <- a or PI P < 2 • This implies that M has an inverse on all the lq spaces, which concludes the proof of Theorem 13. We now return to CZO, and prove the result stated in Theorem 15. Proof of Theorem 15. Recall that T 6 C means that M, the associated matrix belongs to M., or more explicitly,
We must prove that T~~l G £ as soon as T is invertible on If""1 and BMO. Because the estimates on m\^\i are not of convolution type, we cannot directly apply the previous techniques. In fact, the counterexamples show that there are no subalgebras in £ which are characterized by stronger conditions on the locality of the coefficients m,\ r \', such that the stability by taking the inverse would be true. Actually, we must use two unbounded operators. A and A", denned by
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A is a pseudo-version of \/—A, and X is the paramultiplication by the function x. The desired estimates on m^\, will be obtained in three steps:
which is (57) in the end-point case 7 = 0, and finally (57) for a given e > 0. Step one is a result of complex interpolation. If z is a complex number with modulus less than 7/2, the operator i\zTK"z is bounded on jff" 1 , uniformly in z, and depends analytically on z. If we take any / in jET 1 , and a corresponding g in BMO, ||ff||BMO — 1> such that
we also have By Schwarz's lemma
which implies that
as soon as \z\ < £Q, for a given £Q. Because C-2 depends only on CQ, Ci and 7, so does £Q. This is an example of the following more general statement, communicated to us by G. David and S. Semmes. Theorem 16. Let Tz be an analytic family of operators, each operator defined and bounded on a Banach space Ez. Assume that the spaces K? form a family of complex interpolation spaces, and that TO is invcrtiblc on EQ. Then, there exists £ > 0 stick that Tz is inve.riible on Ez if z < e.
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In our own context, we obtain that A.EoT~lA.~Eo and A^TA*0 are continuous on I/1 and BMO, from which it is easy to deduce our first estimate on m\f\i. The proof of step two proceeds as in the proof of Theorem 13, by considering the commutator [X, T~1}. It is not surprising for the following intuitive reason. Denote by x the pointwise multiplication by the function x. If K(x, y) is the kernel of T, then \x — y\K(x, y) is bounded, which means that the commutator [.T,T] maps Ll to L 00 . However, T~l cannot be bounded on Ll and L 00 , and we cannot directly use the formula
Thus, we substitute Hl for i1, BMO for £°° as usual, and the paramultiplication operator X for x. If / € Hl, the derivative of (x — X ) f is also in Hl (just compute the image of the wavelet basis), and hence (x — A")/ is bounded. Since
[x,T] maps Hl to BMO, and so does [A^T"1]. This implies the estimate 1
which completes the step 2. We conclude with the step 3 by proceeding as in the proof of Theorem 13. Beginning with the inequality
+ two other terms, like in (56), we try to derive the relation
We only give an indication of how to treat the first sum which we intrepret as (/17?(7),\,A'i where
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and then prove the continuity of A,B,C on suitable spaces of sequences. We use the spaces
and
Through a wavelet basis, these spaces are isomorphic to and 5^,°°. It turns out that • C is continuous on i^. In fact, this is equivalent to stating that
T is invertible on J?1' , which is a consequence of the continuity on Hl of A.-e°T-1Ae<>. • B is continuous from • A is continuous from We use here the two kinds of estimates given by the steps 1 and 2. This shows that ABC maps B^'1 to B^°° if £ is small enough, which is precisely the desired conclusion. The theorem is thus proven. There exists several variants of this theorem. For example, Theorem 17. Let T £ C, of regularity strictly greater than ~ (this means 6 > ^ Jn Definition 10, or equivalently, 7 > | in (57)). Then, ifT is invertible on H1/2 and H"1/2, the homogeneous Sobolev spaces T~l belong to C, As in the case of the Theorem 15, one can construct counterexamples showing that this statement is the best possible. Finally, we mention that these theorems are also true in dimension n, except that the spaces 7/ 1 / 2 and 7/" 1 / 2 must be replaced by ff n / 2 (IR") and /i-"/ 2 (IR") in Theorem 17.
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8.3.4 Numerical implications
The problem of characterizing the inverses of CZO remains wide open (see Pereyra (1993) for recent work). It has however some importance, even from a numerical point of view. We briefly describe why this is so. Assume that we are numerically solving either an integral equation or a partial differential equation by a Galerkin method, with an approximation space given by a multiresolution analysis. Moreover, assume that we use the non-standard representation for the differential operator (because we demand O(N] operations), and that, eventually taking into account preconditioning, we must invert a matrix, say A, related to the representation of a CZO, denoted by T. Then, the inverse of this matrix is also sparse (to within a given precision as always) if and only if the inverse of T is also a CZO of sufficiently high regularity. This means that to compute the inverse of the matrix, one must assume that the operator T is invertible on a sufficiently wide range of spaces, for example on 1P(IR) for every 5, \s < m, for large enough m. In such a case, some theorems, e.g. 14 and 17 apply, and show that the inverse of the matrix A is sparse. Notice that, even if this inverse is not explicitly computed, its sparsity has important consequences to the resolution of any linear system whose matrix is equal to A.
9
Adapting the Wavelets
Until now, we have discussed how to analyze operators with the help of wavelet basis functions. We shall see that one can proceed further by reversing the process, and adaptingthe wavelets to the operators. We present two particular problems: the continuity of the Cauchy operator on Lipschitz graphs and Kato's conjecture in one dimension, which can both be solved in this way. One important point in what follows, in addition to the explicit construction we shall present, is that the MRA is flexible enough to generate special classes of bases, depending on the particular application.
WaveJets, Functions, and Operators 9.1
165
Wavelets Adapted to an Accretive Function and the Cauchy Operator on Lipschitz Graphs
The analysis techniques developed so far cannot be directly applied to the Cauchy operator
where 4> = $' is bounded, because it is not obvious that C^(l) e BMO. (Notice however that it is precisely by evaluating the BMOnorm of C^(l) that T. Murai (1987) obtained the best estimate on ||CVI|2,2-) However, it is well-known that
The problem which naturally arises is then to adapt the tool, i.e., the wavelet basis, to this cancellation property. This means that we require a basis of L 2 (IR), {#A}, such that / 6 \ ( x ) b ( x ) d x = 0, when 6 = 1 + i
for a given function 9. We will instead require standard estimates on 6\. To state our results we need to define accretive functions. Definition 14. A complex-valued function b is accretive when there exists a constant 8 > 0 such that Re b > 6 almost everywhere. Theorem 18. Let b be a bounded and accretive function, and r an integer. Then, there exists a family of functions 9\ of regularity r, such that
for every integer p, and Cntp not depending on A,
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iv) the collection of 0\ form an unconditional basis o/ J L 2 (IR). In fact, there is more structure with the ^'s because they are generated, and themselves generate, a MRA of regularity r. The starting point of the construction is a given MRA, {Vj}j^, of regularity r. We substitute the bilinear form
for the usual scalar product and try to mimic the construction of wavelets by defining
We face two difficulties. The first stems directly from the fact that B is not a scalar product, but a complex-valued form. However, B is bounded on L 2 , and also accretive, which means that, if / G L 2 ,
where 6 is the constant of accretivity of the function 6. This property implies that Xj and Vj do not intersect, and that they are a direct sum in Vj+i: and furthermore, that this decomposition is uniform with respect to j (i.e., the projections onto Vj and Xj are uniformly bounded). However, it is not at all obvious that
Indeed, let Aj/ represent the projection of / into Xj, parallel to Ufc^j %k- We have / = Y^j Aj/ in the sense that
However, we want this sum to be unconditionally convergent or, in other words, that
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The second difficulty is that we are forced to construct our adapted wavelets without the use of the Fourier transform. Let us begin by seeing how to eliminate this last difficulty, following an idea of Meyer. We proceed in three steps: 1. Construction of an unconditional basis in Vj, orthogonal with respect to 5, and satisfying estimates. 2. This allows the explicit construction of the projection onto Xj parallel to Vj: by projecting Wj onto Xj, we will obtain a first unconditional basis of Xj with estimates. 3. Orthonormalize this basis with respect to B as in step 1. The key point here is the orthonormalization procedure, which is due to Poincare. We demonstrate by explaining the first step. Consider the Gram matrix
If we demand that Vj = Vj (which is the case as soon as the scaling function is real-valued), the matrix Bj is uniformly bounded and invertible on 12(7L). In fact, it is accretive:
-i /Q
This allows us to define B • , for example by the classical formula
and the (f>jk, A; 6 K, form an unconditional basis of Vj (with constants independent o f j ) . How do we get estimates? We want to prove that
for every integer p, and the corresponding estimates on the derivatives of cj>jk, up to order r, with Cp independent of j and k.
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This reduces immediately to proving estimates on {Bj the type
—-1 /9
}ki of
—1/2
for every p. In other words, we want Bto belong to the algebra C of matrices, uniformly in j. But, using Theorem 13, and the fact that
one proves the previous inequalities. With the functions (j>jk at our disposal, we can now construct the projection onto Vj parallel to Xj, denoted by TTJ:
Given a wavelet V^, define
Then, the functions Ojk form an unconditional basis of Xj, with estimates. Finally, we 5-orthonormalize the Oj^s- This is done following the same procedure as for the <^fc's. There is only one difference in the proof that the matrix { B ( 9 j k , 6 j i ) } k l is accretive. This is not straightforward, because Xj ^ Xj. The trick 10 here is to note that 9jk — 9jk G Vj, because t/^ is real-valued, so that B(0jk,@jt) = B(0jf;,&je), and we can proceed as in step 1. Denote by Ojk the functions we obtain. By construction, the family {&jk} satisfies standard estimates as described by i) of Theorem 18. We also have B ( 0 j k , 0 j i k i ) = fi(jk)(j'k')- Since if 0 < n < r, the function xn is in the "closure" of Vj (see Proposition 1), we have f xn0jk(x)b(x}dz — 0. Proving that the fljfc's are dense in L2 is left to the reader. We wish to write for every / € X 2 ,
Which is due to P. G. Lemarie.
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the series being unconditionally convergent. That is to say, we want to prove that the family {Ojk} is an unconditional basis of L 2 . Let T& be the operator defined by
Then, our question is: is Tj, continuous and invertible on L21 Because (at first formally) t T 6 ~ 1 (V>j^) = bOjk, this reduces to knowing whether Tf, is continuous or not. But, thanks to Proposition 3, we know what to do: estimate f Ojk and prove Carleson's condition. Because Ojk = E; o-j(k, l)0ji, with o,j(k, l)\ < Cp(l + \k- /|)~ p for every p and every j, it is sufficient to estimate f Ojk. Then, by definition of Ojk,
Let m,j(x) = ^2i(f <j)jf)<j)ji(x). The estimates on <j>jt imply that m,j G L™ and \\mj\\eo < C, llmj"'^ < C2jn if 0 < n < r. Now let
Thanks to orthogonality, / rnji^jk = 0, and it is easy to see that the functions m^jk satisfies standard estimates. Because b is bounded (here, b e BMO would be enough), the numbers f Ojk satisfy Carleson's condition, and our theorem is proved. We now associate an appropriate non-standard form to the Cauchy operator (70. Let us in fact be more general, and consider an operator T, whose kernel is CZO. The projection onto X j , Aj, is given
by and we recall that the projection onto Vj, TT,-, is
The adapted non-standard form will be the expansion
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which gives rise to the matrices whose coefficients are
We now assume that T satisfies an adapted weak boundedness property of the following form. Property 1. If g ( x ) is r-times differentiable,
with
for every p and for 0 < n < p, then there exists a constant C, depending only on the constants CntP such that
Then, using only this property, the regularity of the kernel and the cancellation property of each 6jk, we can prove the estimates
As for the Tl-theorem, the adapted non-standard form gives a proof of the T(6)-theorem. Theorem 19. T(b)-theorem (G. David, J. L. Journe, S. Semmes) Let T be an operator whose kernel satisfies standard estimates. Then T is continuous on L2 if and only if it satisfies the adapted weak boundedness property 1, T(b] <S BMO, *T(6) € BMO.
The proof follows an outline already used. It can be reduced to Proposition 5. Let {T^} be a family of functions satisfying standard estimates, and U the operator defined by U(b9jk) = Tjk- Then, U is continuous if and only if the numbers f brjk satisfy Carleson's condition.
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Proof. If / brjk = 0, the proof proceeds by showing that the matrix B(Tjk,Oj'k') belongs to the algebra Ai, exactly as in Lemma 2. In the general case, one must use the functions
Then, one writes
and one proceeds as in Proposition 3. We also use the fact that the {Ojk} form an unconditional basis of BMO (although it is not a basis of H1}. However, if our aim is to only prove the T(6)-theorem and nothing more, an even simpler construction is preferred, because it applies to a wider class of functions b (the so-called para-accretive functions). This construction, due to R. Coifman and S. Semmes, is basically an adapted Haar system which allows for a simpler proof of Theorem 11, thanks to the very strong localization properties of the Haar functions. The surprising point is that, despite the lack of regularity of the functions (which is not too strong) it is still possible to analyze
czo.
But our aim was twofold, and we now turn our attention to Kato's conjecture. 9.2
Kato's Conjecture in One Dimension
Let a(x) be an accretive and bounded function, D the self-adjoint operator jj^, and / the bilinear form
The domain of J is the Sobolev space H 1(1R), and J is accretive, i.e.,
if Re a > S. The domain Dom(L) of L — DaD is the space of functions / in H1 for which
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for every g 6 Hl. Then, DaD f is by definition the unique function h in I/2 such that
DaD is a maximal-accretive operator in the sense of Kato. Following Kato, there then exists a functional calculus associated with L. In particular, L 1 ' 2 is the unique maximal-accretive operator such that (L 1 / 2 ) 2 = L and whose domain contains D(L). It may be denned via the formula
Kato's conjecture is that Dom(L 1 / 2 ) = Hl(1&). This has been answered in the affirmative by R. Coifman, A. Mclntosh and Y. Meyer (1982), in the same paper where they prove the i2-boundedness of the Cauchy operator on Lipschitz curves. The corresponding conjecture in dimension n, where L = —V • AV, with a bounded and accretive matrix A, is still open. The adapted wavelets lead to an elementary proof of Kato's conjecture in dimension 1 and relies on the explicit characterization, in terms of wavelets basis, of the domain of L. Step 1: let {6\} be a family of adapted wavelets to the accretive function I/a, and suppose that each 6\ is at least Lipschitz, with f(x/a)0\ = 0. Then, because of
we have
which is one way of splitting the multiplication by a into a sum of localized and regular pieces. From this, we deduce that (at first formally)
Step 2 is specific to the one-dimensional case, because it uses the operator D~^. The functions -9^ being oscillatory, the functions D~l \~9\} are well-localized, and regular by construction. We have the following result.
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Theorem 20. If TX = D l i~6\}, then {T\} form an unconditional basis of L 2 , Hl and Dom(L), with the following characterizations. Iff = E\c\9\, then
Lions (1962) proved that, for every maximal-accretive operator L, the spaces Dom(i s ), 0 < s < 1, with L° = /, form a chain of spaces of interpolation. Then, the previous theorem implies
Proof. If a\ = D0\, then /
Finally, it is a basis of Dom(Z/), because
by construction: but {cr\} is also a basis of L2. This proof is simple enough to serve as a model for other situations. Consider /, an interval in H, V a closed subspace of Hl(I) containing 7?o(7), and J the form
defined on V x V (D denoting the operator t ^ with domain V), where again a is accretive and bounded. The choice of V determines the boundary conditions. The form J defines an operator L = D*aD, maximal-accretive, with domain included in V.
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Theorem 21. For every choice ofV, one has
This result can even be generalized, by standard techniques, to the case of an open set fl in 1R, and of a general second-order operator
where the coefficients are bounded and a is accretive. Proof. The proof mimics that of the whole real line case, and is only sketched very briefly. Whatever A denotes, we construct {0\}, basis of L 2 (J) satisfying estimates and cancellation versus ^, and define r\ = D~l (\0\) ,
9.3
Kato 's Conjecture and Inversion of Singular Integrals
The link between Kato's conjecture in dimension 1 and CZO is known to be profound. Kenig and Meyer (1985) proved the formula
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if - = 1 + i(f> and C^ is the Cauchy operator. A little more generally, one can prove the following (unpublished) result Theorem 22. // a and b are two bounded and accretive functions, then
Proof. The proof shows that Kato's problem is related to questions about some inverses of CZO (or slightly different singular integrals). That inverses of CZO are involved, even in the multidimensional case, can be seen exactly as we will see it for the one dimensional case. The reason why we are able to treat the latter case is that the inverses in question are also CZO; but we have since learned that this may not be true, and probably it is false in the former case. In order to present the argument without too much technical detail, consider again the operator L = DaD, and let us explain how to prove i1/2 = TD, where T is a CZO with T (l) =* T(l) = 0. The operator T is given by the formula
We now define which is continuous on Z/ 2 (IR), and
which is denned and continuous from -Hl to L2. In fact, denoting by St(x, y] the kernel of 5t, we prove the following estimates (where C and s are uniform positive constants):
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Ph. Tchamitchian
Once these results are established, letting
the desired conclusions on T are easily obtained. Proof. The starting point is the algebraic identity
The main task is then to obtain estimates on the kernel of ( ~ + jD 2 J (by a scaling argument, we assume here that t = 1). The idea is to adapt the well-known factorization of partial differential operators to ^ + D2, to bring out the role of CZO's (more precisely, a class of singular integrals studied by Bourdaud), and then to invert the factorization. Remarkably enough, we do not need the adapted wavelets (this is no longer true for (bDaD)1/2, however). Let (Vj) be a MRA of regularity of r > 1. We use the basis made of V>ji, k € TL and j > 0, and of <j)Qk, k 6 7L. We still denote it by {^A}; for brevity, with A G AQ. Let AQ be the operator denned by
and which is a wavelet version of \JI — A. OUT factorization becomes
where, by definition, the operator R is given by its matrix in the {V'AJAe/lo basis, M = {m A ,A'}, where
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The functions 2~ J Dtj)\ satisfy standard estimates, and have a vanishing integral. Thus, we obtain standard estimates on the scalar products /2~ J \D?/;>2~- 7 Dt/)\i. The wavelets have no cancellations versus -, but the renorinalization factors 2~-7 are the required substitutes Finally, one has the following estimate:
for every integer n. We see that the matrix M belongs to Ba (with a = 1), the class of matrices P — {PA,A'} such that
for two constants C, 7 > 0. Clearly Ba is an algebra of matrices, which represents operators having a kernel with a Calderon-Zygmund type singularity near the diagonal, and integrable far from the diagonal. They are continuous on the Sobolev spaces HS(HC), for |s| < a. This algebra has been studied by Bourdaud. As in the Lemarie algebra, there exists results on the inversion of operators in Ba. We will use the following proposition. Proposition 6. // an operator P, represented by a matrix in Ba, where a > |, is invertible on Hl^(HL) and on H-1/2(]R), then P"1 has a matrix in BE, for a given e > 0. Moreover, all the constants appearing in (17) for \P^i\ depend only on the analogous constants for \P\\i\, and on the norms of P, P"1 on jET1/2 and /I"' 1 ' 2 . In our case, because the operator R is related to - + D 2 , it is easy to prove its invertibility on H1/2 and ./7"1/2, and therefore to obtain estimates on M~^.
if we compute < Siftx, t/v >, thanks to the operator A^1 on the right hand side, we have
178
Ph. Tchamitchian
Hence, the functions 2 J Stp\ satisfy standard estimates. Writing the kernel of S as
we obtain the inequalities i), ii) and iii). The cancellation properties of S(x, y) are proved directly, by noticing that the integrals
are absolutely convergent, thanks to the previous estimates. Now, the formula
implies that S ( ^ j = tS(l) — 0. This concludes the proof of the theorem.
10
References
General references and introduction Bony, J.M. 1983. "Propagation et interaction des singularites pour les solutions des equations aux derivees partielles non lineaires." Proc. of the Intl. Congress of Mathematicians, Warszawa, 1133-1147. Coifman, R.R. and Wickerhauser, V.M. 1993. "Wavelets and adapted waveform analysis. A toolkit for signal processing and numerical analysis. In Different perspectives on wavelets, Proc. of Symposia in Appl. Math. Daubechies, I. 1992. "Ten lectures on Wavelets." CBMS-NSF Series in Appl. Math. 61, SIAM, Philadelphia. Meyer, Y. 1990. Ondelettes et operateurs I, II Hermann, Paris. Meyer, Y. 1991. Ondelettes et operateurs III Hermann, Paris. Semmes, S. 1989. "Nonlinear Fourier analysis." Bull. AMS 20(1), 1-18. Stein, E. 1970. Singular integrals and differentiability properties of functions. Princeton University Press. Stein, E. and Weiss G. 1971. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press.
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Wickerhauser, V.M. 1993. "Best-adapted wavelet packet bases." In Different perspectives on wavelets, Proc. of Symposia in Appl. Math. 47, I. Daubechies (ed.), AMS, 155-171. Zygrmmd, A. 1968. Trigonometric series, 2nd Ed., Cambridge University Press. Section 2
Grossman, A. and Morlet, J. 1984. "Decomposition of Hardy functions into square integrable wavelets of constant shape." SIAM J. Math. Anal. 15, 723-736. Holschneider, M., Kronland-Martinet, R., Morlet, J., Tchamitchian, Ph. 1988. "Algorithme a trous." Preprint CPT-1988, Centre de Physique Theorique, CNSRS-Luminy Marseille. Paul, T. 1984. "Functions analytic on the half-plane as quantum mechanics states." J. Math. Phys. 25, 3252-3263. Section 3
Grossmann, A., Holschneider, M., Kronland-Martinet, R., Morlet, J. 1987. "Detection of abrupt changes in sound signals with the help of wavelet transforms." In Inverse problems: an interdisciplinary study, Academic Press, London, 289-306. Holschneider, M., Tchamitchian, P. 1991. "Pointwise analysis of Riemann's non differentiate function." Invet. Math. 105, 157-175. Jaffard, S. 1991. "Pointwise smoothness, two-microlocalization and wavelet coefficients." Publ. Matematiques 35, 155-168. Mallat, S., and Hwang, W.L. 1992. "Singularity detection and processing with wavelets." IEEE Trans. Info. Theory 38(2),617643. Section 4
Delprat, N., Escudie, B. Guillemain, P., Kronland-Martinet, R., Tchamitchian, P., Torresani, B. 1992. "Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies." IEEE Trans. Info. Theory 38(2), 644-664. Section 5
Battle, G. 1987. "A block spin construction of ondelettes, Part I: Lemarie functions." Cornm. Math. Phys. 110, 601-615. Cohen, A. 1990. Ondelettes, analyses multi-resolution et traitement numerique du signal. Ph.D these. Univ. Paris, France.
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Cohen, A., Daubechies, I., Feauveau, J.Ch. 1990. "Biorthogonal bases of compactly supported wavelets." Comm. Pure Appl. Math. 45, 485-500. Dahmen, W. and Micchelli, A. 1990. "On stationary subdivision and the construction of compactly supported orthonormal wavelets." Intl. Series Num. Math. 94 Birkhauser. Daubechies, I. 1988. "Orthonormal bases of compactly supported wavelets." Comm. in Pure and Appl. Math. 41, 909-996. Deslauriers, G. and Dubue, G. 1987. "Interpolation dyadique." In Fractals, dimensions non entieres et applications G. Cherbit (ed.), Manon. Haar, A. 1910. "Zur Theorie der orthogonalen Funktionensysteme. Math. Ann. 69, 331-371. Lemarie, P.G. 1988. "Ondelettes a localisation exponentielle." J. Math. Pures et Appl. 67, 227-236. Mallat, S. 1989. "Multiresolution approximation and wavelets." Trans. AMS 315, 69-88. Section 6 Meyer, Y. 1989 "Wavelets and operators." In Analysis at Urbana, E. Berkson et al., eds., London Math. Soc. Lect Note 137, Cambridge Univ. Press, 256-364. Section 7 Calderon, A.P. 1964. "Intermediate spaces and interpolation, the complex method." Studia Math. 24, 113-190. Section 8 Beals, R. 1977. "Characterization of pseudo-differential operators and applications." Duke Math. J. 44, 45-57. Beylkin, G. 1991. "On the representation of operators in bases of compactly supported wavelets." submitted to SIAM J. Numer. Analysis. Beylkin, G., Coifman, R. and Rokhlin, V. 1991. "Fast wavelet transforms and numerical algorithms I." Comm. Pure. Appl. Math. 44, 141-183. David, G. and Journe, J.-L. 1984. "A boundedness criterion for generalized Calderon-Zygmund operators." Ann. of Math. 120, 371-397.
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David, G., Journe, J.-L. and Semmes, S. 1985. "Operateurs de Calderon-Zygmund, functions para-accretives et interpolation." Rev. Mat. Iberoamericana 1(4), 1-56. Jaffard, S. 1990. "Proprietes des matrices bien localisees pres de leur diagonale et quelques applications." Annales de 1'IHP, Series Problemes non lineaires 7(5), 461-476. JaiFard, S. and Meyer, Y. 1989. "Bases d'ondelettes dans des ouverts de IT." J. Math. Pures et Appl. 68, 95-108. Lemarie, P.G.. 1984. "Algebres d'operateurs et semi-groupes de Poisson sur un espace de nature homogene." Publ. Math. d'Orsay. Pereyra, M.C. 1993, "On the resolvents of dyadic paraproducts." Preprint Yale University, New Haven, CT, USA. Tchamitchian, Ph. 1987. "Biorthogonalite et theorie des operateurs." Rev. Mat. Iberoamericana 3(2), 163-189. Section 8 Auscher, P. and Tchamitchian, Ph. 1991. "Ondelettes et conjecture de Kato." C.R. Acad. Sci. Paris 313, 63-66. Auscher, P. and Tchamitchian, Ph. 1992. "Conjecture de Kato sur les ouverts de 1R." Rev. Mat. Iberoamericana 8(2), 149-199. Auscher, P. and Tchamitchian, Ph. 1993. "Une nouvelle approche de la conjecture de Kato et equations elliptiques complexes en dimension 2." Prepublication 93-19, IRMAR, Universite de Rennes I, Rennes, France. Coifman, R., Jones, P. and Semmes, S. 1989. "Two elementary proofs of the i2-boundedness of the Cauchy integral on Lipschitz curves." J. American Math. Soc. 2, 553-564. Coifman, R.R., Mclntosh, A., Meyer Y. 1982. "L'integrale de Cauchy definit un operateur borne sur les courbes lipschitziennes." Ann. of Math. 116, 361-387. Kenig, C. and Meyer, Y. 1985. "The Cauchy integral and the square root of second order accretive operators are the same." In Recent progress in Fourier analysis. Math. Studies 111, NorthHolland. Lions, J.L. 1962. "Espaces d'interpolation et domaines de puissances fractionnaires." J. Math. Soc. Japan 14, 233-241. Murai, T. 1987. A real variable method for the Cauchy transform and analytic capacity. Lecture Notes in Math. 1307, Springer Verlag. Tchamitchian, Ph. 1988. "Ondelettes et integrale de Cauchy sur les courbes lipschitziennes." Ann. of Math. 129,611-649.
4 WAVELETS, MULTIRESOLUTION ANALYSIS AND FAST NUMERICAL ALGORITHMS G. Beylkin
Contents 1
Introduction
184
2
Preliminary Remarks 188 2.1 The Haar basis 188 2.2 Orthonormal bases of compactly supported wavelets . . 191 2.3 Multi-wavelet bases with vanishing moments 196 2.4 A remark on computing in wavelet bases 198
3
The Non-Standard and Standard Forms 3.1 The Non-Standard Form 3.2 The Standard Form
200 200 206
4
Compression of Operators
209
5
Differential Operators in Wavelet Bases
215
6
Convolution Operators in Wavelet Bases 225 6.1 The Hilbert Transformlghkghkhgfghjklhfdfdgdfljgdgljfkkkkffkk 6.2 Fractional derivatives 230
7
The Two-Point Boundary Value Problem
8
Multiplication of Operators 238 8.1 Multiplication of matrices in the standard form . . . . 238 8.2 Multiplication of matrices in the non-standard form . 241
9
Fast Iterative Algorithms in Wavelet Bases 244 9.1 An iterative algorithm for computing the generalized inverse 244 9.2 Computing the inverse, of periodized second derivative operator 246
231
Fast Numerical Algorithms 9.3 9.4 9.5
An iterative algorithm for computing the projection operator on the null space An iterative algorithm for computing the square root of an operator Fast algorithms for computing exponential, sine and cosine of a matrix
183
249 249 250
10 Product of Functions in Wavelet Bases 251 10.1 Uncoupling the interaction between scales 253 2 10.2 Computing u in the Haar basis 254 10.3 Computing u2 in the wavelet bases 255 10.4 Relations between values of functions and their wavelet coefficientsfkjkjdfjkdslfldsfjldsjfkdjjdsljkfljui87uiojgkklfgjflgkfkgjfdlgjkfk 11 References
259
184
1
G. Beylkin
Introduction
Notwithstanding a short period of time since the word "wavelets" was first used to designate classes of functions which form bases of functional spaces, a set of ideas associated with the wavelet transform is already having a significant impact in science and engineering. To be sure, many of the ideas behind the wavelet transform have appeared independently in different fields of mathematics, electrical engineering, physics and computer science. These ideas arose to address the limitations of the Fourier transform as a tool for analyzing signals and images in applications, and functions and operators in mathematics and physics. For example in image processing (Burl and Adleson 1983) and in seismics (Goupillaud, Grossman and Morlet 1984), multiresolution methods were developed in a search for a substitute for signal processing algorithms based on the Fourier transform. The technique of subband coding using quadrature mirror filters (QMF) with the "exact reproduction property" was introduced in Smith and Barnwell (1986). Coherent states were studied in quantum mechanics (see references in Klauder and Skargerstam 1985, Littlewood and Paley 1937) while Calderon-Zygmund theories (Calderon and Zygmund 1957) were developed in mathematics. For a historical account see Meyer (1990). In numerical analysis the fast multipole method (FMM) for computing potential interactions was constructed (Greengard and Rokhlin 1987, Carrier, Greengard and Rokhlin 1988). But it is the introduction of wavelets and the notion of multiresolution analysis that allows us to develop a unified perspective on these developments. In hindsight it is clear that the stumbling block in developing simpler analysis and fast algorithms in all of these fields was, in fact, a limited variety of orthonormal bases of functional spaces. There were (with some qualifications) only two major choices, the Fourier basis (circa, 1810) and the Haar basis (circa 1910). These two bases are almost antipodes in terms of their time-frequency (or space-wave number) localization. Therefore, it is a remarkable discovery that besides the Fourier and Haar bases, there is an infinite number of various orthonormal bases with controllable localization in the time-frequency domain. Such bases of wavelets wore first constructed by Stromberg (1983) and. later and independently, by Meyer (1985). The notion of Multiresolution Analysis wa.s introduced by Meyer (1986) and Mallat (1987) Since then there appeared many new constructions of or-
Fast Numerical Algorithms -*o
185
thonormal bases with controllable localization in the time-frequency domain, notably orthonormaj bases of compactly supported wavelets in Daubechies (1988),"wavelet-packet" bases in Coifman and Meyer (1989) and Coifman and Wickerhauser (1990), local trigonometric bases in Coifman and Meyer (1990) and Malvar (1990), multi-wavelet bases in Alpert, Beylkin, Coifman and Rokhlin (1993) and wavelet bases on the interval in Cohen, Daubechies, Jawerth and Vial (1992), Cohen, Daubechies and Vial (1992) and Jouini and Lemarie-Rieusset (to appear). In these lectures we will consider fast numerical algorithms based on representations in the wavelet "system of coordinates". We will base our discussion on several papers, namely Beylkin, Coifman and Rokhlin (1989, 1991), Beylkin (1992), Alpert et al. (1993), Beylkin (1993), and Beylkin. (1994) as well as notes made by the author in preparation for lectures at INRIA in the spring of 1991. The main thrust of these papers is to consider wide classes of operators in the wavelet system of coordinates as an approach for both, their analysis and discretization. Indeed, wide classes of operators (CalderonZygmund operators or pseudo-differential operators, for example) which naively would require a dense (full) matrix for their numerical description, have sparse representations in wavelet bases. Sparse representations lead to fast numerical algorithms, e.g. an O(— JVloge) algorithm for the evaluation of N x N matrices on vectors, or an O(— N log c) algorithm for the matrix multiplication, where € is the desired accuracy. The wavelet-based algorithms in numerical analysis are similar to other transform methods in that vectors and operators axe expanded into a basis and the computations take place in the new system of coordinates. On the other hand, using wavelets as elementary building blocks provides a multiresolution structure and an efficient description of transformations on a given scale and of the interactions between scales. Such organization of both linear and non-linear transformations may be traced to the approach developed in Liftlewood-Paley and Calderon-Zygmund theories. Hierarchical subdivision is also used in the fast multipole method (FMM) for computing potential interactions (Rokhlin 1985. Greengard and Rokhlin 1987. Carrier, Greengard, and Rokhlin 1988) which requires order N
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G. Beylkin
operations to compute all the sums
and the number of operations is independent of the configuration of charges. In the FMM the reduction of the complexity of computing the sums in (1) from order N2 to —TV log e, where e is the desired accuracy, is achieved by approximating the far field effect of a cloud of charges located in a box by the effect of a single multipole at the center of the box. All boxes are organized in a dyadic hierarchy enabling an efficient O(N) algorithm. The fast algorithms of Beylkin, Coifman and Rokhlin (1989, 1991) use wavelet representations to provide a systematic generalization of the FMM and its descendants (e.g. O'Donnel and Rokhlin 1989, Alpert and Rokhlin 1991, Greengard 1990) to all Calderon-Zygmund and pseudo-differential operators. Both the subdivision of the space into "boxes" and the approximation of the "far field effect" are provided adaptively by a wavelet basis. The subdivision of the space and its organization in a dyadic hierarchy are a consequence of the multiresolution properties of the wavelet bases, while the vanishing moments of the basis functions make them useful tools for approximation. A novel aspect of representing operators in wavelet bases is the socalled non-standard form (Beylkin et al. 1989, 1991). The remarkable feature of the non-standard form is the uncoupling it achieves among the scales. A straightforward realization, or the standard form, by contrast, contains matrix entries reflecting "interactions" between all pairs of scales. The non-standard form leads to an order N algorithm for evaluating operators on functions, whereas the standard form yields, in general, only an order NlogN algorithm. It is also quite remarkable that the error estimates for the nonstandard form lead to a proof of the celebrated "T(l)" theorem of David and Journe. In addition to the direct links between practical numerical algorithms and abstract results of harmonic analysis established in Beylkin et al. (1989, 1991), the algorithm for multiplication of functions in the wavelet bases (Beylkin 1991) parallels the results of Bony (198.1, 1983, 1984) and Coifman and Meyer (1978) on propagation of singularities for linear and non-linear equations. A "traditional" point of view in numerical analysis has been that integral equations normally loa.d to dense systems of linear algebraic
Fast Numerical Algorithms
187
equations, whereas finite-difference and finite-element methods are devices for reducing a partial differential equation to a sparse linear system where the cost of sparsity is the inherently high condition number of the resulting matrices. For example, the condition number of the matrix of the second order finite-difference operator grows as l/h2 (or TV 2 ), where h is the step size (N is the number of points in a discretization). On the other hand, the condition number of the matrix of the linear algebraic system obtained by discretizing the second kind integral equation does not grow with the reduction of the step size but the matrix is dense. As a response to these problems FMM provides a way to apply a dense matrix (for a number of important operators) to a vector in O(— TV loge) operations (where € is the desired accuracy) thus making integral equations a useful numerical tool in large-scale computations. Various multigrid methods accelerate convergence of iterative methods for solving systems of linear equations resulting from discretization of elliptic partial differential equations. The multigrid approach may be viewed as a device for preconditioning such linear systems. It turns out that representing integral operators (associated with elliptic problems) in wavelet bases yields sparse matrices (Beylkin et al. 1989, 1991). For differential operators wavelets provide a system of coordinates where the preconditioner is a diagonal matrix (Beylkin 1992) , Jaffard (1992), Beylkin (1994). Moreover, since properties of operators represented in wavelet bases apply to a wide class of operators, it is possible to consider numerical calculus of operators (Beylkin et al. 1989, 1991, Beylkin 1991, Alpert et al. 1993), i.e., consider functions of operators computed via fast algorithms. The product of two operators in the standard form requires between —TV log e and —TV log2 TV loge operations, where € is the desired accuracy. An algorithm for the multiplication of operators in the nonstandard form requires —TVloge operations (Beylkin 1991). Among the algorithms requiring multiplication of matrices is an iterative algorithm for constructing the generalized inverse (Schulz 1933, BenIsrael and Cohen 1966, Soderstrdm and G.W.Stewart 1974), the squaring method for computing the exponential of an operator (see, for example, Ward 1977), and similar algorithms for the sine and cosine of an operator, to mention a few. By replacing the ordinary tnatrix multiplication in these algorithms by the fast multiplication in the wavelet bases, the number of operations is reduced to, essen-
188
G. Beylkia
tially, order N operations. For example, if both the operator and its generalized inverse admit sparse representations in wavelet basis, then the iterative algorithm (Schultz 1933) for computing the generalized inverse requires only O ( N l o g K ) operations, where K is the condition number of the matrix. Various numerical examples and applications may be found in Beylkin et al. (1992), Alpert et al. (1993), Beylkin (1994), and Beylkin et al. (1994). Though computing in wavelet system of coordinates does have many advantages, it takes a significant amount of time to develop competitive (e.g., vs FMM) practical numerical algorithms which are applicable in multidimensional problems. Such work is under way. A number of wavelet-like algorithms have been developed as a way of addressing some of the difficulties of computing in wavelet bases. Here we will not consider these issues, in part because the work in these directions is still continuing. In these lecture notes we review the standard and non-standard representations of operators in wavelet bases and the associated fast numerical algorithms in Section 3 and compression of operators in Section 4. In Section. 5 we compute explicitly the non-standard forms of derivative operators and, in Section 6, of several convolution operators, such as fractional derivatives and the Hilbert transform. We consider diagonal preconditioning (such preconditioners turn out to be dense in the ordinary representation) and a solution of two-point boundary value problems in Section 7. Fast matrix multiplication is considered in Section 8 and computing functions of matrices in Section 9. Finally, in Section 10 we describe algorithms for the multiplication of functions represented in the wavelet bases. We start with preliminary remarks in Section 2.
2
2.1
Preliminary Remarks
The Haar basis
The Haar basis (Haar 1910) is the most simple example of a basis that illustrates algorithms which we will describe in these lectures and it provides a useful prototype for numerical experimentation. The Haar basis in L 2 (R), h^k(x] = 2^/2/^2-? a; - fc), where j,keZ, is formed by the dilation and translation of a single function
Fast Numerical Algorithms
189
The Haar function h satisfies the two-scale difference equation,
where ,\'(>T) i g the characteristic function of the interval (0,1). The characteristic function x(x)ssss prototype of the scaling function for wavelets and satisfies the two-scale difference equation,
Given TV = 2ra "samples" of a function, which may for simplicity be thought of as values of scaled averages of / on intervals of length 2-",
we obtain the Haar coefficients
and averages
for j = 0,. . . , n - 1 and fc = 0 , . . . , 2 7l ~- 7 ~ 1 - 1. It is easy to see that evaluating the whole set of coefficients djk, s3k in (6), (7) requires 2(N — 1) additions and 27V multiplications. In two dimensions, there are two natural ways to construct the Haar basis. The first is simply the tensor product h j . j i ^ . k ' f a , y) = h j , k ( x ) h j ' , k ' ( y } i so that each basis function hjtj< ^^(x, y) is supported on a rectangle. The second basis is defined by the set of three kinds of basis functions supported on squares: hj^(x)hj^i(y},sad hj^(.x}Xj,k'(y}i and X},k(x}hjtk'(y}i 1 where x(x) 'ls the characteristic function of the interval (0,1) and Xj,k(*) = 2~j/2xC2~jz1 ~ *)• By considering integral operators with distribution kernels which are smooth (non-oscillatory) away from the diagonal,
190
G. Beylkia
and expanding K in two-dimensional Haar bases, we find that the decay of entries as a function of the distance from the diagonal is faster in these representations than that in the original kernel. For example, kernels K(x, y) of Calderon-Zygrnund operators satisfy the estimates
where x°^k = V(k+\) denotes the center of the support ofhjtk- Thus, entries (3Jkk, decay quadratically as functions of the distance between k and k'. Given a finite precision of calculations, entries j33kk, may be discarded as the distance between k and k' becomes large, leaving only a band around the diagonal. The rate of decay in (12) depends on the number of vanishing moments of the functions of the basis. The Haar functions have only one vanishing moment, f h(x)dx = 0, and for this reason the gain in the decay is insufficient to make computing in the Haar basis practical. To have a faster decay, it is necessary to use basis functions with several vanishing moments. As we will see, the vanishing moments are responsible for attaining practical algorithms, i.e., they control the constants in the complexity estimates.
Fast Numerical Algorithms 2.2
191
Orthonormal bases of compactly supported wavelets
Let us now briefly consider compactly supported wavelets with vanishing moments constructed by Daubechies (1988), following the work of Meyer (1989) and Mallat (1988). For a complete account we refer to Daubechies (1992). We start with the definition of the multiresolution analysis. This notion introduced by Meyer (1986) and Mallat (1987) captures the essential features of all multiresolution approaches developed so far. Definition 1. A multiresolution analysis is a decomposition of the Hilbert space L 2 (R ), d > 1, into a chain of closed subspaces
such that
Since in these lectures we use only orthonormal bases, we replace Condition 4 by
Let us define the subspaces Wj as an orthogonal complement of VjinV,-.!, so that
Selecting the coarsest scale n, we may replace the chain of the subspaces (13) by
192
G. Beylkin
and obtain
If there is a finite number of scales then without loss of generality we set j = 0 to be the finest scale and consider
instead of (16). In numerical realizations the subspace V0 is finite dimensional. Let us consider a multiresolution analysis for L 2 (R) and let {ip(x~ k)}k£i be an orthonormal basis of W0. We will require that the function if) has M vanishing moments,
There are two immediate consequences of Definition 1 with Condition 4'. First, the function
In general, the sum in (20) does not have to be finite and by choosing a finite sum in (20) we are selecting compactly supported wavelets. We may rewrite (20) as
where
and the 27r-periodic function mo is defined as
Fast Numerical Algorithms
193
and, therefore,
and
Using (21), we obtain
and, by taking the'sum in (27) separately over odd and even indices, we have
Using the 27r-periodicity of the function mo and (26), we obtain (after replacing £/2 by ^) a necessary condition
for the coefficients /ij. in (23). On denning the function 1/1 by
where or, equivalently, the Fourier transform of ^ by
where
it is not difficult to show (Mover 1990, Daubechies 1988, Daubechies 1992), that for each fixed scale j g Z. the wavelets { i p l t k ( x ) = 2~1'2u>('2~3x — k)}k£z form an orthonormal basis of W?.
194
G. Beylkin
Equation (29) can also be viewed as the condition for exact reconstruction for a pair of the quadrature mirror niters (QMFs) H and G', where H - {Mfc'o"1 and G = Wk=%~1 • Such exacvt QMF filters were first introduced by Smith and Barnwell (1986) for subband coding. We will not go into the full discussion of the necessary and sufficient conditions for the quadrature mirror filters H and G to generate a wavelet basis and refer to Daubechies (1992) for the details. The coefficients of the quadrature mirror filters H arid G are computed by solving a set of algebraic equations (Daubechies 1988). The number L of the filter coefficients in (23) and (33) is related to the number of vanishing moments M, and L = 2M for the wavelets constructed in Daubechies (1988). If additional conditions are imposed, Beylkin, Coifman and Rokhlin (1989, 1991) for an example, then the relation might be different, but L is always even. Lemma 1. Any trigonometric polynomial solution rao(0 of (29) is of the form
where M > 1 is the number of vanishing moments, and where Q is a polynomial, such that
where
and R is an odd polynomial, such that
and
The proof of this lemma contains an algorithm for generating the coefficients of the quadrature mirror filters H and G and we refer to Daubechies (1992) for details. The decomposition of a function into the wavelet basis is an order N procedure. Given the coefficients ,^, /,; = (}, 1 . . . . , N as "samples"
Fast Numerical Algorithms
195
of the function /. the coefficients s3k and d3k on scales j > I are computed at a cost proportional to N via
and
where sjk and d3k are viewed as periodic sequences with the period 2™~ J . If the function / is not periodic (e.g. denned on an interval), then such straightforward periodization introduces "artificial" singularities. A better alternative is to use wavelets on the interval (Daubechies, Jawerth and Vial 1992, Cohen, Daubechies and Vial 1992, and Jouini, Lemarie-Rieusset 1994). In practice, this approach amounts to modifications near the boundary of both decomposition and reconstruction algorithms. Alternatively, multi-wavelet bases with vanishing moments were constructed in Alpert et al. (1993) and Alpert (1990) and will be described in the next subsection. Computing via (39) and (40) is illustrated by the pyramid scheme
The reconstruction of a function from its wavelet representation is also an order TV procedure and is described by
Computing via (42) is illustrated by the pyramid scheme
196 2.3
G. Beylkin Multi-wavelet bases with vanishing moments
The Haar system is a degenerate case of Daubechles's wavelets with M = 1. There is, however, a different construction of orthonormal bases for L 2 ([0,1]) or L 2 (R) which generalizes the Haar system and yields basis functions with several vanishing moments (Alpert et al. 1993 and Alpert 1990). Recently, it was brought to our attention that Federbush (1981) considered a similar construction. Let us construct M functions, / I , . . . , / M : R —> R, supported on the interval [—1,1] and such that (i) on the interval (0,1) the function /,- is a polynomial of the degree M — 1. (ii) extend /, to the interval ( — 1,0) as an even or odd function according to the parity of i + M — 1. (iii)
We require these functions to be orthogonal:
and have vanishing moments (iv)
The properties (i) and (ii) imply that there are M 2 polynomial coefficients that determine the functions / i , . . . , /M, while the properties (iii) and (iv) provide M 2 constraints. It turns out that the equations uncouple to give M nonsingular linear systems which may be solved to obtain the coefficients and, thus, yielding the functions / i , . . . , /M up to a sign. The functions / i , . . . , /M may be obtained constructively as follows. Let us start with
l and note that the 2M functions 1, x\..., a;M~] , ,/V, /2l,. . ., j'\4 are 2 linearly independent. Then, by the Gram-Schmidt process, we orthogonalize /^ with respect to 1, x,..., xM~i, to obtain f^, for m =
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1 , . . . , M. This orthogonality is preserved by the remaining orthogonalizations, which only produce linear combinations of the /^. First, if at least one of /^ is not orthogonal to x , we reorder the functions,so that (fl* XM) ^ 0. We then define /^ = f^-am- f2,, where am is chosen so that ( f m i x M ) = 0 for m = 2 , . . . , Af, achieving the desired orthogonality to XM. Similarly, we continue to orthogonalize with respect to xM+1,. . ., x2M~2 to obtain /j2, /f, /|,.. ., /^/+1, such that {/™ +1 ,a; i } = 0 for i < m + M - 2. Finally, we perform the Gram-Schmidt orthogonalization on /jjf +1 , /M_I, • • • , /i, in that order, and normalize to obtain /M)/M-I> • • • > / ! • By construction, the functions {/m}m=i^ satisfy properties (i)-(iv). On denoting
we define the space W^, j — 0 , — 1 , — 2 , . . . , as a linear span of functions
We also define V^ to be the space of polynomials on [0,1] of degree less than M. By defining recursively
starting with j = 0, we obtain multiresolution analysis
and
It is easy to see that the orthonormal set
is an orthonormal basis of L 2 (R). We refer to bases constructed in this manner as multi-wavelet bases of order M. Numerical algorithms, various examples and applications utilizing these bases are described in Alpert et al. (1993) and Alpert (1990). It is interesting to note that as a numerical tool the multi-wavelet bases have been constructed only after "ordinary" wavelets had been used in algorithms of Beylkin, Coif man and
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G. Beylkin
Rokhlin (1989, 1991) though their construction addresses the problem of vanishing moments directly and does not require any prior knowledge of the wavelet theory. If the number of vanishing moments M > 2, then, on a given scale j, the multi-wavelet basis functions with different labels n do not have an overlapping support unlike the "ordinary" compactly supported wavelets. On the other hand, multi-wavelets are not smooth functions and their smoothness does not improve with increasing number of vanishing moments. Let us outline the construction of bases for L 2 ([0,1] 2 ) and note that the construction of bases for L 2 [0, l]d and L 2 (R d ), d > 2, is similar. Let us define the space V • ' as
where Vf is given in (48), and the space W- ' as the orthogonal complement of Vf'2 in vf' 2 ,
The space Wy ' is spanned by the orthonormal basis
where {um}™zf is an orthonormal basis for Vgf. Each element among these 3M2 basis elements has vanishing moments, i.e., it is orthogonal to the polynomials xlyl, i, I — 0 , 1 , . . . , M — 1. The space WJ ' is spanned by dilations and translations of the basis functions of W0 '2 and the basis of L2([0,1]2) consists of these functions and the low-order polynomials xlyl, ?', / = 0 , 1 , . . . , M — 1. We note that the two-dimensional multi-wavelet bases require 3M2 different combinations of one-dimensional basis functions, where M. is the number of vanishing moments. 2.4
A remark on computing in wavelet bases
Finally, we observe that once the filter H has been chosen, it completely determines the functions
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to the recursive definition of the wavelet bases, all the manipulations are performed with the quadrature mirror filters H and G, even if they involve quantities associated with . The expressions for the moments,
may be found in terms of the filter coefficients {hk}k=o~l- Applying m jj operator (\d/d^) to both sides of (21) and setting £ = 0, we obtain
where
Thus, we have from (56)
with A-fo = 1Alternatively, using
the moments Mm may be obtained within the desired accuracy as a limit of recursively generated sequence of vectors, {Mm }^^~l for r = 1,2,...,
starting with
Each vector {Xm }^=^f~1 represents Af moments of the product in (59) with r terms, and the iteration converges rapidly. Notice that in both algorithms we computed integrals in (55) without computing values of the function (p.
200
3
G. Beylkin
The Non-Standard and Standard Forms
In this Section we develop representations of operators in wavelet bases. 3.1
The Non-Standard Form
Let T be an operator
with the kernel K(x,y), Denning projection operators on the subspace V ; , j € Z,
as
and expanding T in a "telescopic" series, we obtain
where is the projection operator on the subspace Wj. If there is the coarsest scale n, then instead of (65) we have
and if the scale j = 0 is the finest scale, then
where T ~ TO = Po'TPo is a discretization of T on the finest scale. Expansions (65), (67) and (68) decompose T into a sum of contributions from different scales. The non-standard form introduced in Beylkin et al. (1.991) is a representation of an operator T as a, c h a i n of triplets
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where operators { A j , B j , T j } j £ % are defined as A3 = QjTQj, Bj = QjTP,, and Tj = P3TQ3. The operators {Aj,Bj,Tj}j€Zaaa admit a recursive definition via
and the operator represented by the 2 x 2 matrix in (73) is a mapping
If there is a coarsest scale n, then
where Tra = PnTPn. If the number of scales is finite, then j = 1,2,.. .,n in (76) and the operators are organized as blocks of the matrix (see Figures 1 and 2). Let us make the following observations: 1. The operator Aj in (70) describes interaction on the ddddd independently from other scales. 2. The operators Bj, Tj in (71) and (72) describe interaction between the scale j and all coarser scales. Indeed, the subspace V, contains all the subspaces V ? » with j' > j (see (13)). 3. The operator Tj is an "averaged" version of TJ_I.
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G. Beylkin
Figure 1: Organization of the non-standard form of a matrix. The submatrices Aj, Bj, and Fj, j = 1,2,3, and T3 are the only non-zero submatrices.
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Figure 2: An example of a matrix in the non-standard form (see Example 1).
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G. Beylkin
Figure 3: The non-standard form of the same matrix as in Figure 2 using a basis of wavelets on the interval (Beylkin and Brewster 1994). The vertical and horizontal bands (which are present in Figure 2 due to periodization) do not appear in this representation.
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The operators Aj, Bj and F?- are represented by the matrices a-7, P and 7-',
and
The operator Tj is represented by the matrix s j ,
Given a set of coefficients s°k k, with fe, k' = 0 , 1 , . . . , N — 1, a repeated application of the formulas (39), (40) produces
with i, / = 0 , 1 , . . . , 2n-i - 1, j = 1, 2 , . . . , n. Clearly, formulas (81) (84) provide an order N2 scheme for the evaluation of the elements of all matrices aj', ft, 7J with j = 1,2 . . . , n. A fast scheme for the evaluation of the elements of matrices a j ,/3 J ,7-' with j = 1,2 . . . , n , requires that the locations of the singularities be known in advance. Given the locations of the singularities, only the significant coefficients above a threshold of accuracy are computed, and the only issue is how to compute averages s\ l in a neighborhood of the singularities without (84). We refer to Beylkin, Coifman and Rokhlin (1989, 1991) for the details of such computations using quadrature formulas.
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G. Beylkia
The term non-standard form refers to the fact that the operator is applied to a function which is not in the usual wavelet representation. This representation is redundant and contains both "averages" and "differences" and is illustrated in Figure 1. Such imbedding into higher dimensional space is the price for splitting the interaction between scales. It follows from (68) that after applying the nonstandard form to a vector, we arrive at the representation
where UQ = PQU (see Figure 1). We note, that in order to reconstruct a vector from the first and the second sums in (85), we may apply the same algorithm. The reconstruction from the first sum is just a reconstruction from the wavelet expansion and is accomplished by using the quadrature mirror filters H and G. The reconstruction from the second sum may be accomplished by using the same algorithm but with the pair of filters H and G. Both results are then added. The number of operations required, for this computation is proportional to TV. Depending whether we want to obtain the result on V0 or a wavelet expansion of the vector, there are two possible pyramid algorithms. The first one corresponds to the ordinary reconstruction algorithm,
and the result is a vector in VQ. The second one corresponds to the ordinary decomposition algorithm,
and as a result we obtain wavelet expansion of the vector. 3.2
The Standard Form
The standard form is obtained by representing the operator in the tensor product basis. For a matrix it simply means that the wavelet
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transform is applied to all rows (columns) and then to all columns (rows). We would like to introduce the standard form by emphasizing its connection with the non-standard form. The standard form is obtained by representing
If there is the coarsest scale n, then instead of (86), we have
In this case, the operators {B^ , Fy } for / = j + 1,.. .,ra are the same as in (87) and (88) and, in addition, for each scale j, there are operators {B"+l} and {F™ +1 },
In this notation, F™+1 = Fn and -B"+1 = Bn. If the number of scales is finite and VQ is finite dimensional, then the standard form is a representation of TO = PoTP^ as
The operators (92) are organized as blocks of a matrix (see Figure 4 and Figure 5). If the operator T is a Calderon-Zygmund or a pseudo-differential operator then, for a fixed accuracy, all the operators in (92), except Tn, are banded. As a result, the standard form has several "finger" bands which correspond to the interaction between different scales. For a large class of operators (pseudo-differential, for example), the interaction between different scales characterized by the size of the coefficients of "finger" bands decays as the distance j ' — j between the
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G. Beylkin
Figure 4: Organization of a matrix in the standard form.
Figure 5: An example of a matrix in the standard form (see Example 1).
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scales increases. Therefore, if the scales j and j' are well separated, then for a, given accuracy, the operators Bj , Tj can be neglected. There are two ways of computing the standard form of a matrix. The first consists of applying the one-dimensional transform (see (39) and (40)) to each column (row) of the matrix and then to each row (column) of the result. Alternatively, one can compute the nonstandard form and then apply the one-dimensional transform to each row of all operators Bj and each column of all operators Fj.
4
Compression of Operators
The compression of operators or, in other words, the construction of their sparse representations in orthonormal bases, has been proposed in Beylkin, Coifman and Rokhlin (1989, 1991). Such sparse representations lead to fast algorithms. While the compression of data (of images, for example) achieved without the use of a sparse representation in some basis may be adequate for some applications, the compression of operators calls for a representation in a basis to effectively compute in the sparse form. The standard and non-standard forms of operators in the wavelet bases may be viewed as compression schemes for a wide class of operators. We restrict our attention to two specific classes of integral operators frequently encountered in analysis and applications, Calderon-Zygmund and pseudo-differential operators. Let us label the coefficients ajkki, /3]k k, and 7^ k, in (77)-(79) by the intervals / = I3k and /' = /^, denoting the supports of the basis functions. If the kernel K = K ( x , y ) is smooth on the square I x /', then expanding K into a Taylor series around the center of the square and combining (19) with (81) - (84) and remembering that the functions ^'/ and ?/>// are supported on the intervals / and /', we obtain the estimate
The right-hand side of (93) is small whenever either |/| or the derivatives involved are small, and we use this fact to "compress"' matrices of integral operators by discarding the coefficients that are smaller than a chosen threshold.
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G. Beylkin
For most numerical applications, the following Proposition 1 is quite adequate, as long as the singularity of K is integrable across each row and each column. Proposition 1. // the wavelet basis has M vanishing moments, then for any kernel K satisfying the conditions
the matrices a- 7 , /3J, 7J (77) - (79) of the non-standard form the estimate
satisfy
Similar considerations apply in the case of pseudo-differential operators. Let T be a pseudo-differential operator with symbol
where K is the distributional kernel of T. Proposition 2. If the wavelet basis has M vanishing moments, then for any pseudo-differential operator with symbol a ofT and a* ofT* satisfying the standard conditions
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If we approximate the operator T^ by the operator T0 N.B obtained from TO"" by setting to zero all coefficients of matrices a^, /^';, and 7^ outside of bands of width B > 2M around their diagonals, then it is easy to see that
where C is a constant determined by the kernel K. In most numerical applications, the accuracy e of calculations is fixed, and the parameters of the algorithm (in our case, the band width B and order M) have to be chosen in such a manner that the desired precision of calculations is achieved. If M is fixed, then B has to be such that
or, equivalently,
The estimate (101) is sufficient for practical purposes. It is possible, however, to obtain
instead of (101). In order to obtain this tighter estimate and avoid the factor Iog2 TV, we use some of the ideas arising in the proof of the "T(l)" theorem of David and Journe. The estimates (94), (95) are not sufficient to conclude that aj ; , @ll-> "/i l are bounded for i — l\ < 2M (for example, consider K(x,y) = i~r~r)- Therefore, we need to assume that T defines a bounded operator on L2 or, a substantially weaker condition,
for all dyadic intervals /. This is the so-called "weak cancellation condition" (Meyer 1989). Under this condition and the conditions (94), (95), Proposition 1 may be extended so that the estimate (96) holds for all integers ?', / (Meyer 1989).
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G. Beylkhi
Let us evaluate an operator T in the non-standard form on a function /,
and write (106) as a sum of three terms,
where
and
It is a remarkable fact that by analyzing the functions (113) and (114) (and, therefore, the operators Lp and i*), it is possible to decide if a Ca.lderon-Zyginund operator is bounded.
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Theorem l.(G. David, J.L. Journe) Suppose that the operator (62) satisfies the conditions (94), (95), and (105). Then a necessary and sufficient condition for T to be bounded on L2 is that ft(x) in (113) and 7(V) in (114) belong to dyadic B.M.O., i.e. satisfy condition
where J is a dyadic interval and
Splitting the operator T into the sum of three terms (107) and estimating them separately leads to the estimate (104). We note that the functions T(l) and T*(l) are easily computed in the process of constructing the non-standard form and may be used to provide a useful estimate of the norm of the operator. The compression of operators yields fast algorithms for the evaluation of operators on functions. Example 1. In this example, we consider the matrix
and convert it to the non-standard form using wavelets with six vanishing moments. Setting to zero all entries whose absolute values are smaller than 10~7, we obtain the non-standard form where the nonzero elements are shown in black in Figure 2. The same matrix in the basis of wavelets on the interval is illustrated in Figure 3. The standard form of the matrix A with N = 256 is depicted in Figure 5. The results of experiments in applying the sparse matrix A in the basis of periodized wavelets to a vector are tabulated in Table 1. Column 1 of Table 1 contains the number N indicating the size of NxN matrix Aij, columns 2, 3 contain CPU times Ts, Tw required by the standard order O(N2) and the fast O(N] schemes to multiply a vector by the matrix, and column 4 contains the CPU TJ time used to produce the non-standard form of the operator. Columns 5, 6 contain the LI and L^ errors of the direct calculation, and columns 7,8 contain the same information for the result obtained by computing in the wavelet system of coordinates. Finally, the last column contains the compression coefficients Ccomp, defined by the ratio of N'2 to the number of non-zero elements in the non-standard form of of the matrix.
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G. Beylkia
Time T3
Tw
Td
Error of Single Precision Multiplication LOO - norm Jj-2 - norm
64
0.12
0.16
7.76
1.26 • 10~ 7
3.65 • 10~ 7
128
0.48
0.38
32.62
2.17.10~ 7
8.64 • 10~7
256
1.92
0.80
96.44
2.81 -KT 7
1.12-10- 6
512
7.68
1.80
252.72
4.21 • 10~7
1.75-1Q- 6
1024
30.72
3.72
605.74
6.64-KT7
3.90-10" 6
Input Size N
Table 1: Numerical results for Example 1.
Error of FWT Multiplication LI - norm Loo - norm
Compression Coefficient
c* ^comp
Input Size N
8.89 • 10~8
1.72 • 10~r
1.39
64
1.12- 10~7
9.94-10- 7
2.22
128
1.25- 10~7
5.30 • 10~7
3.93
256
1.23- 10~7
5.16-1Q- 7
7.33
512
1.36- 10~r
5.04 • 10~7
14.09
1024
Table 1: Numerical results for Example 1 (contd.).
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215
Differential Operators in Wavelet Bases
In this Section we compute the non-standard form of differential operators by solving a small system of linear algebraic equations (Beylkin 1991). As an example, we construct the non-standard form of the operator d/dx. The matrix elements cr-;, /?^, and 7^ of Aj, Bj, and Fj, where i,l,j £ Z for the operator d/dx are as follows,
and
where
and
Moreover, using (20) and (30) we have
and
where
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G. Beylkia
Therefore, the representation of d/dx is completely determined by the coefficients r\ in (126) or, in other words, by the representation of d/dx on the subspace VQ. Rewriting (126) in terms of
Thus, the coefficients r\ depend only on the autocorrelation function of the scaling function
and
where
are the autocorrelation coefficients of the filter //. 2. If M > 2. then Equations (128) and (129) have a unique solution with a finite number of non-zero r/, namely, r/ 7^ 0 for -L + 2 < / < L- 2 and
Fast Numerical Algorithmsjf
fgfff
Proof. First we note that any trigonometric polynomialTOQ(£)satisfying (29)is such that
where ara are the autocorrelation coefficients of H — {hk}k=o~ '
and To demonstrate (134), we compute m 0 (^)| 2 using (23) and obtain
and
|m where an are given in (133). Combining (135) and (136) to satisfy (29) we obtain
and hence, (134).
and hence,
Substituting I = k — m, we rewrite (139) as
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G. Beylkia
Changing the order of summation in (140) and using we arrive at
where a n are given in (133). Using (134) we obtain (128) from (141). In order to obtain (129) we use
where
are the moments of the function
where e > 0, and hence, the integral in (127) is absolutely convergent. This assertion follows from Lemma 3.2 of Daubechies (1988), where it is shown that
where
Due to the condition (38), we have Iog2 B — M — 1 — ( with some e > 0. The existence of the solution of the system of equations (128) and (129) follows from the existence of the integral in (127). Since the scaling function
Fast IVumericaJhjfhjjkdfhjkkjd
hgjg
Multiplying (128) by e1^ and summing over I, we obtain
where
and
Noticing that
and
and using (132), we obtain from (147)
Finally, using quadrature mirror condition (29), we arrive at
Setting ^ = 0 in (153) we obtain f(0) = 2f(0) and, thus, (146). Uniqueness of the solution of (128) and (129) follows from the uniqueness of the representation of d/dx. Given the solution r\ of (128) and (129), we consider the operator Tj defined by these coefficients on the subspace Vj and apply it to a sufficiently smooth function /. Since r[ = 2~ J r/, we have
where
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G. Beylkin
Rewriting (155)
and expanding f ( x — Vl) in the Taylor series at the point x we obtain
It is clear that as j -+ — 00, operators Tj and d/cJx coincide on smooth functions. Using standard arguments it is easy to prove that T_oo = d/dx and, hence, the solution to (128) and (129) is unique. The relation (131) follows now from (127). Examples. As examples we will use Daubechies' wavelets constructed in Daubechies (1988). First, let us compute the coefficients a-2k-\i k = 1,..., M, where M is the number of vanishing moments and L = 2M. Using relation (4.22) of Daubechies (1998).
and by computing /J sin2 1 £ d£, we find
where
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Thus, by comparing (159) and (132), we have
where m = 1 , . . . , M. We note that by virtue of being solutions of a linear system with rational coefficients, the coefficients r; are rational numbers. The coefficients r\ are the same for all Daubechies' wavelets with a fixed number of vanishing moments M, while there are several wavelet bases for a given M depending on the choice of roots of polynomials in the construction described in Daubechies (1988). The solution to Equations (128)-(129) for Daubechies' wavelets with M = 2,3 are 1. M = 2
and
We note, that the coefficients (-1/12, 2/3, 0,-2/3,1/12) of this example can be found in many books on numerical analysis as a choice of coefficients for numerical differentiation. 2. M = 3
and
The structure of non-standard and standard forms of derivative operators is illustrated in Figures 6 and 7. Remark. We note that expressions (123) and (124) for a; and fli (li — ~P-i) may be simplified by changing the order of summation in (123) and (124) and introducing the correlation coefficients The expression for a; is especially simple, a; = 4r-2i — r\. As a way of solving Equations (128) and (129), we may use an iterative algorithm. We start with r_i = 0.5 and r\ — —0.5 and iterate using (128) to recompute r/. It is easy to verify using (153)
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G. Beylkin
Figure 6: Sparse structure of the non-standard form of derivative operators. The width of bands depends only on the choice of the basis and is equal to 1L — 3. that (129) and (131) are satisfied due to the choice of initialization. In Table 2 we present the coefficients TI for Daubechies' wavelets with M = 5,6,7,8,9. It displays only coefficients {n}^2 since r_; = — r; and TQ — 0. For the coefficients r\' of dn/dxn, n > 1, the system of linear algebraic equations is similar to that for the coefficients of d / d x . This system (and (128)) may be written in terms of
as
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Coefficients
Coefficients
1
n
M =5
1 2 3 4 5 6 7 8
-0.8259060118501 0.2288201870670 -5.335257193267E-02 7.461396365775E-03 -2.392358200239E-04 -5.404730164475E-05 -2.524117113568E-07 -2.696047942352E-10
M =6
1 2 3 4 5 6 7 8 9 10
-0.8501366615559 0.2585529441415 -7.244058999766E-02 1.454551104200E-02 -1.588561543476E-03 4.296891570995E-06 1.202657519572E-05 4.206912045117E-07 -2.899666805705E-09 6.968651152008E-13
M -7
/
1 -0.8687439145238 2 0.2829650945259 3 -9.018906621779E-02 4 2.268741101465E-02 5 -3.881454657629E-03 6 3.373440477641E-04 7 4.236394680070E-06 8 -1.650167921087E-06 9 -2.187113033190E-07 10 4.183054820375E-10 11 -1.203527399999E-11 12 -6.628390059460E-16
Table 2: The coefficients {r/}j;rf L = 2M and M = 5 , . . . , 9 .
2
r,
M=8
1 2 3 4 5 6 7 8 9 10 11 12 13 14
-0.8834460460910 0.3032593514767 -0.1063640682895 3.129014783949E-02 -6.958379116454E-03 1.031530213376E-03 -7.667706908380E-05 -2.451992110954E-07 -3.993810456389E-08 7.207948238595E-08 9.697184925641E-10 7.252206916650E-13 -1.240078536098E-14 1.585464751684E-19
M =9
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
-0.8953164058370 0.3203120622486 -0.1209536493600 3.995272188669E-02 -1.061693066982E-02 2.103402810656E-03 -2.781207764993E-04 1.962043776364E-05 -4.878246887963E-07 1.036122059148E-07 -1.596686479864E-08 -8.137410829411E-10 -5.402519753363E-13 -4.781400591681E-14 -1.618788001301E-18 -4.850747431075E-24
for Daubechies' wavelets, where
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G. Beylkin
Figure 7: Sparse structure of the standard form of derivative operators. where mo is the 27r-periodk function in (23). Considering the operator MQ on 2?r-periodic functions
we rewrite (163) as so that f is an eigenvector of the operator MO corresponding to the eigenvalue 2~". Thus, finding the representation of the derivatives in the wavelet basis is equivalent to finding trigonometric polynomial solutions of (165) and vice versa (Beylkin 1992). The operator MQ is also introduced in Cohen, Daubechies adn Feaveau (1992) and Lawton (1991), where the problem (165) with the eigenvalue one is considered to decide if a pair of quadrature mirror filters generates a wavelet basis. A very important property of the wavelet representation of the derivative operators (and, in general, pseudodifferential operators with homogeneous symbols) is that these operators have an explicit diagonal preconditioner in wavelet bases. We present here two tables
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Figure 8: An example (n = 5) of the diagonal matrix P used to rescale the matrix of the periodized second derivative D^1 in the wavelet system of coordinates. illustrating such preconditioning applied to the standard form of the second derivative. In the following examples the standard form of periodized second derivative D2 of size N x TV, where TV = 2", is preconditioned in the wavelet basis by the diagonal matrix P, where P,-; = ^-;2J, 1 < j < n, and where j is chosen depending on ? , I so that N - TV/2-7"1 + 1 < i , I < N - TV/2-7, and PTVA~ = 2 n . The matrix P is illustrated in Figure 8. The following Tables 3 and 4 compare the original condition number K of ~D% and KP of D^.
6
Convolution Operators in Wavelet Bases
In this Section we compute the non-standard forms of convolution operators. Quadrature formulas for representing kernels of convolu-
G. Beylkin
226
N
K
Kp
64
0.14545E+04
0.10792E+02
128
0.58181E+04
0.11511E+02
256
0.23272E+05
0.12091E+02
512
0.93089E+05
0.12604E+02
1024
0.37236E+06
0.13045E+02
Table 3: Condition numbers of the matrix of the periodized second derivative (with and without preconditioning) in the basis of Daubechies' wavelets with three vanishing moments M — 3.
N
K
Kp
64
0.10472E+04
0.43542E+01
128
0.41886E+04
0.43595E+01
256
0.16754E+05
0.43620E+01
512
0.67018E+05
0.43633E+01
1.024
0.26807E+06
0.43640E+01
Table 4: Condition numbers of the matrix of the periodized second derivative (with and without preconditioning) in the basis of Daubechies' wavelets with six vanishing moments M — 6.
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tion operators on V0 are of the simplest form due to the fact that the moments of the autocorrelation of the scaling function
which easily reduces to
where the coefficients a^k-i are given in (133). We also have
and, by changing the order of integration, obtain
where $ is the autocorrelation function of the scaling function yj,
Let us verify that
a.nd
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G. Beylk'm
Clearly, we have
Using (173) and the identity <£(£) = <£(£/2)m0(£/2), we observe that (172) holds provided that
or, due to (29),
But (175) follows from the explicit representation in (34). We note that since the moments of the function $ vanish, Equation (169) leads to a one-point quadrature formula for computing the representation of convolution operators on the finest scale for all compactly supported wavelets. Let us consider a,n approach which consists in solving the system of linear algebraic Equations (167) subject to asymptotic conditions. This method is especially simple if the symbol of the operator is homogeneous of some degree since in this case the operator is completely defined by its representation on VQ. Let us consider two examples of such operators, the Hilbert transform and the operator of fractional differentiation. 6.1 The Hilbert Transform We apply our method to the computation of the non-standard form of the Hilbert transform
where p.v. denotes a principal value at ,s = x. , of functions The representation of H on V0 is defined by the coefficients
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which, in turn, completely define all other coefficients of the nonstandard form. Namely, H = {Aj,Bj,Tj}jez, A3 = A0, Bj = B0, and Tj = FO, where matrix elements cc z -_;, /? z -_/, and 7;_; of AQ, BQ, and FQ are computed from the coefficients r/,
and
The coefficients r;, / e Z in (177) satisfy the following system of linear algebraic equations
where the coefficients a^k-i are given in (133). Using (169), (171) and (172), we obtain the asymptotics of r; for large /,
obtain By rewriting (177) in terms of <£(£),
we obtain n = — r_/ and set TO = 0. We note that the coefficient ro cannot be determined from Equations (181) and (182). Solving (181) with the asymptotic condition (182), we compute the coefficients r;, / ^ 0 with any prescribed accuracy. Example. In "Fable 5 we present the coefficients r\ of the Hilbert transform for Daubechies' wavelets with six vanishing moments with accuracy 10~ . The coefficients for / > 16 are obtained using asymptotics (182). We note that r__/ = -r\ and r0 — 0.
G. Bevlkia
230
Coefficients
I
n
] 2 3 4 5 6 7 8
-0.588303698 -0.077576414 -0.128743695 -0.075063628 -0.064168018 -0.053041366 -0.045470650 -0.039788641
Coefficier ' 9 10 11 12 13 14 1.5 16
n -0.035367761 -0.031830988 -0.028937262 -0.026525823 -0.024485376 -0.022736420 -0.021220659 -0.019894368
Table 5: The coefficients r;, / = 1 , . . . , 16 of the Hilbert transform for Daubechies' wavelet with six vanishing moments.
6.2
Fractional derivatives
We use the following definition of fractional derivatives,
where we consider a ^ 1, 2 , . . . If a < 0, then (184) defines fractional anti-derivatives. The representation of d™ on V0 is determined by the coefficients
provided that this integral exists. The non-standard form, d" = {Aj,Bj,Tj}j£z is computed via Aj = 2^a^0, Bj = 2~aj'50, and 1^ = 2-ajr0, where matrix elements a,-_;, pi-i, and 7,-_/ of AQ, BO, and FQ are obtained from the coefficients r/,
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231
and
It easy to verify that the coefficients 77 satisfy the following system of linear algebraic equations
where the coefficients agfc-i are given in (133). Using (169), (171) and (172), we obtain the asymptotics of ri for large /,
Example. In Table 6 we present the coefficients 77 of the fractional derivative with a = 0.5 for Daubechies' wavelets with six vanishing moments with accuracy 10~7. The coefficients for 77, / > 14 or / < — 7 are obtained using asymptotics
7
The Two-Point Boundary Value Problem
Let us consider the two-point boundary value problem
with the Dirichlet boundary conditions w(0) = «(1) = 0. We assume that a is a sufficiently smooth function and a(x] > 0, x € (0,1). Discretizing this problem on a staggered grid, we obtain the following system of linear algebraic equations
232
G. Beylkin
I
-7 -6 -5 -4 -3 -2 -1 0 1 2 3
Coefficients ri -2.82831017E-06 -1.68623867E-06 4.45847796E-04 -4.34633415E-03 2.28821728E-02 -8.49883759E-02 0.27799963 0.84681966 -0.69847577 2.36400139E-02 -8.97463780E-02
I
4 5 6 7 8 9 10 11 12 13 14
Coefficients n -2.77955293E-02 -2.61324170E-02 -1.91718816E-02 -1.52272841E-02 -1.24667403E-02 -1.04479500E-02 -8.92061945E-03 -7.73225246E-03 -6.78614593E-03 -6.01838599E-03 -5.38521459E-03
Table 6: The coefficients {r/};, / = — 7 , . . . , 14 of the fractional derivative ft = 0.5 for Daubechies' wavelet with six vanishing moments.
where u
where the N x N matrix L is as follows
The condition number of L is large. If o.(.r) — 1 in (194) and (197), then matrix L is the central-difference representation of the second derivative d'2/dx2 and has the condition number O(/V2). Also
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233
noting that the size of the function a might be different in the subintervals of (0,1), we observe that the condition number of the operator of multiplication by the function a(x) could be arbitrarily large. Our goal is to construct the matrix L""1 numerically to an accuracy € in 0( —loge TV) operations. This seemingly hopeless task (it is easy to check for small N that the matrix IT1 is dense in the ordinary representation) has, in fact, a simple solution in the wavelet system of coordinates. The kernel of the inverse operator for (194), the Green's function for the Dirichlet problem for an elliptic operator, has a sparse representation in the wavelet bases since it satisfies the estimates of the type in (94), (95). Let us show how to construct L"1 numerically by starting with the matrix L. Any finite-difference matrix representation of periodized differential operators may be rescaled by a diagonal preconditioner. We use this fact to solve (196) which is obtained via the standard discretization scheme. The wavelets play an auxiliary role in our considerations by supplying a system of coordinates where the condition numbers of sparse matrices involved in computations are under control. Such a "mixed" approach provides a simple way to see the advantages of computing in wavelet bases as well as a way to improve the performance of commonly used finite-difference and finite-element schemes. In order to use periodized differential operators, we consider L as a finite rank perturbation of a periodized matrix. Indeed, we have
where
234
G. Beylkin
and the unit vectors ej, ejv,
First let us assume that the size of the function a does not change significantly over the interval (0,1). To illustrate the effect of the diagonal preconditioning in the wavelet system of coordinates, let us set a = 1 and consider A = D,
In the following two examples we compute the standard form Dw qf the periodized second derivative D of size N x N, where TV = 2", and rescale it by the diagonal matrix P,
where, as before, PH = fnV, 1 < j' < n, where j is chosen depending on i,l so that N - N/2j~..l1+ 1
Fast Numerical Algorithms
N
235
K
Kp
32
0.10409 • 103
8.021
64
0.41535 • 103
9.086
128
0.16605 • 10*
10.019
256
0.66405 • 104
10.841
512
0.26562. 105
11.562
1024
0.10625 • 106
12.197
Table 7: Condition numbers of the matrix of periodized second derivative (with and without preconditioning) in the system of coordinates associated with Daubechies' wavelets with three vanishing moments M = 3.
N
K
Kp
32
0.10409 • 103
5.2002
64
0.41535 • 103
5.2610
128
0.16605 • 104
5.2897
256
0.66405 • 104
5.3035
512
0.26562 • 10s
5.3103
1024
0.10625 • 106
5.3137
Table 8: Condition numbers of the matrix of periodized second derivative (with and without preconditioning) in the system of coordinates associated with Daubechies' wavelets with six vanishing moments M = 6.
236
G. Beylkin
where
and e; = We/, / = 1,7V. Computing the discrete periodized wavelet transform of a vector of size TV = 2™ and using n scales, we obtain on the most sparse scale a single coefficient for differences and a single coefficient for averages which we call the total average. We note that the total average of a vector is proportional to the direct sum of entries of the vector. The sum of entries in the rows of the matrix A is identically zero and, therefore, the matrix Aw has the following structure,
where B is an (TV — 1) x (TV - 1) full rank matrix with the condition number proportional to TV 2 . Let us now determine the vector C T . If we compute Aw by first applying the transformation to the columns of A, we obtain the last row of the transformed matrix as
where p is a factor which depends on the size of the matrix A. In order to obtain Aw, we have to transform further by applying wavelet transform to the rows of the intermediate result. Thus, we obtain
Let us introduce the following notation
where r; are vectors of size TV — 1 and p is a scalar factor (common to both vectors),
and
Fast Numerical Algorithms
237
where By eliminating s,
we obtain the (N — 1) x (N — 1) system of linear algebraic equations for d,
Our method for solving the two-point boundary value problem (194) is based on the fact that the matrix B"1 is sparse in the wavelet system of coordinates and could be computed in O(N) operations. We will construct the matrix B""1 in Section 9 and here let us assume that it is available. Given the matrix B"1, we solve (214) using Sherman-Morrison formula for the rank-one update of the inverse matrix. We obtain
where
Remark. The condition number of the sparse matrix B after rescaling by P is 0(1) as is illustrated in Tables 1 and 2. Thus, the linear system (214) may be solved using (215) by the standard iterative methods (e.g. conjugate gradient) in O(N) operations since using (215) only involves finding the solution of the linear system Bx = y.
238
G. Beylkin We look for the inverse operator in the form
and we obtain
where
and
All matrix-vector multiplications in (218)-(224) involve a sparse matrix B"1 and sparse vectors n and r^r. Thus, the problem of constructing L"1 is reduced to that of computing the matrix B"1. The matrix L"1 is illustrated in Figure 9
8
8.1
Multiplication of Operators
Multiplication of matrices in the standard form
The multiplication of matrices of Calderon-Zygmund and pseudodifferential operators in the standard form has an operation count of at most O(7Vlog 2 TV). In addition, it is possible to control the width of "finger" bands by setting to zero all the entries in the product which are below a threshold of accuracy. Let us compute T = 2T, where
Fast Numerical Algorithms
239
Figure 9: An example of standard form of the matrix L l . Entries with absolute value greater than 10~~8 are shown black.
240
G. Beylkin
and
Since the standard form is a representation in an orthonormal basis, the result of the multiplication of two standard forms is also a standard form in the same basis. Thus, the product T must have a representation
Due to the block structure of the corresponding matrix, each element of (227) is obtained as a sum of products of the corresponding blocks of T and T. For example,
If the operators T and T are derived from Calderon-Zygmund or pseudo-differential operators, then all the blocks of (225) and (226) (except for Tn and Tn) are banded and it is clear that Fj is banded. This example is generic for all operators in (227) except for B"1^ , Tj + 1 , (j — 1,. . . , n ) and Tn. The latter are dense due to the terms involving Tn and Tn. It is easy now to estimate the number of operations necessary to compute T. It takes no more than O(7Vlog 2 N) operations to obtain T, where TV — 1n. If, in addition, the scales j and j1 are well separated, and the operators B3- , Fj may be neglected for a given accuracy (as in the case of pseudo-differential operators), then the number of operations reduces asymptotically to0(N). We .note, that we may set to zero all the entries of T below the threshold of accuracy and, thus, prevent the widening of the bands in the product. On denoting Te and Te the approximations to T and T obtained by setting all entries that are less than 6 to zero, and assuming (without a loss of generality) ||T[| = \\T\\ = 1, we obtain
and, therefore,
The right hand side of (230) is dominated by 3c. For example, if we compute T4 then we might lose one significant digit.
Fast Numerical Algorithms
8.2
241
Multiplication of matrices in the non-standard form
Let us describe an algorithm for the multiplication of operators in the non-standard form (Beylkin 1991). This algorithm is remarkable in the way it, decouples the scales in the process of multiplication. Let T and T be two operators
Given the non-standard forms of T and T. {Aj, Bj, Fj}j£Z and
{Aj,Bj,tj}j£z, we compute the non-standard form {Aj,Bj,Tj}j^2, of T = ff. We recall that the operators of the non-standard form are defined using the projection operators Pj (on the subspace Vj), and Qj = Pj-i-Pj (on the subspace W,;), j € Z, as follows Aj = Q^TQj, B1 = Q j f P j , tj = P3TQ3, A3 = Q j f Q j , B3 = Q3TP3 and f, = P3TQr We will also need the operators T3 = P3TP, and fj = PjTPj, j <E Z. These operators may not be sparse but the algorithm requires only a band around the diagonal of their matrices. In order to decouple the scales, we write a "telescopic" series,
or
We rewrite (234) as a sum of two terms,
242
G. Beylkin
where
and
The operators in (236) are acting on the following subspaces,
and the operators in (237),
where j = 1 , . . . , ? ? , and
Wre now describe an O(N) algorithm for the multiplication of the non-standa,rd forms of Calderon-Zygrnund and pseudo-differential operators. 1. We compute operators (238)-(242). This involves the multiplication of banded matrices. There is no problem in obtaining (238) or (241), since all matrices are banded. We are yet to show that in computing (239) and (240) we need only a band around the diagonal of T; and Tj, j = 1,.... n. Since the number of operations to compute (238)-(242) on a given scale j is half of that on the previous finer scale J — 1, the total number of operations is twice that on the finest scale j = I . Thus, the total number of operations at this step is proportional to A r . 2. We compute the non-standard form of all the operators in (241). First, we observe that YjBj, j = l . . . . , n are banded. Starting from the finest scale j = 1. we expand i jI37 to obtain
Fast Numerical Algorithms
243
AJ + I, jBj+i, FJ+I and TJ + I. We then add T,+i to rj + i5 J+ i and expand the sum of the two, etc. As a result, we obtain AJ, BJ, FJ, j — 2 , . . . , r a and Tn. Since at all scales we are expanding a banded matrix, and the number of operations is halved each time we go to a sparser scale, the total number of operations at this step is proportional to N. The resulting operators AJ, Bj, F J 7 j — 2, . . . , n , and Tn are acting on the subspaces,
and
3. At this step we add the corresponding operators computed at step 1 and step 2, to obtain the non-standard form of the operator T = TT,
and This step obviously requires O(N) operations. Let us now show that in computing (239) and (240) we need only a band around the diagonal of Tj and Tj, j = 1 , . . . , n If the product of two operators satisfies the estimates of Section 4, then the operators {Aj^j^Tj}^^cccof the non-standard form of T = TT are banded (for a given precision). Examining (247)-(249), we find that in (248) and (249) there is only one term that may potentially be dense. However, if the multiplicants and the product are banded then all terms a.re banded, and we conclude that BjTj and TjTj must be banded. Thus, we need only a banded version of the operators TJ and Tj. The banded versions of Tj and Tj are computed in the
244
G. Beylkin
process of constructing the non-standard form and, therefore, we only need to store the results of these computations. There is an alternative direct argument to show that BjTj and fjfj are banded. It requires a proof that the first several moments of rows of BJ and columns of Fj are negligible, which may be found in Meyer (1989a) for pseudo-differential operators.
9
Fast Iterative Algorithms in Wavelet Bases
Fast multiplication algorithms of Section 8 give a second life to a number of iterative algorithms. 9.1
An iterative algorithm for computing the generalized inverse
In order to construct the generalized inverse A^ of the matrix A, we use the following result (Schultz 1933): Proposition 3. Consider the sequence of matrices Xk
with where A* is the adjoint matrix and a is chosen so that the largest eigenvalue of a A* A is less than two. Then the sequence Xk converges to the generalized inverse A*. When this result is combined with fast multiplication algorithms of Section 8, we obtain an algorithm for constructing the generalized inverse in the standard form in at most O(N log2 N log K) operations and in the non-standard form in O(Nlog K), where K is the condition number of the matrix. By the condition number we understand the ratio of the largest singular value to the smallest singular value above the threshold of accuracy. Throughout the iteration (251)-(252), it is necessary to maintain the "finger" band structure of the standard form or the banded structure of the submatrices of the non-standard form of the matrices Xk. Hence, the standard and non-standard forms of both the operator
Fast Numerical Algorithms
245
Size N x N
SVD
FWT Generalized Inverse
i 2 -Error
128 x 128
20.27 sec.
25.89 sec.
3.1 • 1CT4
256 x 256
144.43 sec.
77.98 sec.
3.42 • 10~4
512 x 512
1,155 sec. (est.)
242.84 sec.
6.0 • 10~4
1024x1024
9,244 sec. (est.)
657.09 sec.
7.7 • 10~ 4
215 x 21S
9.6 years (est.)
1 day (est.)
Table 9: Timing and accuracy comparison of FWT Generalized Inverse computation vs singular value decomposition.
and its generalized inverse must admit such structures. Since pseudodifferential operators of order zero form an algebra, the operators of this class satisfy this condition. Also, since any pseudo-differential operator may be multiplied by a compressible operator so that the product is a pseudo-differential operator of order zero, it is easy to see that the iteration (251)-(252) is applicable to all pseudo-differential operators. Example. Table 9 contains timings and accuracy comparison of the construction of the generalized inverse via the singular value decomposition (SVD), which is an O(N3) procedure, and via the iteration (251)-(252) in the wavelet basis using Fast Wavelet Transform (FWT). The computations were performed on Sun Spare workstation and we used a routine from UNPACK for computing the singular value decomposition. For tests we used the following full rank matrix:
where i,j = 1...., A^. The accuracy threshold was set to 10 4 , i.e., entries of X& below 10~4 were systematically removed after each iteration.
246
G. Beylkin
9.2
Computing the inverse of pcriodized second derivative operator
Let us apply FWT Generalized Inverse algorithm to construct the generalized inverse of periodized second derivative operator in Section 7. We start by rescaling the (N — 1) x (TV — 1) matrix B by the diagonal matrix P, where P,v = 6^, 1 < j < n, and where j is chosen depending on i,l so that N - TV/2-'-1 +1 < i,l < N - N/2-1 (see Figure 8). We have
and as seen from Tables 7 and 8, the condition number of the matrix Bp is O(l). Also, B and Bp are full rank sparse matrices (see Figure 10). Our main tool in computing the inverse matrix B"1 is the iterative algorithm, in (251) and (252). For full rank matrices the iteration (251) converges to B"1. The number of iterations is proportional to the logarithm of the condition number of the matrix Bp and, thus, is of 0(1). For full rank matrices the iteration (251) is self-correcting and we use this property as described below. The iteration (251) provides an 0(N) algorithm to compute the inverse matrix if B p , B^l and all the intermediate matrices X;, ha,ve a sparse representation in the wavelet basis. Since we know in advance that B~ x is sparse in the wavelet basis (for a given accuracy c), we only need to maintain sparsity of intermediate matrices X;. Since the iteration is self-correcting, we first compute the lowaccuracy inverse by removing all entries with absolute value less than a given threshold (e.g. 10~ 2 ) after each iteration. Once the iteration converges we improve the accuracy of the inverse matrix by continuing the iteration and decreasing the threshold of accuracy. The sparsity of the resulting matrix B"1 is illustrated in Figure 11. Finally, to obtain B"1, we have
We note that since the matrix P is a diagonal matrix, there is no loss of accuracy in computing via (253) or (254). since only the operation of multiplication is involved. In our particular case the multiplication is by the powers of 2 only and, thus, no rounding errors are introduced.
Fast Numerical Algorithms
247
Figure 10: Matrix B (in the case A = D) of size 255 x 255 in the system of coordinates associated with the basis of Daubechies' wavelets with 3 vanishing moments. Entries with the absolute value greater than 10~14 are shown black.
248
G. Beylkia
Figure 11: Matrix B 1 computed via iterative algorithm of this Section with diagonal rescaling. Entries with the absolute value greater than 10~9 are shown black and the matrix verifies HBB""1 — I| , 9
HB^B-IH ~ i(r .
Fast Numerical Algorithms 9.3
249
An iterative algorithm for computing the projection operator on the null space.
Let us consider the following iteration
with where A* is the adjoint matrix and a is chosen so that the largest eigenvalue of a A* A is less than two. Then / — Xk converges to PnuiiIt can be shown either directly or by combining an invariant representation for PnUU = / - A*(AA*^A with the iteration (251)-(252) to compute the generalized inverse (AA*)^. Fast multiplication algorithms make the iteration (255)-(256) fast for a wide class of operators (with the same complexity as the algorithm for the generalized inverse). The important difference, however, is that (255)-(256) does not require compressibility of the inverse operator but only of the powers of the operator. 9.4
An iterative algorithm for computing the square root of an operator.
Let us describe an iteration to construct both /I1/2 and A" 1 / 2 , where A is a self-adjoint and non-negative definite operator. We consider the following iteration
with
where a is chosen so that the largest eigenvalue of j(A + /) is less than \/2The sequence A"; converges to A 1 / 2 and F/ to /I"'1/2. By writing A = V*DV, where D is a diagonal arid V is a unitary, it is easy
250
G. Beylkin
to verify that both Xi and YI can be written as Xi = V*PiV and YI = V*QiV, where PI and Qi are diagonal and
with
Thus, the convergence must be checked only for the scalar case, which we leave to the reader. If the operator A is a pseudo-differential operator, then the iteration (257) - (258) leads to a fast algorithm due to the same considerations as in the case of the generalized inverse in Section 9.
9.5
Fast algorithms for computing exponential, sine and cosine of a matrix
The exponential of a matrix (or an operator), as well as sine and cosine functions are among the first to be considered in any calculus of operators. As in the case of the generalized inverse, we use previously known algorithms (see e.g. Ward 1977), which obtain a completely different complexity estimates when we use them in conjunction with the wavelet representations. An algorithm for computing the exponential of a matrix is based on the identity
First, exp(2~ L A) is computed by, for example, using the Taylor series. The number L is chosen so that the largest singular value of '2~L A is less than one. At the second stage of the algorithm matrix 2~LA is squared L times to obtain the result. Similarly, sine and cosine of a matrix may be computed using the elementary double-angle formulas. On denoting
Fast Numerical Algorithms
251
we have for / = 0 , . . . , L — 1
where / is the identity. Again, we choose L so that the largest singular value of 1~L'A is less than one, compute the sine and cosine of 2~LA using the Taylor series, and then use (265) and (266). Ordinarily such algorithms require at least O(N3) operations, since a number of multiplications of dense matrices has to be performed. Fast algorithms for the multiplication of matrices in the standard form reduce the complexity to no more than O(Nlog2 N) operations, and in the non-standard form to O(N) operations (see Section 8). To achieve such performance, it is necessary to maintain the "finger" band structure of the standard form or the banded structure of the submatrices of the non-standard form throughout the iteration. Whether this is possible depends on the particular operator and, usually, may be verified analytically. Unlike the algorithm for the generalized inverse, algorithms of this subsection are not self-correcting. Thus, it is necessary to maintain a sufficient accuracy initially so as to obtain the desired accuracy after all the multiplications have been performed.
10
Product of Functions in Wavelet Bases
The wavelet bases provide a system of coordinates in which wide classes of linear operators are sparse. As a result, the cost of evaluating Calderon-Zygmund or pseudo-differential operators on a function is proportional to the number of significant wavelet coefficients of this function, i.e., the number of wavelet coefficients above a given threshold of accuracy. In order to use the wavelet bases for solving partial differential equations, one is led to consider differential operators and operators of multiplication by a function. It turns out that differential operators require O(l) coefficients for their description in wavelet bases (in the non-standard form). On the other hand, the operator of multiplication by a function a ( x ) , x 6 R d , seems to require O(Nd) coefficients for its description independently of the properties of a ( x ) . Indeed, the operator of multiplication by a(x) has the gen-
252
G. Beylkin
eralized kernel a(x)6(x - y) with the singularity along the diagonal x = y even if a(x) is smooth and non-oscillatory. Heuristically, if the solution of a partial differential equation is smooth and non-oscillatory on most of its support but is singular or oscillatory at a few locations, then solving in the wavelet bases should lead to fast and adaptive algorithms where the number of operations is proportional to the number of the significant coefficients in the representation of the solution in the wavelet bases. However, if in the process of solving this equation it is necessary to multiply the solution by a smooth function then, due to our previous remark, the algorithm will be insensitive to the smoothness properties of functions involved. Indeed, this point is most clear for nonlinear equations. For example, considering Burgers equation we observe that it is necessary to compute the square of the solution at each time step and, therefore, gains in the sparsity of the representation in wavelet bases are lost. Let us address the problem of pointwise multiplication of functions in wavelet bases. We will consider computing u2 in the wavelet bases since the product of two functions may be written as uv — \[(u + f) 2 — (u — v)2]. It appears that the straightforward algorithm which would require computing the expansion of the products of the basis functions, storing and using them to perform the multiplication is inefficient. Such an algorithm requires computing the coefficients
where ip3k(x) = 2~- 7 / 2 t/'(2~ ; a: — k) are basis functions. While computing Cfc3kl™ does not present a problem, the number of nonzero coefficients is large and, what is more important, the number of operations to compute u2 is proportional to Nf, where Ns is the number of significant coefficients in the representation of u. In many applications functions of interest are singular or oscillatory at only a few locations. The number of significant wavelet coefficients of such functions is 0(1) on each scale so that Ns is proportional to log(JV). Asymptotically the straightforward algorithm will perform better that an O(N) algorithm, but for problems of practical size it is inefficient. Efficient multiplication of functions represented in wavelet bases requires going back to the physical coordinates. There are several
Fast Numerical Algorithms
253
strategies to accomplish it. Let us consider an algorithm for pointwise multiplication of functions in the wavelet bases based on uncoupling the interactions between scales. The complexity of this algorithm is automatically adaptable to the complexity of the wavelet representation of u and is proportional to Ns (Beylkin 1991, Beylkin 1993). The algorithm permits a generalization for computing F(u) directly in the wavelet basis, where F is a smooth function and u is represented in a wavelet basis. 10.1
Uncoupling the interaction between scales
Let us consider the projections of u 6 L 2 (R) on subspaces Vj,
where {Vj}j£Z> 'ls a multiresolution analysis of L 2 (R). In order to uncouple the interaction between scales, we write a "telescopic" series,
or
In (270) there is no interaction between different scales j and j', j ^ j'. Let consider each term of (270) as a bilinear mapping
and
and note that in (271) and (272) we select the representation of L 2 (R) depending on the scale j.
254
G. Beylkin
Remark For the numerical purposes we need formulas (269) or (270) with a finite number of scales, though it is clear that by taking limits j —> oo and j —> — oo we have
which is essentially the para-product of Bony (1979,1981 1983) and Coifman and Meyer (1978). 10.2
Computing u2 in the Haar basis
Let us start by considering an example of (270) in the Haar basis. We have the following explicit relations,
where Xk(x) = 2~j/2x(2~jvjhsgdkjgjksgdsjksgdjksgkjgdkjdjkjkgjkj the characteristic function of the interval (0,1) and h is the Haar function, h ( x ) = 7\'(2x) — x(^x ~ !)• Expanding u0 into the Haar basis,
and using (274), we obtain from (270)
On denoting
Fast Numerical Algorithms
255
we rewrite (276) as
Remark. We note that if the coefficient d3k is zero then there is no need to keep the corresponding average s:ik. In other words, we can use coefficients d}k as templates to decide if we need to keep averages near the singularities, i.e., where the wavelet coefficients djk ( or prodiicts ^i^fc) are significant for a given accuracy. Finally, to compute the coefficients of the wavelet expansion of the function u(j, we need to expand the second sum in (278) into the wavelet basis. Starting from the scale j = 1, we compute the differbbb kk +l ences and averages d?k and sj(.+1. We then add sk+l to s3^1 before expanding it further according to the following pyramid scheme
It is clear, that the number of operations for computing the Haar expansion of MQ is proportional to the number of significant coefficients d3k in the wavelet expansion of UQ. In the worst case, if the original function is represented by a vector of the length TV, then the number of operations is proportional to TV. If the original function is represented by 0(log2 TV) significant Haar coefficients, then the number of operations to compute its square is proportional to Iog2 TV. The algorithm in the Haar basis easily generalizes to the multidimensional case.
10.3
Computing u2 in the wavelet bases
We now return to the general case of wavelets and derive an algorithm to expand (270) into the wavelet bases. Unlike in the case of the Haar
256
G. Beylkin
basis, the product on a given scale "spills over" into the finer scales and we develop an efficient approach to handle this problem. We use compactly supported wavelets though our considerations are not restricted to such wavelets. In order to expand each term in (270) into the wavelet basis we are led to consider the integrals of the products of the basis functions, for example
where j' < j. It is clear, that the coefficients M^yWW(k^k'J} are identically zero for \k — k' > k®, where k0 depends on the overlap of the supports of the basis functions. The number of necessary coefficients may be reduced further by observing that
so that
We also observe that the coefficients in (283) decay as the distance r = j — j' between the scales increases. Rewriting (283) as
and recalling that the regularity of the product tp(2~rx)tp(2~rx — k + k') increases linearly with the number of vanishing moments of the function ^/>, we obtain
with some A (Daubechies 1988, and Daubechies and Lagarias 1991). Let us define j0 as the distance between scales such that for a given f all the coefficients in (284) with labels r = j — j', r > j'0, have absolute values less t h a n c. For the purpose of computing with accuracy e, we replace the mappings in (271) and (272) by
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and
Since
and we may consider the bilinear mappings (285) and (286) on Vj 0 _i x Vj0-\. For the evaluation of (285) and (286) as mappings
we need significantly fewer coefficients than for the mappings (285) and (286). Indeed, it is sufficient to consider only the coefficients
Though it is a simple matter to derive and solve a system of linear equations to find Mo(p,q), we advocate a different approach to evaluate (289) in the next subsection. Let us now explain the reasons for considering (285) and (286) as mappings (289). On a given scale j the procedure of "lifting" the projections P3u, Q?u into a "finer" subspace is accomplished by the pyramid reconstruction algorithm. Let us assume that only a small number of the coefficients of QJU are a.bove the threshold of accuracy. We note (see Remark for the Haar basis) that only those coefficients of PJU that contribute to the product ( P j u ) ( Q j u ) (above the threshold (} need to be kept. In fact, one may consider the function Qjii as a "cutoff function" for PJU. Computing QJU and the corresponding part of PJU in the "finer" subspaces V ? _ i . V ; _ 2 ; • • • • V / 0 _ j via the pyramid reconstruction algorithm roughly doubles the number of the coefficients on each scale. The procedure is similar to that of "oversatnpling" by the factor of two (from scale to scale) and. as we reach V? 0_|, we have increased
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the number of coefficients by the factor of approximately 2 J °. This factor is a dominant constant in the complexity estimate of the algorithm. As an example, to maintain the single precision accuracy (c ?s 10~ 6 ), we have j0 = 6 and the "over-sampling" factor 2J° = 64 (for wavelets with six vanishing moments). Examples of functions for which it is indeed necessary to use j0 = 6 are the basis functions themselves. It is clear that the smoother are the basis functions, the smaller is the distance JQ. The compactly supported wavelets, however, are not very smooth since there is a trade-off between the number of vanishing moments and the smoothness of the function (Daubechies 1988). Fortunately, in many applications, projections P3u and QJU may be smoother than the basis functions. In. this case the scale distance jo may be chosen according to the smoothness of PJU and QJU and, thus, will be less that j'0. If we use the coefficients in (281) for computing the product, then we cannot make use of this observation in an adaptive manner. On the other hand, by computing QJU and the corresponding pa.rt of PJU in the "finer" subspaces we may decide to terminate the process adaptively. We note that the number of operations for this step of the algorithm is proportional to the number of the significant wavelet coefficients of the function.
10.4
Relations between values of functions and their wavelet, coefficients.
The result of "oversampling" PJU and QJU is that their product (for a given accuracy e) is in the same subspace as the multiplicands (see (289)) . To compute the product, one may use the coefficients (291) but such approach does not lead to an efficient algorithm in the multidimensional case. Instead of (289), it is sufficient to consider the mapping
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where <j)mk = <$>(m — k ) . Computing ~Jj.. allows us to evaluate the coefficients /; from the values of the function at integer points,
Formula (295) is the quadrature formula for / € V 0 . For most wavelets the entries of the matrix 4>~^k = $~l(m — k) decay fast away from the diagonal and for a given accuracy € the sum in (295) has very few terms. The algorithm to evaluate the product via mapping in (292) is now clear. First, we compute the values of the multiplicands at the integer points using (294). Then we compute the product via ordinary multiplication at these points. Finally, we use (295) to obtain the coefficients of the product. The virtue of this approach is that both operations (294) and (295) are convolutions. In the multidimensional case, the convolutions are separable and, therefore, lead to a faster algorithm than the algorithm using coefficients in (291).
11
References
Alpert, B.. 1990. "Sparse representation of smooth linear operators," Ph.D. thesis, Yale University Alpert, B., Beylkin, G., Coifman, R. R., and Rokhlin, V., 1993. "Wavelets for the fast solution of second-kind integral equations." SIAM J. of Sci. and Stat. Comp. 14(1). 159-174. B. Alpert, B. and V. Rokhlin, V., 1991. "A Fast Algorithm for the Evaluation of Legendre expansions." SIAM J. on Sci. Stat. Comput. 12(1), 158-179. A. Ben-Israel, A. and D. Cohen, D., 1966. "On iterative computation of generalized inverses and associate projections." SIAM J. Numer. Anal. 3(3), 410-419. G. Beylkin, G., 1991. "Wavelets, Multiresolution Analysis and Fast Numerical Algorithms." A draft of INRIA Lecture Notes. Beylkin, G., 1992. "On the representation of operators in bases of compactly supported wavelets." SIAM J. Numer. Anal. 29(6), 1716-1740. Beylkin, G., 1993. "On the fast algorithm for multiplication of functions in the wavelet bases." In Yves Meyer and S. Roques, editors, Progress in Wavelet Analysis and Applications, 5362. Proceedings of the International Conference Wavelets and Applications, Toulouse, France, June 1992, Editions Frontieres.
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Beylkin, G., 1994. "On wavelet-based algorithms for solving differential equations.'' In John J. Benedetto and Michael W. Frazier, editors, Wavelets: Mathematics and Applications, 449466. CRC Press. Beylkin, G. and Brewster, M. E. "Fast Numerical Algorithms using Wavelet Bases on the Interval." In progress. Beylkin, G., Coifman, R. R., and Rokhlin, V. 1989. "Fast wavelet transforms and numerical algorithms I." Yale University Technical Report YALEU/DCS/RR-696. Beylkin, G., Coifman, R. R., and Rokhlin, V. "Fast wavelet transforms and numerical algorithms II." In progress. Beylkin, G., Coifman, R. R., and Rokhlin, V., 1991. "Fast wavelet transforms and numerical algorithms I." Comm. Pure and Appl. Math. 44, 141-183. Beylkin, G., Coifman, R, R., and Rokhlin, V., 1992. "Wavelets in Numerical Analysis." In Wavelets and Their Applications, 181-210. Jones and Bartlett. Bony, J. M., "Interaction des singularites pour les equations aux derivees partielles non-lineaires." Sem. e.d.p., 1979/80, 22, 1981/82, 2 et 1983/84, 10, Centre de Mathematique, Ecole Poly technique, 91128-Palaisau, France. Bony, J. M., 1981. "Calcul symbolique et propagation des singularites pour les equations aux derivees partielles non-lineaires." Ann. Scient. E.N.S. 14, 209-246. Bony, J. M., 1983. "Propagation et interaction des singularites pour les solutions des equations aux derivees partielles nonlineaires." In Proceedings of the International Congress of Mathematicians, Warszawa, 1133-1147. Burt, P. J. and Adelson, E. H., 1983. "The Laplacian pyramid as a compact image code." IEEE Trans. Communications 31(4), 532-540. Calderon, A. P. and Zygmund, A., 1957. "Singular Integral Operators and Differential equations." Amer. J. Math. Soc. 84, 901-921. Carrier, J., Greengard, L., and Rokhlin, V., 1988. "A fast adaptive multipole algorithm for particle simulations." SIAM J. of Sci. and Stat. Comp. 9(4). Cohen, A., Daubechies, L, and Feauveau, J.-C., 1992. "Biorthogonal bases of compactly supported wavelets." Comm. Pure and Appl. Math. 45(5), 485 560. Cohen, A., Daubechies, L, Jawerth, B., and Vial, P., 1992. "Multiresolution analysis, wavelets and fast algorithms on an interval." Comptes Rendus Acad. Sc. Paris.
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Cohen, A., Daubechies, I., and Vial, P., 1992. "Wavelets on the interval and fast wavelet transforms." preprint. Coifman, R. R. and Meyer, Y., 1978. "An dela des operateurs pseudo-differentiels." In Asterisque, 57, (seconde edition revue et augmentee), Societe Mathematique de France. Coifman, R.. R. and Meyer, Y., 1989. "Nouvelles bases orthonormees de £2(R.) ayant la structure du syseme de Walsh." preprint. Coifman, R. R. and Meyer, Y., 1990. "Nouvelles bases orthogonales." C.R. Acad. Set., Paris. Coifman, R. R. and Wickerhauser, V., 1990. "Best-adapted wave packet bases." Daubechies, I., 1988. "Orthonormal bases of compactly supported wavelets." Comm. Pure and Appl. Math. 41, 909-996. Daubechies, I., 1992. Ten Lectures on Wavelets. CBMS-NSF Series in Applied Mathematics. SIAM. Daubechies, I. and Lagarias, J. C., 1991. "Two-scale difference equations I. Existence and global regularity of solutions." SIAM J. Math. Anal. 22(5), 1388-1410. Federbush, P., 1981. "A Mass Zero Cluster Expansion." Communication Math. Phys. 81, 327-340. Goupillaud, P., Grossman, A., and Morlet, J., 1984. "Cycle-Octave and related transforms in seismic signal analysis." Geoexploration 23, 85-102. Greengard, L., 1990. "Potential flow in channels." SIAM J. Sci. Stat. Comput. 11(4), 603-620. Greengard, L. and Rokhlin. V., 1987. "A fast algorithm for particle simulations." J. Comp. Phys. 73(1), 325-348. Haar, A., 1910. "Zur Theorie der orthogonalen Funktionensysteme." Mathematische Annalen, 331-371. Jaffard, S., 1992. "Wavelet methods for fast resolution of elliptic problems." SIAM Journal on Numerical Analysis 29(4), 965986. Jouini, A. and Lemarie-Rieusset, P. G.. "Analyse multi-resolution biorthogonale sur 1'intervaJle et applications." Annales de I'lnstitut Poincare, Analyse Non-lineaire, to appear. Klauder, J.R. and Skargerstarn, 1985. "Coherent States." World Scientific, Singapore. Lawton, W. M., 1991. "Necessary and sufficient conditions for constructing orthonormal wavelet bases." J. Math. Phys., 32(1). Littlewood, J. and Paley, R., 1937. "Theorems on Fourier Series and Power Series." Proc. London Math. Soc. 42(2), 52-89.
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Mallat, S. 1987. "Multiresolution approximation and wavelets." Technical report MS-CIS-87-87, GRASP Lab 80, Dept. of Computer and Information Science, University of Pennsylvania. Mallat, S., 1988. "Review of multifrequency channel decomposition of images and wavelet models." Technical Report 412, Courant Institute of Mathematical Sciences, New York University. Malvar, H. S., 1990. "Lapped Transforms for Efficient Transform/ Subband Coding." IEEE Trans. Acoust., Speech, Signal Processing 38(6), 969-978. Meyer, Y., 1989a. "Le calcul srientifique, les ondelettes et filtres miroirs en quadrature." CEREMADE, Universite ParisDan phine. Meyer, Y., 1986. "Principe d'incertitude, bases hilbertiennes et algebres d'operateurs." In Seminaire Bourbaki,, 662. Societe Mathematique de France, 1985-86. Asterisque. Meyer, Y., 1986. "Ondelettes et fonctions splines." Technical report, seminaire edp, Ecole Polytechnique, Paris, France. Meyer, Y., 1989. "Wavelets and operators." In N.T. Peck E. Berkson and J. Uhl, editors, Analysis at Urbana. London Math. Society, Lecture Notes Series 137, 1989. Meyer, Y., 1990. "Ondelettes et Operateurs." Hermann, Paris. O'Donnel, S. T. and Rokhlin, V., 1989. "A fast algorithm for the numerical evaluation of conformal mappings." SIAM J. Sci. Stat. Comput. 10(3), 475-487. Rokhlin, V., 1985. "Rapid solution of integral equations of classical potential theory." J. Comp. Phys., 60(2). Schulz, G., 1933. "Iterative Berechnung der reziproken Matrix." Z. Angew. Math. Mech. 13, 57-59. Smith, M. J. and Barnwell, T. P., 1986. "Exact reconstruction techniques for tree-structured subband coders." IEEE Transactions on ASSP 34, 434-441. Soderstrom, S. and Stewart, G. W., 1974. "On the numerical properties of an iterative method for computing the Moore-Penrose generalized inverse." SIAM J. Numer. Anal. 11(1), 61-74. Stromberg, J. 0., 1983. "A Modified Franklin System and HigherOrder Spline Systems on Rn as Unconditional Bases for Hardy Spaces." In Conference in harmonic, analysis in honor of Antoni Zygmund, Wadworth math, series, 475 493. Ward, R. C., 1977. "Numerical computation of the matrix exponential with accuracy estimates." SIAM. J. Numer. Anal. 14(4), 600 610.
5 SOME WAVELET ALGORITHMS FOR PARTIAL DIFFERENTIAL EQUATIONS J. Liandrat
Contents 1 Introduction
265
2 Approximation of Functions 265 2.1 Quality of the Approximation Using Wavelet Expansions 266 3
Fast Algorithms
3.1 3.2 3.3
270
Tree Algorithms Related to Orthonormal Decomposition^ 0 Tree Algorithms Related to Biorthogonal Decomposition27l Fast Algorithms on Adapted Spaces 272
4 Differential Operators 4.1 Action on Wavelets 4.2 Action on Multiresolutions
272 273 278
5 Approximation of Elliptic Operators 279 5.1 Linear operators with constant coefficient\jh hjch 5.1.1 Algorithm I: Convergence and Stability . . . . 280 5.1.2 Implementation 282 5.1.3 Algorithm II: Convergence and Stability . . . . 285 5.1.4 Implementation 286 5.2 Parabolic Equation 287 5.3 Linear Operator with Non-Constant Coefficientjdj kldkd 5.3.1 Construction of the first estimate 289 6 Adaptive Algorithms 6.1 Adapted Approximation Spaces 6.2 Control of the Approximation Spaces
291 291 294
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J. Liandrat Numerical Implementation 7.1 Wavelets on an Interval .
7.1.1 7.1.2
295 295
Periodized Wavelets on [0,1] . 296 Other Boundary Condition Wavelets on [0,1] . 299
7.2
Wavelets in Multidimensions
7.3
Nonlinear Terms . 303 7.3.1 Collocation algorithm . . . . . . . . . . . . . . 303
7.4
Numerical Tests on Burgers Equation
7.4.1 7.4.2 7.4.3 7.5
302
. . . . . . . . . 305
General Approach 305 Computational Space . 307 Comparison with Spectral Method and Exact Solution 308
Non-Constant Coefficient Elliptic Equation
310
8
Conclusion
312
9
References
313
Wavelet Algorithms for PDE'S
1
265
Introduction
The use of wavelets in numerical analysis is at first glance attractive for two basic reasons. First, the wavelet approach and its associated multiresolution analysis provide nice approximation spaces Vj and Wj suitable for the computation of an approximate solution to problems in which small-scale structures are localized in space and whose location may vary in time. Second, the multiresolution spaces lead to fast hierarchical algorithms with 0(JVlog N) operation count. These two properties alone are not sufficient to fully exploit the potential of wavelets. On must understand the action of operators and their inverses on wavelets to insure that the implicit systems of equations that are a part of many discrete numerical methods have the required nice properties for efficient inversion. These include diagonal dominance, low condition numbers (or the existence of good preconditioners), and banded structure. In this paper, we review recently developed algorithms devoted to the numerical solution of partial differential equations (PDEs) that are fully based on the wavelet approach. This is in contrast to various other methods, which although they make use of wavelets, do not intrinsically depend on them. Such methods include finite-element type methods which use scaling functions as basis functions, and methods which use wavelets as a preconditioner. Although these alternatives may be efficient, they are not considered here. Additional information on numerical algorithms and related issues are discussed in the two articles of Jameson and Beylkin in this book. After a short review of basic results related to the approximation of functions, we focus on the approximation of operators and on the design of efficient wavelet-based numerical algorithms for various model PDEs. Finally, examples based on Burgers equation and the heat equation wiD be presented.
2
Approximation of Functions
Let us consider problem (P}'
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where fi is an open set of IR,™ with boundary 9Q, / a given function of L 2 (fi), g a given function defined on d$l and B the boundary condition operator. Generally, the first step in solving problem (P) numerically is to replace eq. (1) by the approximate version (PN):
where EN is the finite-dimensional approximation space, and fa and gN are approximations to / and g in EN and £^($(1.) respectively. It is well-known that the departure of the solution of the approximate problem (PN) from the exact solution of problem (F) is generally controlled by the quality of the approximation provided by EN- More precisely, there is a constant C independent of EN such that
where || • || defines a suitable norm. Since minv^EN ||tt — v\\ defines the quality of the approximation by EN, it is useful to review the properties of wavelets as they relate to the approximation of functions. These properties will prove useful to solve efficiently ODE's and PDE's.
2.1
Quality of the Approximation Using Wavelet Expansions
Orthonormal bases of wavelets are good bases for many spaces other than L 2 , a feature not shared by Fourier basis functions. For example, Fourier basis functions are incapable characterizing in a simple manner multiple local singularities. Among the different results on the characterization of functional spaces by wavelets, we present here those most pertinent to numerical applications. A space well-known to numerical analysts and engineers alike is the L2 space of square integrable functions. That is, if / G £ 2 ,
This integral defines a norm in L2. However, one is often confronted with the need to characterize the first s derivatives of a function, and
Wavelet Algorithms for PDE'S
267
to establish bounds on their magnitude. This leads to a more precise norm, defined in a space called Hs:
Note that H° = L 2 . The Sobolev space Hs is simply a Hilbert space with the norm || • ||#» denned above. Wavelets are well-known for their ability to characterize functions and determine the spaces which they belong. This is done through an analysis of the function's wavelet coefficients. For example, if a function is known to be in i2, it is possible by constructing an appropriate series involving the wavelet coefficients of / to determine whether / also belongs to the space Hs, without explicitly computing derivatives. The precise manner in which this check is accomplished is given by theorem 1. However, we must first define the concept of r-multiresolution which characterizes the rate at which the wavelets and its r — 1 first derivatives decay at infinity. Definition 1. To be more precise, a multiresolution with r-regularity generates r—regular wavelets which satisfy the following two properties:
Theorem 1. Given a function f 6 L2, and a r—multiresolution Vj, j € 2Z o/Z-2(]R), then
if and only if where {sj} £ / 2 (1N). Moreover, \\f\\H* is equivalent to the norm
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In other words, given a function
at scale j, the bounded L 2 norm of the projection Qjf of / in the wavelet space Wj is*
The subscripts j and k refer to a function evaluation at the point x = 2~->(k + ^). The collection of these points for all k and j form a dyadic grid. Theorem 1 states that / 6 Hs if and only if
It essentially gives a measure of the error committed in approximating /(x) and its first s derivatives when all wavelet coefficients at a scale j — jo and all finer scales j > jo are removed. When constructing adaptive algorithms, it will be necessary to remove a subset of wavelets at a given scale. Therefore it is necessary to obtain a local version of Theorem 1. To do so, consider a finite interval / C IR and redefine the norm ||<5j/||2 by
The results of Theorem 1 then carry over with IR replaced by the finite interval /. Thus we can precisely measure the error incurred when removing all wavelets at scales j > j'0 which intersect the finite interval /. Examples of the quality of wavelet approximations can be found in Daubechies (1992). The following examples have been selected to illustrate specifically the generation of low dimensional approximation spaces for PDE's. "In our notation, the imbeded spaces are refined as j increases. The projection operator which projects a function from L2 to Vj is called P}, while the projection operator from L2 to the wavelet space Wj is called Q3.
Wavelet Algorithms for PDE'S
269
Example. The following plots are a 3-D representation of the wavelet coefficients Wf(b, a) related to a spatial "redundant wavelet transform" (Tchamitchian chapter in this volume) of a function f ( x , t ) corresponding to a numerical approximation to Burgers equation (see Section 7.4) at various times. We remind the reader, that given a time t, the wavelet coefficient of the wavelet ip((x — b}/a) (translate b and scale a) of a function /(#,£) is
where
Note that the scale axis is not linear but logarithmic in the figures, i.e. s = —/O(/2(o)Since the chosen wavelet ifj(x) generates the orthogonal spline wavelets on a dyadic grid, the orthogonal wavelet coefficients can be directly measured at the points of this grid. The solution analyzed here exhibits a strong gradient region which moves from left to right and sharpens as / increases. It is clear from the plots that the wavelet decomposition adapts itself to the scales of the functions by focusing around the singularities. For example, at t = t$, there is a strong increase of small scales near j = 5 and x = 0.6 (see Figure Id). Although not plotted here, the global Z/2 norm ||/||2 decreases with time (property of Burger's equation), however, in the neighborhood of the shock, the norm ||/||#2 can increase faster than ||/||2- Thus it is important to consider Hs norms in adaptive algorithms if all the scales of the functions and a fixed number of derivatives must be resolved while at the same time the number of degrees of freedom must be reduced. One can also characterize the local regularity of the solutions at different times from the wavelet plane data, but that is beyond the scope of this chapter. However, it is intuitively clear that "energy" accumulates near singularities, and therefore it seems well-suited to trace out the critical regions, which play an essential role in the approximation theory and and in the computation (see Section 6).
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270
A
Figure 1: Solution of Burgers equation at times to = .0, ti = I/TT, t
3
3.1
Fast Algorithms
Tree Algorithms Related to Orthonormal Decomposition
A fundamental property of wavelets, of direct relevance to fast numerical algorithms, is their hierarchal structure which naturally leads to fast algorithms. The algorithms used to compute the wavelet coefficients, i.e. < u, ipjk > for a function u(x) 6 Vp are tree algorithms. Thanks to the orthogonality property of the transform, if
one has
and
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271
where A(t) and B(i) are two discrete filters (see Strang, this text). The sums involved in tree algorithms are rapidly evaluated via convolution in O(N) or O(NlogN) operations: the former count if the wavelets (and filters) have compact (i.e. finite) support, and the lat-
ter count if the filters are more general. Thanks to scaling covariance, this property is available at every scale. Therefore, the computation of the coefficients < u,ifjjk >-, j < J can be rapidly accomplished from the scaling coefficients cj^ =< u,(/>ji >. We wish to emphasize that the existence of a tree algorithm to compute scalar products < u,r)jk > (if fjjk(x) = t]jo(x — &2~- ? )) is ensured if there exists a constant p, such that rjjk is orthogonal to the set generated by ^-/p j > j + p. Indeed, with this hypothesis one gets
where A • ./(i) is a discrete filter associated with rjjk but now a function of both the scales j and j'; that is rjjk need not possess the property of scale covariance. A tree algorithm structure can then be automatically derived. For example, we will see in Section 7 that this property is true to within a prescribed accuracy for the functions
and
for a reasonably small value of p (i.e. p & 2). Here, L is a member of a certain class of elliptic operators. 3.2
Tree Algorithms Related to Biorthogonal Decomposition
Scale covariance and translation invariance are key notions for tree algorithms. However, orthogonality is not really necessary. This has led to the construction of biorthogonal multiresolution frameworks (Cohen 1990). From the point of view of numerical resolution of partial differential equations, the biorthogonal multiresolution framework may provide more freedom in the choice of wavelet basis functions and therefore allow the construction of wavelets with more suitable properties for the numerical discretization of functions and operators:
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compact support, regularity, or vanishing moments. As explained in Section 4.2, biorthogonal multiresolution spaces are also obtained through the action of homogeneous operators on orthogonal multiresolution spaces, and thus lend themselves to the solution of differential equations. 3.3
Fast Algorithms on Adapted Spaces
The introduction of adapted spaces of approximation leads directly to the redefinition of the various tree algorithms, for which, as we have seen above, translation invariance on a dyadic grid and scaling covariance play key roles in a general approach. Usually the scaling covariance will be maintained; but the dyadic translation is obviously lost on adapted spaces. Therefore, some modifications to the algorithms are required. It turns out that fast tree algorithms are still possible on conestructured spaces of approximation (Section 6.1) and that starting from an orthogonal multiresolution analysis on a full dyadic grid one can construct a biorthogonal multiresolution analysis in an adapted space. Decomposition algorithms with O(N) operations are still possible if the wavelets have compact support. The interested reader should refer to Ponenti (1994) for a careful analysis of this problem. We emphasize that such algorithms are quite essential to make optimum use of adapted spaces to solve PDE's.
4
Differential Operators
The next step in the use of wavelets to solve PDE's is to understand the action of a large class of operators on the wavelet themselves and on the associated multiresolution. For example, given a multiresolution composed of wavelets and scaling functions defined in spaces Vj and Wj, one might ask, what is the structure of the collection of functions once an operator is applied to them. Furthermore, do the spaces spanned by these new functions still form a multiresolution? We answer these questions in the following sections.
Wavelet Algorithms for PDE'S
4.1
273
Action on Wavelets
Given a differential operator, the simplest basis to work in is one which diagonalizes the operator. This is one reason why Fourier methods are so popular: they exactly diagonalize linear differential operators with constant coefficients. However, an operator's eigenvectors cannot always be computed. This is in particular true when the operators have variable coefficients. Wavelets are not the eigenvectors of differential operators, nor are they the eigenvectors of any Sturm-Liouville problem. Thus they cannot be directly used to diagonalize the operators, nor can spectrally accurate algorithms be constructed from them. However, they can still be used to approximate efficiently differential operators and to construct algorithms to the desired accuracy. Despite the constraint of the Heisenberg uncertainty principle, the r—regular wavelets (see Tchamitchian, this volume) also have a "good localization" in the Fourier space. Consequently, wavelets nearly diagonalize differential operators: i.e, wavelets are not too widely spread out by differential operators and the image of a wavelet by a differential operator still retains the nice properties of regularity and localization. We will have much use for the two families of derived functions {rjk} and {0jk}, defined by the action of the operator L:
(For T and 0 the subscripts jk have the standard meaning in tensor analysis and do not refer to translation and scale indices.) In the case of const ant-coefficient elliptic differential operators, the following theorem, which describes the action of elliptic operators on wavelets, is a particular application of a more general result proved in Meyer (1990b). We define the symbol of the differential operator
as where
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Theorem 2. // L is a constant coefficient elliptic mth order differential operator with a strictly positive symbol (CT(U;) > 0 for all u> € 1R), if ij)jk> (Jity G TL, is a family of r—regular wavelets, then the functions Tjk = Li/jjk satisfy the same estimates as ifjjk as soon as r > m and the functions Ojk — L~1tl>jk satisfy the same estimates as i(jjk up to a normalization factor, i.e.:
and all integers p. Furthermore,
and all integers p < r + a. The key point of this result is that the constants Cp and Cp are independent of the scale parameter j. Therefore, the localization of the functions Ojk and Tjk follows homogeneously with respect to j the localization of tf)^ in both the Fourier and the physical domains. Following Meyer (1990b) the families {Tjk} and {Ojk} are called "vaguelettes" or pseudo-wavelets. If Lu = /, we obtain directly the exact formal solution
where and
(L* stands for the adjoint of L.) The role played by the functions Bjk and Tjk to solve problem (P) is clearly important. Their properties of localization, described by Theorem 2 will turn out to be essential for the practical implementation of algorithms to solve problem (-P/v)However, if one has
Wavelet Algorithms for PDE'S
275
then an alternative expression for u is
where the family of functions {£p/t} is denned by
Why should we prefer a decomposition using the functions Ojk and tfrjk rather than one using £p/t and (f>pk^ In other words, why use wavelets rather than scaling functions? Figure 2 gives the clue. We have plotted, together for various values of p and j, the functions |VVfc(w)|, \0jk(u)\, |<£ p fc(w)|, \^pk(u)\ corresponding to spline wavelets of order m = 6 and the operator
of symbol
Notice that while the numerical supports of ^>, ip and 6 increase with the scale j, the numerical support of £ is constrained by the support of the Greens function Gr(u>}. Figure 3 shows a plot of the Greens function Gr(x) associated with the operator L as well as its Fourier transform Gr(u>). The functions t/)jk(x), Ojk(x),
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o2
Figure 2: Spline wavelet and pseudo-wavelets related to L = I — —^ in the Fourier domain - \$jk(u)\, - - \^jk(u}\, ... |£/fc(w)|, .- |0jfc(w)| at various scales (j = 0 to j = 3).
Figure 3: Green's function associated with L = I — -jj-^z- (a) G r (w), (b) GT(x).
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Figure 4: Spline wavelet and pseudo-wavelets related to L = I — -j^ on the real line at various scales, (a) jk(x), (b) tf)jk(x), (c) Ojk(x), (d) ^(^)-
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Action on Multiresolutions
The following section, borrowed from Ponenti (1994) provides a precise description of the action of elliptic operators on the multiresolution structure. It is a generalization of a result from Lemarie-Rieusset (1992). Let {Vj} the family of approximation spaces connected to the multiresolution analysis associated with the wavelet t/? and consider a homogeneous elliptic operator L. The following theorem explains that the biorthogonal functions Oj^ and TJ& are connected to a biorthogonal multiresolution framework related to the initial mutiresolution structure (of -t/>) and to the symbol ff(w) of the operator, assuming that a specific relationship between the regularity of the multiresolution and the order of the operator is satisfied. A multiresolution framework for Ojk and TJJ. allows the use of pyramidal algorithms to evaluate scalar products involving these functions and to perform fast summations involving scalar products with these functions. For completeness we give the precise theorem which states the conditions under which a biorthogonal multiresolution analysis is possible. Moreover, the theorem explicitly provides the filters mj, mp, TOJ and m^ that replace the filters mo, rn\ of the initial orthogonal multiresolution. Theorem 3. Consider L, a constant coefficient elliptic operator of symbol a(u>) and {Vj}, a multiresolution analysis of L 2 (IR). If L is s-homogeneous (i.e.
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and
More details on the choice of 5(w) and the implementation of biorthogonal multiresolutions can be found in Ponenti and Liandrat (1994).
5
Approximation of Elliptic Operators
This section is devoted to the third step that justifies the use of wavelets to discretize partial differential equations. Recall that the first two steps were the approximation of functions and the action of operators on multiresolutions and wavelets. Here we discuss the approximation of the differential operator L in the finite-dimensional space VJv- Although prevalent and easy to implement, Galerkin type algorithms are not the only viable algorithms within the framework of wavelets. We now begin the discussion of specific efficient algorithms which exploit the localization of wavelets in the physical and Fourier spaces. Note that when there are N discrete points, an algorithm is considered efficient if it can be implemented in CN log N or CN operations with reasonable values for C independent of N. We first present two algorithms devoted to the inversion of constant coefficient elliptic operators and strongly based on the results of Theorems 2 and 4. The case of parabolic operators is then considered. Finally we present a fourth algorithm related to the inversion of non-constant coefficient elliptic operators.
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5.1
Linear operators with constant
coefficients
From the formal inversion of Lu = /, we derive two useful formulas to approximate the solution u. In both cases, fast tree algorithms transform these theoretical formulas into efficient numerical algorithms linked to Theorem 2. 5.1.1
Algorithm I: Convergence and Stability
Consider an elliptic equation Lu = f where L is a linear, constant coefficient elliptic operator with strictly positive symbol (T(OJ) and /e£2.
We have already seen that
for all A E A, from which
The algorithm consists in computing a projection of u = L~l f into a suitable approximation space V. It is therefore not an algorithm of Galerkin type (which "projects" the operator itself into V). Once V, the space of approximation, is chosen, with
an approximation of u in V can be estimated as
where "P is the orthogonal projection into the approximation space V. As denned, uy is the best approximation ofu available in V. This is in contrast with methods which first approximate L in V. In that case, the resulting u in V is suboptimal. This algorithm is obviously convergent in the L2 norm since we are computing a projection of the exact solution into the approximate space. Therefore if the approximate space converges in L 2 , so must, the function converge to the exact solution in L2. It follows that the solution is also convergent in the suitable IIs norm (e.g. H^ norm if
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L is of second order and / € £ 2 ), provided that the wavelet basis is also an unconditional basis of Hs (see Tchamitchian, this volume). The following theorem (Liandrat and Tchamitchian 1993) summarizes some useful estimates on the convergence of the algorithm: Theorem 4. With the above notations, with L a nih order differential operator and with (^\) being an unconditional basis of Hs,
and where C depends on the operator L and the multiresolution analysis only. The convergence estimates concerning algorithm I are the same as for classical algorithms (e.g. Galerkin), and are directly linked to the approximation properties of the spaces Vj. For constant coefficient problems, there is no special gain in using wavelets, as far as convergence is concerned. It is not clear however, that the numerical accuracy obtained by the present algorithm is the same as in Galerkin methods due to the fact that the computed approximation is the best one available for a chosen approximation space. Moreover, when one turns one's attention to stability estimates, the situation is completely different: The use of wavelet bases provides a uniform preconditioning of the algorithm. First, a preliminary observation is in order: since PL~l is not invertible (it has only a right inverse), the condition number is not clearly defined. Remembering that the family {T\ = Lt}}\} is the biorthogonal family associated with {9\} and denoting QT the oblique projection
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where
we have, by construction
and
The restricted operator PL~lQr from span AgAv ,{rA) to V is invertible and we can define its condition number. Of course, exactly as for a Galerkin method, the condition number can be quite high (for instance, if L = I — A, then
where Fj, and T are defined by
for all A € A. Then, since TA and a(2J)'>p\ are two unconditional bases of Hl, T is continuous and invertible. Hence, the condition number is bounded independently of the approximation space V. Thus, the ill-conditioning is entirely transfered to the operator TL which is by definition, diagonal in the wavelet basis. The operator TL is well known in the wavelet literature (JafTard and Laurencot 1992, Beylkin chapter in this volume). 5.1.2
Implementation
Starting from the set of coefficients < f^Ojk >•, we know that using classical tree algorithms, the solution of Lu = f can be rapidly reconstructed from the set of coefficients
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The single difficulty is then to compute efficiently the scalar product
where P± is the projection into the space VI, orthogonal to V. Therefore, if V is generated by the family {t/)\,\ G Ay), the scalar products < Ojk^x >, A ^ Ay (or A 6 Ay ± ) govern the error made by replacing < f,0jkkjdhfkjdhfjkdhfddkjfhsdjkdjfhdskjfhjkdskjds.fd numerical estimate of HQp^jfcllHQj^jfcll" 1 (where Qp is the orthogonal projection onto Wp] for p > j and for various values of j (to show the scaling invariance). Note that the norm of the projection decreases as j —p increases and is independent of the scale j. This is a numerical check that the localization of Ojk homogeneously follows that of ij}^. Here, the Ojk function is the one used in the numerical resolution of Burgers equation (Section 7.4.1).
Figure 5: Estimates of l|^p^).lll for j = 6, 7, 8, 9,10. iNWfcll
The computation of the scalar products < /, 0\ > can then be
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performed using a tree algorithm. Indeed, if one writes
then, a good approximation of calling that
it follows that
Since we have
and
we obtain
where {otj} and {(Pe)j} are two families of finite sequences parametrized by j and depending on the multiresolution analysis and on the constant coefficient operator L. However, these filters cannot be deduced, as classical multiresolution analysis filters can, from a single finite sequence. They are generally different at every scale due to the lack of scaling covariance for the operator itself. In practice, otj and (0l)j are computed before the calculation proceeds:
Similar to the classical tree algorithms', the calculation of < /, 9jk > for 0 < k < 2 j , 0 < j < J -I is reduced to p(J -p+ 1) + Q.5p(p- 1) convolutions involving the finite sequences otj and (/3;)j. Figure 6
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Figure 6: Tree algorithm filters connected to our first algorithm, (a)
a», (b) «,-(*), (c) A», (d) (A)>(*)shows the finite sequences «j and (fli)j used for the numerical tests of Section 7.4.1 where j = 8 and 1=1. For a classical tree algorithm, p = 1, and ctj is independent of j, while (/3;)j = 0 for all /. In the periodic case with V C Vj, the storage array is of size 0(2J). 5.1.3
Algorithm II: Convergence and Stability
We start again from
The second algorithm consists again in truncating the above series:
We have again, with "P the orthogonal projection on V:
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Following the same approach as for the first algorithm, one can prove the convergence of uy in the L2 norm and also in suitable Hs norms. Along the same lines as in Section 5.1.1 we obtain
and
Finally, uy satisfies the same convergence estimates as uy once uy is replaced by uy in the theorem statement. To study the stability of the algorithm, we proceed as in Section 5.1.1 and introduce the oblique projection Qr associated with 9. We get Again, QrL~^Tr can be factorized and a diagonal preconditioner can be exhibited. We conclude as in Section 5.1.1 that the wavelets provide a uniform preconditioning of the inversion algorithm and the algorithm is therefore stable. 5.1.4
Implementation
In this case,
and the difficulty is in computing the wavelet coefficients < uy, i/)\ > from the sequence of known coefficients < /, ^\ >. In reality, the values of < My,V>A > are only required when the algorithm is to be applied recursively (time-dependent problems). For a stationary problem, one directly computes the sum in Equation (5.1.3) using a tree algorithm. As was the case for algorithm I, one could take into account the rapid decrease of the scalar products < Ojki^j'k1 > with increasing scales (j,j') and position (&,&'), but we choose an alternative, which involves the biorthogonal functions (f>6-k related to the set {#,-&} when the biorthogonal framework is available (see Section 4.2). Indeed, one has then
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Figure 7: Biorthogonal filters connected to the second algorithm. and following Section 4.1, a fast tree algorithm can then be used to compute the set of coefficients < u,(j)TJk >. From these coefficients it is then possible to compute the final set of coefficients < uv,^\ > using another tree algorithm. Figure f 3 shows the biorthogonal filters connected to a biorthogonal mutiresolution derived from spline wavelets of order m = 6 and where the operator
5.2
Parabolic Equation
The numerical solution of parabolic evolution equations allow a comparison of the present algorithms to the classical approach (Canute, Hussaini, Quarteroni and Zang 1987). Let us consider the following
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where LO is a linear differential operator with
where u^ is the approximation of u at time nAi. At each time step, we solve the elliptic equation
Once the right hand side has been evaluated, an approximation of w( n+1 ) is computed as shown in the previous section. Finally, the numerical solution of the parabolic equation leads to a cascade of elliptic problems. We would like to note that this procedure does not fall under the general framework of Canute et al. (1987) for spectral methods. Indeed, the introduction of the so-called semi-discrete approximation of the continuous problem using a space of approximation (Vjvr), and discrete operators (LM and GM) leads to
The same time-discretization scheme as above gives rise to the implicit system
For example, if we take the simple case G — 0, the latter method yields
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bbb
while our method gives
Notice that one can replace P(I + AtL0}~1 by P(I + AtLo^P in the (n — 1) first terms of the product because the first step produces Pu. The difference between the two approaches is best seen by only considering the first time step (n — 1). Whereas the spectral algorithm projects both u and L onto VM before inversion, the present algorithm projects [7 + At L]~lu into VM- Thus the present algorithm produces an optimal solution for a given space VM after one time step. The differences between the algorithms after multiple time steps as they relate to accuracy and stability is less clear. Details on the convergence and stability of the present parabolic algorithm can be found in Liandrat and Tchamitchian (1993). 5.3
Linear Operator with Non-Constant
Coefficients
In a similar vein, (Tchamitchian 1994; Lazaar, Liandrat and Tchamitchian 1994) efficient algorithms to estimate the inverse of non-constant elliptic operators have been proposed. Without going into details, we present the main features of their construction for the simple example of the operator defined by
where b(x) is a strictly positive and bounded Lipschitz function on JR.. The general idea is to provide a first estimate of the inverse L~1 and then to correct this estimate using a residual correction algorithm. Good convergence properties of this correction algorithm is obviously related to a proper definition of the first estimate of L~l. 5.3.1
Construction of the first estimate
The first estimate of the inverse is constructed from a cross between a variational approach for large scales and a collocation approach for small scales. More precisely, given a level of approximation p, and
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an integer q < p, we decompose Vp in the standard way:
We first introduce the operator Apq defined in Vp as
where Lp is the restriction of L to V^,, II? is the orthogonal projection from Vp to Vq and II* the associated extension from Vq to Vp. Then, for q < j < p — 1, we introduce the functions Tjk £ L2 as
where and define the operator Ppq: P
The first estimate of L~^ is the sum Apq+Ppq. Defining the correction operator Upq through the expression
the following theorem (Lazaar et al. 1994) ensures that an iterative correction of the first estimate will converge. Theorem 5. There exists a constant C, depending on b(x) and on the mutiresolution such that
Note that C does not depend on p or q. For large enough values of q, the spectral radius of U is less than one. Then a classical residual correction algorithm will converge. Finally, the inverse of Lp is given for instance by
Details of the implementation of this algorithm can be found in Lazaar et al. (1994).
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291
Adaptive Algorithms
The adaptivity problem in the numerical resolution of time-dependent partial differential equations is related to the following issues: 1. Given a function and a prescribed accuracy £, is it possible to define a suitable space adapted to the approximation of this function to within el 2. Given this adapted space, can we derive a fast numerical algorithm within this space? 3. Is it possible to control the time evolution of the adapted space so that it remains adapted to the exact solution? We show next that one can introduce (e,s)-adapted wavelet spaces such that the class of algorithms derived in the previous sections are well designed for these purposes. 6.1
Adapted Approximation Spaces
From a numerical perspective, approximation spaces of small dimension are clearly desirable. Given two positive constants (£,s), we then propose the following two-step definition for a finite-dimensional adapted space V C Vp. Definition 2. A space is called a finite-dimensional adapted space V C Vp if it has the two properties 1. Choose the smallest p such that
2. For each scale j < p — I ,
where £ controls the global level of approximation while s controls the amount of energy in the different scales away from j. The first property of adapted spaces determines the coarsest scale p such that the total energy of all the wavelets in all finer scales is
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below the threshold specified by (1). The second property of these spaces relates to the wavelet coefficients that are neglected in the scales j < p. In effect, (2) states that the energy in the neglected wavelets at each scale j is less than a specified value which is a function of the total energy of the function / at scales j' close to j. In fact, there is a certain degree of scale mixing in (2) for reasons of stability. At this time, the parameter s is determined from experience, and is typically a function of the evolution operator. For a certain class of operators, these adapted spaces are stable, i.e. if Lu = / and if V is adapted to / then it is also adapted to u on a related space (e1, s')-adapted, independent of the scale p and depending only on the operator. To show the interest in the notion of adapted space, we have built several adapted spaces V, (e,s)-adapted to the analytical solution /(a;) of a Burgers equation (see Section 7, second case). For different values of (s, s) we have then estimated the precision of the approximation in terms of the L2 norm of f ( x ) and the L°° norm of its gradient (see table 1 and Figure 5). Knowing that a non-adapted space Vj approaching / within an accuracy of 10~7 requires N > 256, the results of table 1 clearly show that the (e,s)-adapted space provides a good approximation of the initial function with a reduction of dimension up to 65% while keeping the Z/2 and £°° errors below 5 10-7.
Dim.ffgkffgfgf 256 128 64
ll/-nv(/)H 7 . a
||/'-(iM/))'IUoo
ll/llr.2
7.25 10~7 1.32 1(T6 2.29 1(T4
ll/'lUoo
9.01 10~6 2.80 10~3 2.30 10"1
Table la: Non-adapted spaces of approximation
Dim.sndsdsdss 89 62 49
II/-IM/)II,,2
II/MIM/H'HL,,,
ll/llr,2
H/'lUoo
7.63 10-7 4.39 10~5 4.46 ID" 4
9.28 10~6 2.42 10~5 8.87 10~5
£
io-7 10~5 10~4
Table Ib: Adapted spaces of approximation
mv b(fm Pv(/)ll
9.04 10~5 1.00 10~3 3.16 10~3
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Figure 5 shows that adapted spaces exhibit a cone-like structure around each singularity that reflects the fact that scales focus around the singularities. On such cone-structured adapted spaces, we define a boundary space V& in the following way. Given V and Ay, V& is generated by the wavelets of indices (j, k) 6 Ay6 C A which (in a sense) form the boundary of V towards the finer scales. Figure 5 illustrates the boundary space Vt, by plotting filled circles.
Figure 8: Adapted spaces of approximation D and boundary space in the dyadic plane for various values of (s,s) and related precision, (d is the dimension of the adapted space). We will see in the next section that the distribution of the energy 11Pi/11 denned by
is suitable to control the evolution of V in the sense that the small scales that form outside of V are expected to appear first within Vj. In the above definition for Evb(x), the interval I\ — Ij/, is the interval [&2~ J ,(fc + 1)2~J'], which is the minimum dyadic interval supported
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by the dyadic grid at scale j and centered about 2" J (A; + |). This is valid for a quadratic nonlinear term but must be adapted when the equation is strongly nonlinear, has time-dependent source terms, or has strongly variable coefficients (in space or time). Note that this result has not been proven for general cases. In these cases, it is quite possible that small scales might form outside V, without significant influence on the magnitude of Evb(x). One should note that this problem is present in any adaptive method which remains entirely within its own space, without ever returning to the finest grid. 6.2
Control of the Approximation Spaces
According to Section 5.2, the numerical resolution of a parabolic problem can be considered as a sequence of elliptic problems. In the general form introduced in Section 5.2 these problems are nonlinear. Therefore, there is no hope for a space V initially (e, s)-adapted to the elliptic problems at early times to remain adapted for all time with satisfactory values of £ and s. The space V must adapt in response to the solution as it evolves in time. As a result the notion of adaptivity becomes dynamic and some form of control of the space of approximation V must be defined. Among the various options available, keep in mind that to be practical, the control algorithm must be fast and independent of the discretization scheme (Berger and Oliger 1994). Given the data and a numerical time discretization scheme, we define the notion of (s — s) adaptivity: Definition 3. A family of spaces {]/(")} is (e,s)-adapted to the numerical resolution of the parabolic problem up to time N Ai if for each value n, 0 < n < N — I , V^ is (e,s)-adapted to the corresponding elliptic problem at time nAt. The control of the space V^ is achieved through the "boundary kkkk space", Vj, defined above. As soon as the function Eyb(x] exhibits maxima larger than a chosen constant £& at any cc, the global space V^n> is modified. Moreff precisely, additional wavelets surrounding the boundary space V^ are introduced or some wavelets of V^ are removed according to the location of the maxima in a manner consistent with the definition of Vfc. Figure 6 shows the distribution of energy in the boundary space
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Vj just before and after a refinement procedure in the simulation of the Burgers equation. Note the gap in Figure 9b. This corresponds to the wavelets in the redefined section for Vj, which has not yet built up any energy. This energy will be generated by the nonlinear terms or by the convection of the small scales already within V.
Figure 9: Distribution of energy in Vb. (a) Before refinement, (b) after a refinement around x = 0.25.
7
Numerical Implementation
We now address some particular issues related to the practical implementation of the previously defined algorithms. The last part of this section is devoted to numerical tests. 7.1
Wavelets on an Interval
For numerical applications, wavelets defined on 1R, i.e. on the infinite real line, are generally not suitable and must be replaced by wavelets supported on a finite interval, say [0,1]. The simplest, and also computationally efficient, approach is based on the periodized
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wavelets. These are ideally suited to the decomposition of periodic functions, which permeate the field of numerical analysis. We recall in the next subsection the basic ingredients of their construction. However, other specific boundary conditions are often required. In the special case of homogeneous boundary conditions, the construction of Cohen, Daubechies and Vial (1993) can be suitably modified (Chiavassa 1994) and is introduced below. 7.1.1
Periodized Wavelets on [0,1]
As pointed out by Meyer (1990), the complete set of tools built in L2(IR) can be adapted to periodic functions of £ 2 ([0,1]) by introducing a standard periodization technique. Definition 4. If ^p\x) is the periodized form of £(x), then the periodized wavelet and scaling functions are
and
This kind of folding provides a multiresolution analysis of £ 2 ([0, l]^). Due to the finite size of the interval, j takes only positive values (for J < 0? Vj = VQ) and k takes a finite number of values at each scale, more precisely: 0 < k < 2J: — 1. Moreover, the periodic function <%Q
cc
that generates VQ is constant and equal to unity over the entire interval. The Fourier transforms of the periodic scaling functions and wavelets are
and (f)jhshand ifiX.' generate respectively V^p' and WJP . We wish to emphasize that although the scaling covariance is lost during the periodization process, invariance by translation is preserved. Fortunately, the scaling covariance is in general asymptotically recovered at small scales, and is exactly recovered above a
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fixed value of j for wavelets of compact support. Plots of the different wavelets and scaling functions at different scales for a m — 6 order spline multiresolution are shown in Figures 10 and 11. For notational simplicity, superscript (p) is henceforth removed.
Figure 10: Periodic (m = 6) spline scaling functions at scales 0 to 7.
7.1.2
Other Boundary Condition Wavelets on [0,1]
Wavelets on the finite interval and useful for non-periodic functions have been constructed by Meyer (1992) with a variant proposed by Cohen et al. (1993). These basis functions correctly handle the regularity properties at the edges of the interval and should then be
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Figure 11: Periodic (m = 6) spline wavelets at scales 0 to 7.
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preferred to periodic wavelets for general (non-periodic) problems. This is similar to prefering Chebyshev functions over Fourier functions to approximate non-periodic functions. The r—multiresolution derived by Cohen satisfy orthogonality of wavelets ty at every scale, orthogonality of translated i/) and in the interior, and r vanishing moments of ty. One should emphasize that the scaling covariance is separately maintained at the edges and interior domain, while invariance by translation is only true in the interior. Regarding implementation, the difference with the periodic case stems from the loss of invariance by translation at the edges, which now leads to individual two-scale relations for every edge function. Memory storage is thus slightly increased, and the use of convolutionbased algorithms for fast summation is lost only at the edges for a constant number of coefficients at every scale. Therefore the O(N) operation count can be maintained, the exact operation count being approximately proportional to the support of the wavelet ^ooThe above construction can be adapted to handle functions of s # ([0,1]) satisfying homogeneous boundary conditions at the edges of the interval (i.e. u = 0 or |^ = 0 for x = 0 or 1, when s > 2). The corresponding bases are obviously useful for the numerical resolution of problems with homogeneous boundary conditions. Figures 12-13 show the wavelets and their related scaling functions which form a multiresolution analysis of #Q([O, 1]) t They are based on compact support wavelets of maximum number of vanishing moments on fft, in this case 4 (Daubechies 1992) and have been derived in Chiavassa (1994). There is currently no purely wavelet-based algorithm to handle nonhomogeneous boundary conditions, even in one dimension. In the work of Cai and Wang (1994) which is based on a wavelet collocation approach, finite-difference approximations must be used to express function derivatives in terms of function values, which adds a nonwavelet component to the algorithm.
7.2
Wavelets in Multidimensions
Efficient numerical algorithms which make use of genuine multidimensional wavelets are not yet available on general domains of 1R". However, a useful approach to consider is the tensorial prodV 6 H£[0, 1] <»e Hl[0, 1] and /(O) = /(I) = 0.
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Figure 12: Multiresolution analysis of 5o([0,l]). (a) Scaling functions at the left edge, (b) scaling functions at center, (c) scaling functions at the right edge.
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Figure 13: Multiresolution analysis of -H"o([0, 1]) (b) Wavelet functions at the left edge, (b) wavelet functions at center, (c) scaling functions at right.
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uct wavelets (Meyer 1990). In two dimensions, for instance, these wavelets are produced from a 2-D multiresolution analysis obtained from two 1-D regular multiresolution analyses. This approach provides a scaling function $(x\,x-i) such that at every scale j, the set of scaling functions
generates the 2-D scaling space
whose rescaled and translated versions combine to form the 2-D wavelet space
W = W] ® Wf + W] ® Vf + Vl ® Wf. The corresponding functions associated with a spline mutiresolution of same order TO = 6 in both directions are plotted in Figure 14. One should emphasize that a mutiresolution approach is required to generalize the numerical algorithms presented in the previous sections to multidimensions. Note that this generalization is not possible if one uses the alternate 2-D orthonormal basis provided by the set
That however, does not mean that that basis could not be interesting for numerical analysis.
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Figure 14: Two-dimensional spline multiresolution with ra = 6 in both directions. 7.3
Nonlinear Terms
As for spectral methods (Canuto et al. 1987), the approximation of general nonlinear operators in a wavelet basis is not straightforward if numerical efficiency is of primary concern. Various algorithms which address nonlinear terms of the form u2 with an operation count of KN and KN log N are not yet of practical use because of the large values of the constants K. To the best of our knowledge, the most efficient estimate of the operator 5 is performed using a collocation approximation. Here, S is a general nonlinear operator defined on ^([0,1]) (e.g. S(u) - u2 or S(u) = eu). 7.3.1
Collocation algorithm
When the space of approximation V is equal to Vj, the collocation approach can be defined very easily as soon as the function <j) satisfies a suitable relation leading to the existence of a translation invariant family of Lagrangian interpolants. Indeed, it is natural to link to Vj the sequence of indices k, 0 < k < 2J and in the same way, a sequence of 1J evenly spaced points x^ of [0,1] in physical space. Then, the collocation approach consists in approximating S by CJ^ScCj
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where Cj (when it exists) establishes a one to one correspondence between the elements u of Vj and the sequences of length 2 J ,
and Sc is the approximation of S on the finite-size sequences, Applying Sc on a sequence of length 2J is generally rapid and straightr orward (it cost 2J operations) arid the entire cost for the estimate of Su is driven by the cost of the collocation projection Cj. Tree algorithms (see Section 3) axe available and it is then possible to reduce the cost of the collocation projection from (2 ) 2 operations to O(J2J) or 0(2^) operations, depending on the wavelet basis used ( Cai 199.4). Numerical tests performed in the case of spline wavelets of order 6 gives an operation count of (4/2^). An example serves to illustrate the above. Consider an even order spline multiresolution and
and let
In the case of an even order spline, Xi = i1~ is the collocation grid. Sc is defined by Sc(ai) = a?, and C is the operator from the approximation space V to the sequences of length 2 defined by
and is clearly a discrete convolution operator. The full approximation of 5* is obtained by the sequence of operations
with d'jk the numerical estimate of < u 2 ,i/)jfc >. Effectively, we have computed the wavelet coefficients of a projection of u(x)2 given the wavelet coefficients of u(x). One should note that the fact that a collocation projection exists for any multiresolution analysis is not proven.
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When the space of approximation is not a classical multiresolution analysis space Vj (this is usually the case when the space of approximation is adapted) the collocation approach is less straightforward. However, an adapted family of points can be defined (Ponenti 1994) in the case of cone structure spaces that preserves fast collocation projection algorithms of tree algorithm type. Therefore, it is again possible to compute an approximation of 5 as Cyl ScCy, where Cv is the collocation projection related to the non-regular grid associated with V. This technique generates a set of collocation points equal in number to the dimension of V. Other algorithms have been proposed recently by Beylkin (1993). They are also based on, or closely related to the collocation methodology. 7.4
Numerical Tests on Burgers Equation
We now solve Burger's equation using several techniques: adapted wavelets, non-adapted algorithms and Fourier collocation. 7.4.1
General Approach
Consider the regularized Burgers equation
on the periodic domain x 6 [0,1] for t > 0 and i> — 10"~2/7r. This is one of the simplest equations combining both nonlinear propagation and diffusive effects. Moreover, exact solutions are known thanks to the Hopf-Cole (Basdevant et al. 1986) transformation which is used to check the accuracy of the global method. To solve system (15) we use the multiresolution analysis and related elements (e.g. filters, scaling functions, etc.) associated with cardinal spline spaces of even order (TO = 6) periodized on the interval [0,1]. Following our discussion on the numerical resolution of evolution equations in Section 5.2, we replace the initial problem by the
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sequence of elliptic problems,
where F only depends on the discretization scheme. For clarity, we continue the presentation of the algorithm using an Euler scheme, explicit for the nonlinear term and implicit for the second order term keeping in mind that the numerical results presented herein use an Adams-Bashforth time dicretization scheme for the nonlinear term and a Crank-Nicholson scheme for the second order term with a constant integration time step Ai = 10~3 at every scale. For Euler's scheme,
and
The straightforward application of algorithm I (Section 5.1.1) leads to
where the suffix * denotes adjoint. Therefore,
since L is self-adjoint, and
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First, the nonlinear term is computed using a collocation approximation following Section 7.3. Then, the calculation of the scalar products < un,0jk > and < (ti n ) 2 ,6K > is performed according to the fast algorithm described in Section 5.1.2. We reiterate that we do not use the functions Ojk or 6*-k but the filters defined in the previous section. For the presented results p fa 2, i.e. the scalar products < un,0jk > and < (it™) 2 ,^. > are computed using Pj+2(un) and therefore we use two families of filters {ctj} and {(/?i)j} to compute the scalar products. Filter coefficients are precomputed before the beginning of the computation. Note that when the coefficients of the equations are time-dependent, or if the timestep were variable, the filters would have to be recalculated which is an added expense. Whether this expense is justified depends on the cost of the implicit solver and the number of time steps between successive computations of the derived basis functions. The cost of this computation is approximately that of one application of the tree-algorithm per scale j. 7.4.2
Computational Space
A first computation was carried out with
The initial space of approximation is V&. Every ten time steps, the control of the approximation space is performed as described in Section 6.2. Figure 15 shows the time evolution of the energy in W$ and clearly brings into evidence the localization of energy around sharp gradients. There remains the question of how to choose the wavelets to add as scales are refined or convected. This is done based on the energy content of the boundary space V\>. The number of wavelets we add is a function of the numerical support of the wavelets. In the results presented here, we only allow for the addition of two additional scales past the scales present at the begining of the calculation. When required, we add or subtract 20 wavelets on the appropriate scales. Although we fix the number of wavelets to add, they can be convected along the position at the scale where they live. Thus, kQ(n) is the leftmost wavelet at scale j = 6, while k'Q(ri) is the leftmost wavelet at scale 7. Obviously, this must be generalized on problems in which more than one singularity is expected.
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Figure 15: Time evolution of EW&, the energy distribution in W$. Table 2 and Figure 16 give details of the time-evolving family of adapted spaces {V^}. The data presented in Table 2 corresponds to fci(ra) - fc0(n) = fci(n) - k'Q(n) = 20. The functions fc0(n), 161 < n < 360 and fc£,(n),221 < n < 290 are also plotted on Figure 16. The time evolving approximated solution is plotted on Figure 1. Note that the prescription k\ - k0 = k{ - k'0 = 20 is only valid if there is only a single singularity. There is not yet a general approach which could be used blindly given any distribution of singularities.
Table 2: Adapted spaces on two scales.
7.4.3
Comparison with Spectral Method and Exact Solution
A second computation corresponds to
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Figure 16: Time evolution of the space of approximation V (see Table 2).
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Table 3 compares the approximate solution with a pseudospectral solution using the ABCN time discretization scheme (see also Figure 17). As expected, a Gibbs phenomenon is observed in the pseudospectral solution but not in the solution computed using the present algorithm. Note that this is not linked to wavelets but to the spline approximation. Table 3 gathers some comparisons with theoretical and pseudo-spectral results (Basdevant et al. 1986). Tmax is the time needed to reach the maximum slope at x = 0.5, and 5max is the value of this slope. It shows that a quality of resolution comparable to that of a spectral method with a space of approximation of dimension N = 256 can be reached with adapted spaces of less than 104 wavelets, and consequently with smaller computational times (at this time, the ratio between the wavelet algorithm with 104 points versus a Fourier algorithm with 256 points is around 90%, which is not bad considering we have only adapted on two scales). Algorithm Present y( n ) = V8 Present V^ = V7 Present V(n\ adapted Fourier "Fourier
27rTmax
5Imax/2.
1.64
150.3
1.63
135.0
1.64 1.62 1.64
None 150.3 134.8 ~Spread 150 ' None ~
Oscill. None
deg freed. 256
Localized
128 < 104 128 256
Table 3: Comparison of different calculations with results of Basdevant et a/.(1986), time scheme is Adams-Bashford, Crank-Nicholson. Exact value of 2xTmax - 1.6037, exact value of Smax/2 = 152.005.
7.5
Non-Constant Coefficient Elliptic Equation
The test problem is related to the numerical resolution of the heat equation denned on an interval with periodic boundary conditions and with a non-constant thermal diffusivity coefficient v. Indeed if one starts from the problem
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Figure 17: Burgers equation (1) solution at various times. Comparison with Fourier spectral algorithm, (a) Fourier pseudo-spectral N - 128 (b) First algorithm N < 124.
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A finite-difference approximation to the operator d./dt leads to the set of elliptic problems:
For instance, choosing an implicit Euler scheme one gets
and
At each time step we apply the algorithm described in Section 5.3. Figure 18 reports the convergence of the algorithm for
and
which shows that the algorithms converge quickly (in less than 12 iterations) towards the best approximation of the solution to the discrete implicit system at each time step, i.e. to the orthogonal projection of the solution in Vp. Other computations (Lazaar et al. 1994) have been performed with functions v ( x ) not quite so smooth. Rapid convergence was also exhibited in these cases.
Figure 18: Convergence of the iterative algorithm.
8
Conclusion
To conclude, we would like to summarize what is, from our point of view, the key points related to the use of wavelets for the design of
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efficient numerical algorithms for the resolution of nonlinear partial differential equations. Theoretical results on wavelets and operators are essential since they lead to estimates for convergence and stability of the algorithms. However, the multiresolution set is really essential for numerical applications as it is the keystone part for the design of fast algorithms. The algorithms presented herein have been designed in the framework of fast and adaptive algorithms (i.e. 0(N] or O(NlogN) operations), and they were specifically developed to exploit to the maximum the properties of multiresolution spaces in general, and wavelets in particular. A large variety of algorithms exploiting at a lower level the properties of wavelets have been derived and seen to be quite efficient in intricate configurations (Jameson 1994, Cai 1994). Although much theoretical and implementation-related work has already been achieved, there is much left to accomplish. It seems clear that for smooth problems, demonstrating the advantages of wavelet-based algorithms over existing methodologies will be very difficult. It is only for the problems with many spatial scales, distributed unevenly over the domain, that wavelet algorithms should present their strength. Only time will tell if wavelet-based algorithms will ever replace the best current algorithms in use today.
9
References
Auscher, P., 1992."Wavelets with boundary conditions on the interval." in "Wavelets: A Tutorial in Theory and Applications", Edt: CHUI, Accademic press Inc. Bacry, E., Mallat, S. Papanicolaou, G., 1991. " A wavelet based space-time adaptive numerical algorithm for partial differential equations." Courant Institute of Mathematical sciences, Technical report 591. Basdevant, C., Deville, M., Haldenwang, P., Lacroix, J.M., Ouazzani, J., Peyret, R., 1986. "Spectral and finite difference solution of the Burgers equation." Computers and Fluids, 14(1), 23-41. Berger, M., Oliger, J., 1984. " Adaptive mesh refinement for hyperbolic partial differential equations." Journal of computational Physics 53, 484-512. Beylkin, G., 1991. "Wavelets, Multiresolution Analysis and Fast Numerical algorithms." INIIIA lecture, INRIA, Paris France.
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Beylkin, G. 1993. "On the fast Algorithm for Multiplication of Functions in Wavelet Bases." In Y. Meyer and S. Roques, Ed., Progress in Wavelet Analysis and Applications, 53-62. Cai, W. and Wang, J-Z, 1994. "Adaptive Wavelet Collocation Methods for Initial Boundary Problems of Nonlinear PDE's. To appear in SIAM Numerical Analysis. Canute,C., Hussaini, M.Y., Quarteroni, A., Zang, T., 1987. "Spectral Methods in Fluid Dynamics." Springer Series in Computational Physics. Chiavassa, G., 1994. Ph.D. Thesis, in progress. Cohen, A., 1990. "Ondelettes, analyses multiresolution et traitement numerique du signal." Ph.D. Thesis, Universite Paris IX Dauphine. Cohen, A., Daubechies, I., Vial, P., 1993. "Wavelet on the interval and fast wavelet transforms." preprint AT$T Bell laboratories. Daubechies, I., 1992. "Ten Lectures on Wavelets." CBMS-NSF Series in applied mathematics, SIAM 61. Holschneider, M., Tchamitchian, P., 1990. " Regularite locale de la fonction 'non differentiable' de Riemann." pp 102-124 in "Les Ondelettes", Lecture Notes in Mathematics 1438, Ed. Lemarie, Publ. Springer Verlag. Jaffard, S., Laurencot, P., 1992. " Orthonormal wavelets, Analysis of operators and Applications to Numerical Analysis." In "Wavelets: A Tutorial in Theory and Applications." Edt: CHUI, Accademic press Inc. Laurencot, P., 1990. "Resolution par ondelettes ID de 1'equation de Burgers avec viscosite et de 1'equation de Korteweg-De Vries avec conditions aux limites periodiques", D.E.A report, E.N.S Cachan. Lazaar, S., Liandrat, J., Tchamitchian, Ph., 1994. " Algorithme a base d'ondelettes pour la resolution numerique d'equations aux derivees partielles a coefficients variables." CRAS Serie 1. Lemarie-Rieusset, P.G. 1992. "Sur 1'existence des analyses multiresolution en theorie des ondelettes." Revista Mathematica Iberoamericana 8(3), 457-474. Liandrat, J., Tchamitchian, Ph., 1990. "Resolution of the 1-D regularized Burgers equation using a spatial wavelet approximationAlgorithm and numerical results." ICASE Report N° 90-83. Liandrat, J., Perrier, V., Tchamitchian, Ph., 1991. "Numerical resolution of nonlinear partial differential equations using the wavelet approach." In "Wavelets and their applications", Ed. B. Ruskai, Publ. Jones and Barlett.
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Liandrat,J., Tchamitchian, Ph., 1993. "Elliptic operators, Adaptivity and Wavelets." submitted to SIAM Journal of Numerical Analyzed. Maday, Y., Perrier, V., Ravel, J.C., 1991. "Adaptivite dynamique sur base d'ondelettes pour 1'approximation d'equations aux derivees partielles." C.R.Acad.Sci. Paris Serie I. Meyer, Y., 1990."Ondelettes et Operateurs 1: Ondelettes." Herman. Meyer, Y., 1990b. "Ondelettes et Operateurs II: Operateurs de Calderon-Zygmund." Herman. Meyer, Y, 1992. "Ondelettes sur Pintervalle." Rev. Math. Iberoam, to appear. Perrier, V., 1991. "Ondelettes et simulations numeriques." Ph.D. Thesis, University of Paris 6. Ponenti, Pj., 1994. "Ondelettes et equations aux derivees partielles." Ph.D. thesis, University of Aix Marseille I. Ponenti, Pj. and Liandrat, J. 1994. "Biorthogonal multiresolution and related fast algorithms for the numerical resolution of elliptic partial differential equations. In preparation. Strang, G., 1973."A Fourier Analysis of the Finite-Element Variational Method." in Constructive Aspects of Functional Analysis, Edizioni Cremonese, Rome. Tchamitchian, Ph, 1994. "Inversion de certains operateurs elliptiques a coefficients variables." Prepublication 94-5, Laboratoire APT, CNRS URA 225, MarseiUe.
6 SOME WAVELET ALGORITHMS FOR TURBULENCE ANALYSIS AND MODELING J. Liandrat
Contents 1
Introduction
317
2
Analysis Algorithms of the Wavelet Plane
319
2.1 The Wavelet Transform as a Starting Point 319 2.2 Local Energy Decomposition . . . . . . . . . . . . . . . . 320 2.3 Analysis of the Energy Plane , 321 2.3.1 Integral Quantities 5(a) 323 2.3.2 Integral Quantities Q(&) 324 2.3.3 Further Local Norms and Integral Quantities . 330 2.4 Segmentation Algorithms 331 2.5 A-Averaged Quantities 332 3 Application to Transition and Turbulence 332 3.1 Experiment, Theoretical Results and Available Data . . 333 3.2 Application of the Energy Surface Description Algorithms 337 4
Conclusion
346
5
References
347
Turbulence Analysis and Modeling
1
317
Introduction
It is well recognized that the physics of transition and turbulence in shear flows is associated with nonhomogeneous data fields due to the occurrence of localized phenomena. For example, intermittency related to high frequency components in fully-developed turbulence, turbulent spots in some transition processes, coherent structures of various types in turbulent shear flows are indeed classical concepts in fluid mechanics. Their analysis, understanding and modeling remain, however, one of the current active topics of research. In all these examples, the definition of local phenomenon is never clear, mainly because of the lack of a good analytical or computational tools suited for their study. For similar reasons, the dynamic (i.e. time-dependent) analysis of these phenomena is still an open problem. In our quest for a good analysis tool, we are driven to a problem of approximation theory. The reason is that a global approximation method such as a decomposition into Fourier modes, is ill-suited to the study of phenomena whose behavior is local in space and/or time. Fourier expansions are ideally suited for the study of homogeneous flows, which are defined by statistical properties of the small scales which are independent of position. Alternatives to Fourier methods exist for the analysis of nonhomogeneous flows. For example, the Karhunen-Loeve decomposition, otherwise known as the Proper Orthogonal Decomposition (Berkooz, Holmes and Lumley 1993; Aubry, Chauve and Guyonnet 1994) is widely used in the study of turbulent and transitional flows. However, their use as a general tool for analysis, understanding, and prediction of these phenomena has not yet been clearly proven. The complete problem does not only involve function approximation, but also (and perhaps more importantly) operator approximation. Indeed, a good understanding of turbulence cannot be reached without a good understanding of its dynamics. The practical implication is that the approximation problem must be studied for a family of signals, i.e. signals at different times and at different positions. This collection of signals will allow the functional approximation problem and the evolution operator approximation to be addressed together. Until a good understanding of the dynamics of turbulence is achieved, and until it can be reliably modeled, there is clearly no hope of effectively controlling it, either to enhance it or to
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suppress it, except of course through empiricism. Since its emergence in the 1980's, the wavelet transform has been presented as an alternative to the Fourier approach for the decomposition of turbulence (Beylkin et al. 1991, Farge 1992, Bertelrud et al. 1992). Although several types of decompositions of data obtained from turbulent and/or transition experiments or simulations have been advanced, only a few approaches have generated insightful information on the flow under study. Wavelets provide a new strategy for examining the flow fields, and offer the flexibility which results from the introduction of well-suited basis functions. This allows the data to be manipulated in many ways not previously accessible, within the context of a known mathematical theory. In this chapter we focus on what we call the higher level algorithms. These algorithms are those that manipulate the wavelet coefficients to provide or extract quantitative information about the flow. Their definition and speed of execution are real issues as far as analysis and modeling are concerned. The transform itself and the algorithms related to it is not considered here. In the framework of this paper, they serve as a starting point. We begin our discussion from the so-called "wavelet plane", i.e from a set of wavelet coefficients associated with the analyzed phenomenon. Details of the algorithms used to compute and fill up that plane can be found in the chapter by Tchamitchian in this volume. Finally, for the type of analysis algorithms proposed below, the continuous redundant wavelet decomposition is preferred over other complete decompositions (e.g. dyadic decompositions). Indeed, good accuracy is required for the wavelet plane coefficients, therefore the property of translation invariance, independent of the scale, is necessary. Dyadic approaches for instance are not appropriate. This does not mean, however, that other approaches, such as nonredundant wavelets, or adapted methods cannot be useful for the analysis and modeling of signals related to fluid mechanics. Details comparing different approaches (mainly the continuous wavelet decomposition and the orthogonal wavelet decomposition) in the framework of turbulence modeling can be found in Basdevant et al. (1993).
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319
Analysis Algorithms of the Wavelet Plane The Wavelet Transform as a Starting Point
Before describing the analysis algorithms, a short review of the main properties of the so-called "wavelet plane" is useful. For clarity of exposition, we only consider wavelets on the real axis (IR), but the modifications necessary for spaces of larger dimension and for more complex geometries (e.g. the interval in 1-D) can easily be derived from this general approach. Indeed, we are interested here in constructing algorithms based on the the wavelet coefficients which are already at our disposal. The real difficulty when confronted with boundaries or higher dimensions will be in the actual calculation of wavelet coefficients, a problem which is not addressed here. Given a function s ( x ) in L 2 (IR) and an analyzing wavelet V>(a;), the wavelet plane is the set of coefficients
where b € IR and a € H {0} Assuming that if) satisfies the admissibility condition
the wavelet transform preserves energy with the appropriate weighting function:
From a general standpoint, the theorems that characterize functions from their wavelet coefficients involve their modulus, so it is natural to consider energy-related variables as an analysis tool, since the energy is built from the modulus of the wavelet coefficients. This leads to additional sets of coefficients related to the energy in a suitable space. Following a short discussion on the derivation of the "energy plane" we return to the description of the various analysis algorithms.
320 2.2
J. Liandrat Local Energy Decomposition
In this section, we analyze wavelet coefficients from the point of view of energy using the I? norm. Following (2) it is possible to define for each value of the dilation (a) and translation parameter (6) an energy
in the surface element AaA6. Recalling that Ws(b, a) is linked to the wavelet ipab that numerically lives on the dyadic interval
where C is a constant depending on the wavelet i/j. One should define'(Moret-Bailly, Chauve, Liandrat, Tchamitchian 1992) the local energy at dilation a and point b as
where Xab is a positive "bump" function with a compact support equal to 7aj, and such that
The function x ensures that the energy at a point b is not only dependent on the wavelet coefficients W(6, a) for all a, but in fact on all the coefficients in the cone defined by (3) and the constant C (see Figure 1). One example where this more general definition of energy is clearly necessary comes about when the signal is antisymmetric about x = 0 and the wavelet is symmetric. In this case, W s (0,a) = 0 at all scales which would imply that there is no small-scale structure in the signal at x — 0. The presence of Xab insures non-zero energy at this point, thus insuring a finite amount of energy at each scale. In turn, this reflects a nonconstant structure of the signal at the antisymmetric point. Usually, Xab is the dilated and translated version of the unit bump function x afl d
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nnnnnbkmjkg;fngjnggn ddddddd
The function Xab ensures a definition of the scale decomposition compatible with the Heisenberg uncertainty principle and, moreover provides a stable definition of the energy Es(b,a). This function can be traced back in all the theorems which provide a local characterization of functions using wavelet coefficients. Figure 2b shows contour lines of the energy function J5S(6, a) computed from the signal plotted in Figure 2a and using a spline wavelet of order m = 4. As we shall see later, the definition of the local energy decomposition Es(b, a) is a particular case of more general local energy decompositions related to a large family of extended norms.
2.3
Analysis of the Energy Plane
The analysis of the energy plane, or more precisely the surface -Bs(6, a), can be performed using various description algorithms. We can separate them into 3 classes: 1. Statistical multiscale algorithms. 2. Algorithms based on a proper selection of subsets.
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^3k > Tjk, V/l € Vjkhgjkfjkghfjkghjfjkfgfdjgfjkghfjkgjkfjgfgjkhghhfhd ^3k > Tjk, V/l € V 3. Algorithms based on integral quantities. We shall not review the class of statistical multiscale algorithms (Benveniste, Nikoukhal and Willsky 1993) that refer to estimation, identification or testing theory for multiscale signals. To our knowledge, this class of algorithms has not yet been used in the framework of turbulence or fluid mechanics modeling but they should certainly not be ignored. Another description concept is based on the selection of a family of points either from Es(b, a) or from the (6, a) plane. They are chosen to allow the analysis or reconstruction of Es(b, a) from a reduced amount of information. The class of "ridge and skeleton" algorithms (Tchamitchian and Torresani 1992), "spectral line" algorithms (Guillemain 1994), "local maxima" or "zero crossings" algorithms (Mallat and Zong 1992) belong to this second family. The selected sets can be used to characterize the global surface and therefore the analyzed function. One refers to the articles of Tchamitchian and Arneodo in this volume for a complete description of these methods and related applications. Here we restrict ourselves to a class of description algorithms that involve the definition of integral variables Q(b) or S(a). These
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quantities can be used to provide information about the energy function Es(b,a). This is done by considering the properties along lines parallel to the a = 0 or b — 0 axes. 2.3.1
Integral Quantities5(a)
We now introduce various integral quantities 5(c) useful for the analysis of signals with a transitional or turbulent character from the points of view of energy. Average scale decomposition. is defined as the integral
The average scale decomposition
It is obtained through an integration of the local energy Es(b, a) over the whole line. Since
the average scale decomposition can be directly interpreted as an energy spectrum filtered by a transfer function (a(71/,)~l|'i/>(au;)|2, which is clearly a function of the chosen wavelets. This can be interpreted in another way: if we restrict our attention to lines of constant a0 in the wavelet plane, we have at our disposal a function Ws(6, a<j) which is simply the signal s(x) filtered by the wavelet, considered as apassband filter. The energy of this filtered signal is
Scale intermittency and Scale intermittency factor. A simple thresholding algorithm (Farge 1992) is sometimes useful to quantify the significant contributions of the dilation GO to the signal s ( x ) : Given a threshold t^, one defines Iao,th(x) as Iaojh(x) = I if \Ws(x,aQ) > th\, otherwise Iao,th(x) = 0.
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Iao,th(x) is an integral variable and is called the scale intermittency function related to OQ and t^. It can be used to compute the corresponding scale intermittency factor
which is another integral variable. Here, meas(E) is the measure of the set E. For applications, the choice of the threshold th is obviously very important and should be based on either the mathematical or the physical properties of s(x). 2.3.2
Integral Quantities Q(b)
Local energy decomposition. 5 S) f, 0 , defined by
For every 60 € 1R-? the function
is a function of a. The graph of this function is the intersection of the surface Es(b,a) with the plane 6 = b0. The function SStb0(a) is called the local energy decomposition, and satisfies
Local energy.
The function
is the local energy at point 6 and is the local contribution of the point b to the total energy of the signal since
From the definition of Es(b,a), Ao(b] can be interpreted as the amount of energy provided by all the scales that "contribute" to the point 6, since Es(b, a) takes into account all the wavelet coefficients in the support cone pointing towards b.
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Local scaling. It is well-known that wavelet coefficients can be used to characterize regularity properties of a large class of functions (Meyer 1990, Daubechies 1992). In the context of this chapter we recall results concerning 1. The analysis of C°° functions, 2. The characterization of Lipschitz functions, 3. The analysis of functions with singularities in the complex plane. Analysis of C°° functions Wavelets with a finite number of zero moments, say r, cannot be used to characterize functions more regular than the one belonging to CT. For every function in CT the asymptotic behaviour of the wavelet coefficient is (Arneodo, Tchamitchian, this volume)
Characterization of a Holder functions Given a real 0 < a < 1, and a position XQ, we define the Holder space (Jaffard 1989): Definition 1. The Holder space Ta is the set of functions s(x) such that there exists a constant K < oo such that
The characterization of such a space is controlled by the two following theorems, assuming that some minor hypotheses on -0 are satisfied. Theorem 1.
and conversely, Theorem 2. //
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One can also characterize Holder functions when
but some extra conditions on the wavelet are required, such as regularity and zero moments. These theorems permit the use of asymptotic power laws (i.e Ws(b, a) ~ aa when the scale goes to zero) to characterize the local regularity of functions. It is clear that these mathematical properties can be exploited to model the flow. Therefore, we introduce a new integral variable Sr(b) on the set of points where a power law behaviour is observed as a —> 0:
where Sr(b) is called the local asymptotic scaling which directly measures the "intensity" of the singularity at b. Obviously there is no reason for the function Sr to be denned on a non-empty set and some knowledge on the analyzed function s(x) is required before performing such an analysis. Analysis of singularities in the complex plane Another interesting situation arises when confronted by functions s(x) such that if s(z) is an extension of s to the complex plane C then s ( z ) = (z — z0)a with z, z0 £ C. We will write z0 = XQ + iy0 with z 2 = — 1 and (0:0,3/0) € Bl2. Indeed, one can prove the following theorem (Belleville 1992): Theorem 3. such that
for a » ao and for a « ao with ao/yo — O(l), where r is the number of zero moments of the analyzing wavelet. For this class of functions there are two relevant asymptotic limits, one when a —> 0, and another when a —> oo. The relevant analysis is not only related to the occurence of asymptotic power laws when
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Figure 3: (a) Test signal built from one sine wave and one singularity in the complex plane, (b) local energy decomposition (contours of Es(b,a), (c) cuts of Es(b,a) around 6 = 2000 near (1) in (b), and (d) ten cuts of Es(b, a) in the neighborhood of the singularity at b = 4000, near (2) in (b). a —>• 0, but also to the occurrence of local power laws around specific scales (i.e, WS(XQ,O) — 0(aaJrl/2) for a » eto)- To characterize the first regime (a —> 0), one can then use the previously defined function Sri when a —>• oo (in practice, it is a » ao), we introduce the two new integral quantities Sc and Dc as
and defined for each value b such that a local power law of exponent a is observed for a dilation parameter larger than aoIn Figure 3a, we show a test signal constructed from a sine wave defined on the whole interval with a complex singularity added at the position 60 = 4000. The energy Es(b,a) (Figure 3b) and the plot of the functions SSfb for 1995 < b < 2005 (Figure 3c) and 3995 < b < 4005 (Figure 3d) clearly reveal the occurrence of the singularity. In Figure 4, three isolated singularities of identical exponents (Figure 4a) are analyzed using the same procedure. The
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Figure 4: (a) Test signal built from 3 singularities in the complex plane, (b) position of the singularities in the complex plane, (c) contour lines of Es(b, a). functions 5 S) t(a) corresponding to the 3 singularities are shown in Figure 5. Local scale decomposition parameters. The following parameters are devoted to a description of the local scale decomposition function Sp(b], which after renormalization can be interpreted as a probability density function (pdf). The normalized scale decomposition is the integral quantity
which provides at each position b an energy pdf function. This normalization is necessary to compare the structure of the local energy decomposition S3tb(a) at several points. A first characterization of JVj is given by the local mean scale
Since a scale is the logarithm of a dilation parameter, a(b) is actually the mean scale related to the pdf jVj,(a).
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Figure 5: Local scale decompositions corresponding to the signal of Figure 4. A second integral quantity, the normalized standard deviation of the scale distribution at point b describes the distribution of scales around the mean scale a(b):
An illustration of the local parameters a(b) and a(b] is shown in Figures 6b and 6c. On these figures, the test signal (Figure 6a) is built from three parts: the first is the modulated sine wave a(i) sin^i), the second is white noise, and the third one is a modulated pulsation between two frequencies, i.e. (s'm(u>it) + sin(u>2t))b(t) where a(t) linearly increases in time, while b(t) linearly decreases in time. The local mean scale and standard deviation are plotted in Figures 6b and 6c respectively. From these plots one can for instance exhibit very precisely the three parts of the signal. The white noise can be isolated, thanks to the random variations of a(6) and 17(6) while the two other parts are characterized by constant or periodic evolutions of these parameters. Note that
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Figure 6: (a) Test signal, (b) integral a(i»), (c) cr(b}. frequency from a(6). Furthermore, if one assumes that the signal is the sum of two sine waves, both frequencies can be recovered from a knowledge of a(b) and
Further Local Norms and Integral Quantities
Other parameters of the type Q(b) can be defined by introducing more general energy surfaces, different from Es(b,a) but still related to the initial signal s(x). Indeed, up to now we have considered the most natural norm, the L2 norm, and have derived from it the energy surface S and all the related integral quantities S(a) and Q(b). Wavelets can be used to characterize many other spaces than L2 and therefore a large family of norms can be constructed from the wavelet coefficients. An example of physically interesting norms is provided by the Sobolev norms defined as
Equivalent quantities can be constructed from the wavelet coefficients
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as
These norms provide additional information on the function s(x] as they take into account not only the function itself but also its derivatives (due to the weighting a~ 2p ). One can then define a new energy surface by the two variable functions (1 + a~2p}\Es(b, a)| 2 from which follows the corresponding integral quantities Q(b} or S(a). The corresponding local energy amount Np(b) can for instance be defined as
In comparison to the previously defined L2 energy, Np(b) preferentially reveals the energy located in the small scales. Since the characterization of Lp spaces or the so-called BMO space (Tchamitchian, this volume) is also possible using wavelets, the corresponding procedure leads to the introduction to other energy surfaces and then to the corresponding integral quantities. One should notice that the BMO corresponding local amount of energy is directly related to the quantity N0(b) and corresponds to the specific choice Xab(x), the characteristic function of Ia\,. 2.4
Segmentation Algorithms
The different integrals Q(b) can be used to perform a segmentation of the real axis. Let p(b] represent any member of the family {Q(b)}. A segmentation of the real axis can then be performed using a segmentation of the interval More precisely, writing
where
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is a family of disjoint intervals included in /, one can define a segmentation of the whole line IR, as
with Ei(x) — {x such that p(x) e /;}. Figure 7 shows a segmentation of the test signal in Figure 6a performed with a three part segmentation [min
A-Averaged Quantities
Given a subset A of the real axis, and p(b) any integral quantity, we will denote p^ the A-averaged value
For the specific case where (J(&) is the local scale decomposition function, the corresponding A-averaged quantity is the average scale decomposition function noted EA(O,). Figure 8 shows the functions EA(O) for the spaces A = E\,E2,Es obtained in Section 2.4. The original signal is shown in Figure 6a.
3
Application to Transition and Turbulence
There are numerous applications of wavelets to the analysis of transition and turbulence, and one must refer to Farge (1992) for a review. In this section, we focus our attention on recent results obtained using the family of algorithms presented in the previous sections. They are applied to the analysis and modeling of transition processes in the boundary layer of a rotating disk. We close this section with some comments on the use of these and other wavelet-based algorithms in the framework of turbulence modeling.
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Figure 7: (a) Segmentation based on a(6) for the test signal of Figure 6b-6d: three part segmentation. 3.1
Experiment, Theoretical Results and Available Data
The general framework of this study is the analysis and modeling of the fluid motions that develop in the vicinity of a solid wall. It is indeed generally admitted that the generation of the so-called turbulent motions is triggered inside a thin layer close to the solid boundary and called the boundary-layer. The case of a rotating flat boundary is very interesting as it provides a simple but very complete configuration for the study of the different regimes of transition and turbulence (Jarre 1993). At the IMST (Institut de Mecanique Statistique de la Turbulence, Marseille, France) a rotating disk experiment (Jarre, Legal and Chauve 1991) is set up and is devoted to the analysis and the modeling of transition in a boundary layer (see Figures 9 and 10). Among the available measurements, the time history of the tangential velocity were chosen to apply the algorithms just discussed. The experiment is set up by a flat disk rotating in water and the data is provided by a non-rotating hot film probe placed at 0.6mm from the surface of the disk. The disk radius is 50cm, the angular frequency u> = Irev/s, and the boundary layer thickness is 1.2mm. In the data considered in this work the sampling rate / satisfies
512Hz < f < 8192Hz.
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Figure 8: Average scale decompositions for the test signal defined in Figure 6a and the segmentation of Figure 6. (a)
Turbulence Analysis and Modeling
Figure 9: Experimental setup.
Figure 10: Rotating disk boundary layer seen from above.
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Figure 11: Hot film data.
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The interest of the rotating disk experiment is that the different flow regimes can be observed simultaneously in different regions of the disk. Indeed, for some rotating velocities it is possible to reach a fully turbulent regime at the external boundary of the disk while the boundary layer in the center zone remains laminar. Geometrically the different regions (laminar, oscillating, intermittent and turbulent) can be characterized by the distance r to the center of the disk. Therefore, given that the governing parameters of the experiment are the water viscosity // and the angular velocity w, different regions on the disk are discriminated using the Reynolds number
Linear theory, which estimates the amplification rate of perturbations linearly added to the laminar solution, can predict correctly the first two regions and particularly the critical Reynolds number Rrc corresponding to the first appearance of the oscillating regime. The dominant frequency of the oscillating regime can also be predicted. In the present experiment \/RTc = 250 and the dominant frequency is fc = 32Hz. No theoretical results are available for the intermittency and turbulent regions of the disk. We now present some results derived by applying the above analysis algorithms to the tangential velocity time series. 3.2
Application of the Energy Surface Description Algorithms
Our first goal is to derive an algorithm that discriminates the oscillation from the turbulent regimes and to therefore provide a quantitative estimate of the critical Reynolds number Rc from the experimental data. It is natural to define the critical Reynolds number as the value of Rr which corresponds to the radius rc at which an equal amount of time is spent in the oscillating and turbulent regimes. According to the remarks in the last section, a segmentation of the whole set of points using a description parameter of type Q(b) has been used to achieve this. The particular diagnostic chosen is a(b). This quantity is indeed particularly relevant for the specific discrimination of oscillating and nonoscillating regimes. Indeed, it can be easily shown that cr(b) is a constant (= crc), independent of the frequency and only dependent on the chosen analysis wavelet, when
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the analyzed signal is a sine wave. Therefore, a very simple segmentation of the interval [min<7(&),max
provides a two part separation of the whole line:
All results presented are based on the spline wavelets of order ra = 6,
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Figure 12: Intermittency factor and transition Reynolds number Rc corresponding to // — 0.5.
Figure 13: Average scale decompositions at different Reynolds numbers related to the oscillating regime (solid line), and non-oscillating
regime (dotted line) segmentation. Rl1"1 = (a) 428, (b) 452, (c) 466, (d) 488, (e) 508, (d) Corresponding most energetic scales.
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Figure 14: (a) Signal and intermittency function, (b) mean scale a(6), (c) standard deviation
Figure 15: Three part segmentation for the signal of Figure 14.
341
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Figure 16: Averaged scale decomposition, the laminar part is unchanged. One also notes an increase in the 1 /2 dominant turbulent scale for Rr > 508. This phenomenon is unexpected and cannot yet be explained. To better illustrate the different structures of the signal, we add one segment to isolate the oscillatory regime. The new segment edge is &c + 0.2, so the new segmentation is now
1 /2
which is plotted in Figure 14 for a Reynolds number of Rr = 450. The resulting segmentation of signal is plotted in Figure 15, with the three averaged scale decompositions in Figure 16. By considering a finer segmentation of the signal and of the interval [min tr, max u] at each Reynolds number, a more precise description of the transition procedure can be obtained. Figure 17 shows the average scale decomposition corresponding to a five subset segmentation of the whole line, induced by a segmentation of [min(r(6),max(T(6)] into five intervals of equal length. From Figure 17, it is clear that the average scale decomposition of the five subsets has a shape almost independent of Reynolds number, at least in the smaller scales. We say that the corresponding segmentation defines five local regimes. We hypothesize that this independence of Rr becomes more and more exact as the number of segments increases. Note however that we are only concerned with the independence of the small scales, since Figure 17 shows that the large scales depend on Rr. From these-results, one concludes that the Reynolds number Rr is not the correct parameter to describe the local regimes since they are all present at each Reynolds number. Therefore, the distinction between the various global regimes
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Figure 17: Average scale decomposition for a five subset segmentation of the whole line and for different Reynolds numbers: (a) 0.445 < a < 0.545; (b) 0.545 < a < 0.645; (c) 0.645 < a < 0.745; (d) 0.745 < a < 0.845; and (e) 0.845 < a < 0.945.
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Figure 18: Generalized intermittency functions for the spaces E? and E5. (laminar, oscillatory, intermittent, turbulent) is not achieved through knowledge of the different local regimes, but rather through their distribution at each Reynolds number. In other words, one should consider the Reynolds number as the parameter that controls the measure of the different elementary subsets of the signal, but not as the parameter that controls the definition of the local regimes. This concept can be quantified by defining a generalized intermittency function which measures the fraction of time spent by the signal in each of these local regimes. When there are only two segments, one recovers the standard notion of intermittency and the plot of Figure 12. In Figure 18, we have reproduced the general intermittency functions corresponding to the sets E5 = {b such that
0.845 < cr(b) < 0.945}
E2 = {b
0.545 < a(b] < 0.645}.
and
such that
The set £5 is represented by an asterisk, while E% is plotted with circular symbols. As expected, the intermittency function of E$ increases with Rr as the large values of a correspond to "turbulent"
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Figu dffdd9: Analytic signal built from singularities in the complex plane (see also Figure 4). local regimes. One should note that the local regime associated with the set E-2 is primarily present in the global intermittent regime of the flow and asymptotically disappears at large Reynolds numbers. At this point, we can summarize the following results. It is possible to associate with the flow a family of local regimes. This family is not Reynolds number dependent since all the local regimes are simultaneously present over the entire disk. However, the distribution of these local regimes in time is controlled by the Reynolds number. The "transition operator" can now be seen as a redistribution of a generalized intermittency factors for each member of the family of local regimes. The next step is then to assume that these regimes are intrinsic to the flow and therefore to the equations that govern its dynamics. That would provide an intrinsic model for the "transition operator" for the rotating disk configuration. We are then led to a situation already found in turbulence, as discussed by Frisch and Morf (1981) when they studied, from a theoretical point of view, the intermittency of the high frequencies in fully-developed turbulence. Along the same lines, an analysis of experimental signals can be found in Gagne (1987), and Sulern, Sulem, and Frisch (1983) describe a numerical implementation. Application to model equations is described in Thual and Frisch (1985). Frisch and Morf (1985) observed that the high frequency components of a signal related to fully-developed turbulence were really similar to the test signal of Figure 19 which is built from singularities in the complex plane. They then conjectured that the intermittency of the small scales in fully-developed turbulence was related to the intermittent occurrence of singularities in the complex plane. In a second step they tried to relate those singu-
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Figure 20: (a) Signal, (b) local energy decomposition (contours of Es(b, a), (c) cuts of Es(b, a) near b = 3000 (1) in (b), and (d) cuts of Es(b,o) near b - 4000 (2) in (b). larities to model equations governing the dynamics of fully-developed turbulence. In the next section, we use the wavelet paradigm to check a similar conjecture for the intermittent bursts occuring during transition on a rotating disk (Moret-Bailly et al. 1992). Characterization of complex singularity algorithm, a test. According to the previous sections, we try now to compare the scale decomposition of the most energetic elementary regimes to the scale decomposition of a signal whose extension to the complex plane is singular at isolated locations. Figure 20 shows the local scale decompositions around points characterized by 0.845 < cr(6) < 0.945 in a portion of the signal corresponding to a global Reynolds num1 /2
ber Rr = 528. The behaviour at small scales exhibits a succession of bumps (at a given position). According to the results of Section 2.3.2, these bumps could be modeled using isolated singularities in the complex plane as can be seen from Figure 3. When there are several isolated singularities at a given position 6, the presence of multiple bumps makes it difficult to estimate the power law in a. At this time, it is reasonable to conjecture that the elementary regimes isolated by the segmentation correspond to singularities or sets of
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isolated singularities in the complex domain. More investigation is obviously required, in particular using higher sampling rates for the data acquisition. Technical problems however have to be faced since obviously, uncontrolled filtering in the data acquisition process degrades the signal, thus preventing a reliable analysis to be conducted. At very high sampling rates, retrieving reliable data is very difficult.
4
Conclusion
We have presented a family of analysis algorithms based on the continuous wavelet decomposition of signals. A particular application of these algorithms was also illustrated with data taken from a rotating disk experiment in which the measured signals had both laminar and turbulent characteristics. At least in that particular case, wavelets allowed one to extract quantitative information not possible to obtain by classical methods. Moreover, the analysis performed at different Reynolds numbers allows one to conjecture a model for the transition operator on a rotating disk. There are several other "wavelet-type" algorithms that are used for turbulence analysis and modeling: One of these is the wavelet packet decompositions coupled to best basis research algorithms (Coifman et al. 1990, Farge, Goirand, Wickerhauser 1991). More recently, the so-called matching pursuit algorithms (Mallat and Zhang 1993) have also been tested (Dussouillez 1994). In both these examples, significant information is provided by the algorithms, but the ultimate modeling step has always suffered due to a lack of quantitative results. Most algorithms have been applied to single 1-D or multi-D signals. This alone cannot possibly describe transition or turbulence since they are both evolution phenomena in space and time. It seems evident that research must be conducted to understand the dynamical time evolution of the various diagnostic data extracted from the flow field. This could be done by modeling the evolution operators which act for example on the family Q(b) or any other pertinent representation of the signal. Only then is there a chance of deriving a simplified form of the equations while retaining the important dynamical characteristics of the flow. To reiterate, whatever the approach used to compress or represent the data, one must always return to the dynamical operators
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(in space and time) which govern the evolution of the flow. Since wavelets are also known to be very efficient to represent operators, it seems natural to extend the use of wavelet analysis to understand or better model the evolution operators which govern laminar or turbulent flow. Hopefully, this will lead to an integrated framework in which both the physics and the mathematics can be described efficiently by the same set of basis functions. Note that this has already been achieved for homogeneous turbulence using Fourier basis functions.
5
References
Aubry, A., Chauve, M.P., and Guyonnet, R., 1994. "Transition to Turbulence on Rotating Flat Disk." Phys Fluids, 6 (8). Basdevant,C., Perrier, V., Philipovitch, T., Do Khac, M., 1993. "Local spectral analysis of turbulent flows using wavelet transform." Preprint ENS. Belleville, M., 1992."Analyse de singularites par ondelettes et applications a la turbulence." D.E.A Universite Paul Sabatier/Sup Aero, IMST Marseille. Benveniste, A., Nikoukhal, R., Willsky, A., 1993. "Multiscale system theory." Progress in wavelet analysis and applications, Ed. Y. Meyer, S. Roques. Berkooz, G., Holmes, P., Lumley, J.L, 1993. "The POD Decomposition in the Analysis of Turbulent Flows." Ann. Rev. Fluid Mech. 25, 537-576. Bertelrud, A., Erlebacher, G., Dussouillez, P., Liandrat, M.P., Liandrat, J., Moret-Bailly, F., Tchamitchian, P., 1993. "Development of wavelet analysis tools for turbulence." ICASE Interim report 23 July 1992. Beylkin, G., Coifman, R., Daubechies, I., Mallat, S., Meyer, Y., Ruskai, M.B., 1991. "Wavelets and their Applications." Jones and Barlett. Coifman, R., Meyer, Y., Quake, S., Wickerhauser, V., 1990. "Signal processing and compression with wavelet packets." Numerical Algorithms Research group, Yale University. Daubechies, I., 1992. "Ten Lectures on Wavelets." CBMS-NSF Series in applied mathematics, SIAM 61. Dussouillez, Ph., 1994. "Algorithmes de type matching pursuit orthogonaux." Rapport de D.E.A, Universite d'Aix Marseille III. Farge, M., Goirand, E., Wickerhauser, V., 1991: "Wavelet packets analysis, compression and filtering of two-dimensional turbulent flows." LMD, Ecole Normale Superieure, Paris, Preprint.
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Farge, M., 1992. "Wavelet transforms and their application to turbulence." Ann. Rev. Fluid Mech., 395-458. Frisch, U., Morf, R., 1981. "Intermittency in nonlinear dynamics and singularities at complex times." Phys. Rev. A 23, 2673. Gagne, Y., 1987. "Etude Experimental de 1'Intermittence et des Singularites dans le Plan Complexe en Turbulence Developpee." Memoire de These, universite de Grenoble. Guillemain, Ph., 1994. "Analyse et modelisation de signaux sonores par des representations temps-frequence lineaires." These de Doctorat, Universite d'Aix Marseille II. Jaffard, S., 1989. "Exposants de Holder en des points donnes et coefficients d'ondelettes." CRAS serie I. Jarre, S, Legal, P, Chauve, M.P., 1991. "Experimental analysis of the instability of the boundary layer over a rotating disk." Europhys. Letters,14(7). Jarre, S., 1993. "Etude experiment ale des instabilites sur disque tournant." These de Doctorat, Universite d'Aix Marseille II. Mallat, S., Zong, S.: "Wavelet transform maxima and multiscale edges", in Wavelets and Applications, Ruskai et al. ed., Jones and Barlett (1992) Mallat, S., Zhang, Z., 1993. "Non linear adaptive time-frequency decomposition," Progress in wavelet analysis and applications, Ed. Y. Meyer, S. Roques. Meyer, Y., 1990. "Ondelettes et Operateurs 1: Ondelettes." Herman. Moret-Bailly, F., Chauve, M. P., Liandrat, J., Tchamitchian, Ph: "Determination du nombre de Reynolds de transition dans une etude de couche limite sur un disque tournant." C. R. Acad. Sci. Paris, t. 313, Serie II, p. 591-598. Moret-Bailly, F., Belleville, M., Dussouillez, Ph., Liandrat, J., Tchamitchian, Ph., 1992. "Local scale analysis and modelisation of transition." Wavelets and applications conference, Toulouse. Sulem, C., Sulem, P.L., Frisch, U., 1983. "Tracing Complex Singularities with Spectral Methods." Journal of Computational Physics 50(1). Thual, 0. and Frisch, U., 1985. "Natural boundary of the Kuramoto Model." Tchamitchian, Ph., Torresani, B., 1992. "Ridge and skeleton extraction from the wavelet transform." In Wavelets and Applications, Ruskai et al. ed., Jones and Barlett.
7 WAVELET ANALYSIS OF FRACTALS: FROM THE MATHEMATICAL CONCEPTS TO EXPERIMENTAL REALITY A. Arneodo
1
Introduction
352
2
The Multifractal Formalism 357 2.1 Fractal Sets 357 2.1.1 Hausdorff dimension 357 2.1.2 Box dimension, capacity 358 2.2 Fractal Measures 361 2.2.1 Spectrum of singularities 361 2.2.2 Generalized fractal dimensions 366 2.2.3 The multifractal formalism 368 371 2.3 Fractal Functions 2.3.1 The concept of self-affinity 371 2.3.2 A first step towards a multifractal formalism for fractal functions: the structure function method 376
3
Singularity Detection and Processing with Wavelets 380 3.1 The Continuous Wavelet Transform 381 3.1.1 Definitions 381 3.1.2 Inversion formula and reproducing kernel . . . 382 3.1.3 Some examples of analyzing wavelets 384 3.2 Singularity Analysis with the Continuous Wavelet Transform 386 3.2.1 Local regularity and Holder exponents of a distribution 386 3.2.2 Wavelet analysis of local Holder regularity . . . 388 3.2.3 Detection and identification of singularities with the wavelet transform. Wavelet transform modulus maxima and maxima lines 390
350 4
Mult if racial Formalism for Distributions Based on Wavelets
4.1
4.2 4.3
5
5.2 5.3
6.2
7
417
Generalized Devil Staircases 417 5.1.1 Deterministic and stochastic signals 417 5.1.2 The structure function approach versus the wavelet transform modulus maxima method . . 424 Not Everywhere Singular Fractal Functions 432 Brownian Signals 436
Wavelet Analysis of Fully-Developed Turbulence Data
6.1
394
Wavelet Transform of Multifracial Measures . . . . . . 394 4.1.1 Wavelet analysis of the local scaling properties of singular measures 394 4.1.2 Wavelet transform and renormalization . . . . 397 The Multifractal Formalism for Singular Measures Revisited with Wavelets 409 Generalization of the Multifractal Formalism to Fractal Distributions: the Wavelet Transform Modulus Maxima Method 413
Numerical Applications of the Wavelet Transform Modulus Maxima Method
5.1
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Multifractal Approach of the Intermittency Problem . . 441 6.1.1 Multifractal description of the energy dissipation 442 6.1.2 Multifractal description of the velocity field using the structure functions 443 6.1.3 The Kolmogorov's refined similarity hypothesis . 444 Wavelet Analysis of Single-Point Turbulence Data . . 445 6.2.1 Local scaling exponents of a turbulent velocity signal 447 6.2.2 Singularity spectrum of a turbulent velocity signal using the wavelet transform modulus maxima method 451
Wavelet Analysis of Fractal Growth Phenomena
452
7.1
452
Laplacian Growth Phenomena
Wavelet Analysis of Fractals 7.2
7.3
7.4
Wavelet Analysis of Fractal Measures over IR™ . . . 7.2.1 Definitions 7.2.2 Wavelets and local scaling properties of fractal aggregates 7.2.3 Determination of the singularity spectrum of fractal aggregates from the wavelet analysis . Wavelet Analysis of Snowflake Fractals 7.3.1 Deterministic and random snowflake fractals 7.3.2 Multifractal snowflakes Wavelet Analysis of Diffusion-Limited Aggregates . . 7.4.1 Self-similarity of DLA clusters 7.4.2 Structural five-fold symmetry of DLA clusters 7.4.3 Fibonacci hierarchical ordering in the DLA azimuthal Cantor sets
7.4.4
Golden mean arithmetic in the fractal branching of DLA clusters
351 . 454 454 458 . 461 462 . 462 465 . 466 466 . 470 471
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Prospects: Solution to the Inverse Fractal Problem from Wavelet Analysis 480 8.1 A Wavelet-Based Tree Matching Algorithm to Solve the Inverse Fractal Problem 482 8.2 Wavelet Transform and Renormalization of the Transition to Chaos 486 8.3 Uncovering a Multiplicative Process in OneDimensional Cuts of Diffusion-Limited Aggregates . . 488
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References
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Introduction
Fractal and multifractal concepts (Mandelbrot 1977, 1982; Halsey et al. 1986; Paladin & Vulpiani 1987) are now widely used to characterize multiscale phenomena that occur in a variety of physical situations (Stanley & Ostrowski 1986, 1988; Pietronero & Tosatti 1986; Guttinger & Dangelmayr 1987; Feder 1988; Aharony & Feder 1989; Vicsek 1989; Family & Vicsek 1991). In its present form, the multifractal approach is basically adapted to describe statistically the scaling properties of singular measures (Benzi et al. 1984; Vul et al. 1984; Halsey et al 1986; Badii 1987; Collet et al 1987; Feigenbaum 1987; Jensen et al 1987; Paladin & Vulpiani 1987; Mandelbrot 1989a; Rand 1989). Notable examples of such measures include the invariant probability distribution on a strange attractor (Halsey et al. 1986; Collet et al. 1987; Rand 1989), the distribution of voltage drops across a random resistor network (Stanley & Ostrowski 1986, 1988; Feder 1988; Bunde & Havlin 1991), the distribution of growth probabilities on the boundary of diffusion-limited aggregate (Feder 1988; Meakin 1988; Vicsek 1989) and the spatial distribution of the dissipation field of fully developed turbulence (Mandelbrot 1974; Paladin & Vulpiani 1987; Frisch fe Orszag 1990; Meneveau & Sreenivasan 1991). The multifractal formalism involves decomposing fractal measures into interwoven sets which are characterized by their singularity strength a and their Hausdorff dimension /(a) (Halsey et al. 1986). The so-called f ( a ) singularity spectrum has been shown to be intimately related to the generalized fractal dimension Dq (Grassberger 1983; Hentschel & Procaccia 1983; Grassberger fe Procaccia 1984; Grassberger et al. 1988). Actually, there exists a deep analogy that links the multifractal formalism with statistical thermodynamics (Sinai 1972; Bowen 1975; Ruelle 1978). The variables q and T(q) = (q — l)Dq play the same role as the inverse of temperature and the free energy in thermodynamics, while the Legendre transform /(a) = min?[ga — r(g)] indicates that instead of the energy and the entropy, we have a and /(a) as the thermodynamic variables conjugated to q and r(q) (Badii 1987; Collet et al. 1987; Feigenbaum 1987; Bohr & Tel 1988). Most of the rigorous mathematical results concerning the multifractal formalism have been obtained in the context of dynamical system theory. It has recently been developed into a powerful technique accessible also to experimentalists. Successful applications have been reported in various fields and the
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pertinence of the multifractal approach seems, nowadays, to be well accepted in the scientific community at large. However, in physics as well as in other applied sciences, fractals appear not only as singular measures, but also as singular functions. The examples range from plots of various kinds of random walks, e.g., Brownian signals (Mandelbrot & Van Ness 1968; Peitgen & Saupe 1987), to financial time series (Mandelbrot 1967; Mandelbrot & Taylor 1967; Li 1991), to geologic shapes (Mandelbrot 1977; Goodchild 1980), to rough interfaces developing in far from equilibrium growth processes (Family & Vicsek 1991), to turbulent velocity signals (Anselmet et al. 1984; Gagne 1987; Gagne et al. 1988) and to DNA "walk" coding of nucleotide sequences (Peng et al. 1992). There have been several attempts to extend the concept of multifractality to singular functions (Frisch fe Parisi 1985; Barabasi & Vicsek 1991). In the context of fully-developed turbulence, the multiscaling properties of the recorded turbulent velocity signals have been investigated by calculating the moments Sp(l) — < <5uf >~ fa of the probability density function of longitudinal velocity increments fivi(x) = v(x + l) — v(x) over inertial separation (Anselmet et al. 1984; Gagne et al. 1988). By Legendre transforming the scaling exponents (p of the structure functions Sp of order p, one expects to get the Hausdorff dimension D(h) = mmp(ph - (p + 1) of the subset of 1R for which the velocity increments behave as 6vi ~ lh (Frisch & Parisi 1985). In a more general context, D(h) will be defined as the spectrum of Holder exponents for the singular signal under study and thus will have a status similar to the f ( a ) singularity spectrum for singular measures (Muzy 1993). But there are some fundamental limitations to the structure function approach which intrinsically fails to fully characterize the D(h) singularity spectrum (Arneodo et al. 1991; Muzy et al. 1993a). Actually, only the singularities of Holder exponent 0 < h < 1 are potentially amenable to this method (singularities in the derivatives of the signal are not identified). Moreover it has fundamental drawbacks which may introduce drastic bias in the estimate of the D(h) singularity spectrum (e.g. divergences in Sp(l) for p < 0). Even though the structure function method was an interesting preliminary step towards a multifractal theory of singular functions, this theory was still lacking and there was a need for an appropriate powerful technical tool to deal with fractal functions. Our purpose here, is to elaborate on a novel strategy that we have recently proposed and which is likely to provide a practical
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way to determine the entire D(h) singularity spectrum from any experimental signal (Arneodo et al. 1991; Muzy et al. 1991, 1993a, 1993b; Bacry et al. 1993). This approach is mainly based on the use of a mathematical tool introduced in signal analysis in the early eighties: the wavelet transform (Combes et al. 1989; Lemarie 1990; Meyer 1990, 1992; Daubechies 1992; Ruskai et al. 1992; Meyer & Roques 1993). The wavelet transform (WT) has been recently advocated as a very efficient technique to collect microscopic information about the scaling properties of multifractal measures (Arneodo et al. 1988, 1989a, 1989b, 1990, 1992a; Holschneider, 1988a, 1988b; Argoul et al. 1989a, 1990). Extensive applications to various multifractal measures' including the invariant measures of some wellknown discrete dynamical systems have clearly demonstrated the fascinating ability of this mathematical microscope to reveal the underlying hierarchy that governs the spatial distribution of the singularities. What makes the wavelet transform such an attractive tool in the present study is that its singularity scanning ability applies to any distribution including singular measures and singular functions (Holschneider 1988a, 1988b; Jaffard 1989, 1992; Holschneider & Tchamitchian 1990; Arneodo et al. 1991; Muzy et al. 1991; Mallat & Hwang 1992). The simplest and classical way of performing a multifractal analysis of a singular measure is to partition its support using boxes of size e (Halsey et al. 1986; Grassberger et al. 1988). Then the measure in each e-box can be characterized by a singularity strength a according to its scaling behavior //;(e) ~ e"*, where the index i denotes the box location. The number Na(e) of occurrences of a particular a defines the /(a) singularity spectrum: Na(f) ~ e~^a\ Moreover, the generalized fractal dimensions Dq = r ( q ) / ( q — 1) can be extracted from the power law behavior of the partition function Z(q,e) = E;M;(e) ~ cT(
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355
structure function method in the case of functions). Then, from the Legendre transform of the scaling exponents r(q) of these wavelet based partition functions, one can extract the whole D(h) spectrum of Holder exponents. At a given scale, instead of using a continuous integral over space (like in the structure function method), one sums discretely over the local maxima of the modulus of the wavelet transform so that, on the one hand the divergences of the negative order moments are removed and on the other hand the multiplicative structure (if there is any) of the singularity arrangement is directly incorporated into the calculation of the partition functions (Arneodo et al. 1991; Muzy et al. 1991). According to Mallat & Hwang (1992), each connected line of local maxima is likely to emanate from a singularity of the considered signal. Along these maxima lines, the wavelet transform behaves, at small scales, as a power-law with an exponent h ( x ) which is equal to the Holder exponent of the signal at the point x. The number of such lines corresponding to the same /t, defines the D(h) singularity spectrum: N(h) ~ a~DW in the limit a —* 0 + . For a large class of fractal distributions, it can be rigorously established that the so-obtained D(h) spectrum corresponds to the Haussdorf dimension of the set of singularities of Holder exponent h. The content of this tutorial paper is a detailed description of what is called the wavelet transform modulus maxima (WTMM) method and which is likely to be a good candidate to achieve a unified thermodynamical description of singular distributions including measures and functions. Moreover, beyond this statistical characterization of the scaling properties of fractal objects, one can hope to take further advantage of the wavelet transform microscope to address the fundamental issue of solving the inverse fractal problem. Actually there is a need to get deeper insight into the complexity of such subjects and eventually to extract some "microscopic" information about their underlying hierarchical structure. In many cases, the self-similarity properties of fractal objects can be expressed in terms of a dynamical system which leaves the object invariant. The inverse fractal problem consists in recovering this dynamical system (or its main characteristics) from the data. In this context, the wavelet transform modulus maxima can potentially be used to extract a one-dimensional map which accounts for its construction process (Arneodo et al. 1993a, 1993b). In a concluding section devoted to prospects, we will elaborate on a wavelet-based tree matching algorithm which provides a very attractive alternative method-
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ology to the approaches developed in the theory of iterated function systems (IFS) (Barnsley & Demko 1985; Barnsley 1988; Handy & Mantica 1990). The paper is organized as follows. Section 2 contains some background material on fractal sets, fractal measures and fractal functions. The foundations of the multifractal formalism, originally introduced for fractal measures, are described and some basic ingredients of the thermodynamics of fractals are discussed. In Section 3, we review some basic definitions concerning the one-dimensional continuous wavelet transform.. We present the continuous wavelet transform as a mathematical microscope which is well-suited to scanning the singularities of fractal functions as well as singular measures. In Section 4, we revisit the multifractal formalism using the wavelet decomposition. We describe the WTMM method within the mathematical framework of the continuous wavelet transform and its modulus maxima representation. We emphasize the WTMM method as a natural generalization of the multifractal formalism to fractal distributions. Section 5 is devoted to numerical and experimental applications of the WTMM method. We illustrate our theoretical considerations on pedagogical examples including recursively generated fractal signals and fractional Brownian motions. We report on a systematic comparison between the structure function approach and the WTMM method which enlightens the fundamental drawbacks and insufficiencies of the former approach. In Section 6, we report on further applications of the WTMM method to fully developed turbulence data. They provide unambiguous quantitative evidence for the multifractal nature of a turbulent velocity signal at inertial range scales. The so-obtained singularity spectrum is compared to previous estimates based on the structure function approach. The validity of the Kolmogorov refined similarity hypothesis is addressed by generalizing the WTMM method to dissipative variables. In Section 7, we extend the wavelet analysis to fractals living in higher dimensional spaces. We define the wavelet transform associated with the n-dimensional Euclidean group with dilations. We illustrate the efficiency of the 2D wavelet microscope in revealing the construction rule of deterministic and random snowflake fractals. Then we use this microscope to explore the intricate geometry of fractal aggregates. In particular we show that the statistical self-similarity of Witten and Sander's diffusion-limited aggregates (DLA) is related to the existence of a predominant structural five-fold symmetry. This study also reveals
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the existence of Fibonacci sequences in the DLA morphology with a branching ratio which converges to the golden mean. We comment on a tentative interpretation of the geometrical complexity of DLA clusters as a "quasi-fractal" architecture, intermediate between the well-ordered hierarchy of snowflakes and the disordered structure of random aggregates. We conclude in Section 8 with some perspectives for future research. A wavelet-based tree matching algorithm is proposed as a very promising tool for solving the inverse fractal problem. Preliminary applications of this algorithm to concrete physical situations are discussed.
2
The Mlilt ifractal Formalism
2.1
2.1.1
Fractal Sets
Hausdorff dimension
The usual notion of dimension of a set corresponds to the number of degrees of freedom, i.e., the number of parameters one has to use to indicate the position of a point in this set. The so-called topological dimension df takes only positive integer values. In the 19th century, the mathematician Peano (1973) built a curve which uniformly "covers" the plane i.e., he gives a one to one map between a curve of topological dimension 1 and a set of topological dimension 2. Thus, the notion of topological dimension is not well-adapted to this particular case. In 1919, Hausdorff suggests another definition of dimension based on the generalization of the notion of length. The following definition has first been stated by Besicovitch (1935) (see also Falconer (1985); Kahane (1985)). Let S be a set in a metric space E (for the sake of simplicity, we will always consider E = IK1}. We then define the Hausdorff measure of S, indexed by the parameter S € #?, in the following way:
where
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and where the lower bound is taken over all the coverings A'(e) of the set S made of balls {Bi} of diameter smaller than c. The Hausdorff dimension of 5, d#(S), is then defined as the unique value of 6 such that l$(S) is finite:
Let us note that dpj can take noninteger values. The Hausdorff measure associated with the dimension dfj is l d H . It is a generalization of the Lebesgue measure (length, surface, volume,...) in the sense that for any positive integer n, ln is the Lebesgue measure in Mn. Thus, in order to evaluate the relative "size" of two given sets, one just needs to compare their Hausdorff dimensions and, if they are equal, the values of their Hausdorff measures. Although the Hausdorff dimension is very well-defined mathematically, it is generally hard to estimate numerically. To circumvent this difficulty, a more practical definition of the dimension of a set is generally used. 2.1.2
Box dimension, capacity
The capacity (also called the box dimension) has been introduced by Kolmogorov (1958). Let 5 be a subset of JR™ and K(f) a covering of 5 with balls of size e. Let N(e) be the number of balls in K(e). The capacity of 5, dc(S), is then denned as the limit:
Thus the box dimension quantifies how the "size" of a set varies when one changes the unit measure. Like the Hausdorff dimension, it classifies sets according to their relative "dimensions". However, let us note that it does not give any way of "measuring" (as we did with the Hausdorff measure) two sets with the same dimension. In order to understand the differences between the 3 dimensions we have just defined (the topological dimension dx, the Hausdorff dimension dj-j and the capacity d^\ we present below a list of sets for which we indicate their respective dimensions. Example.
The Hausdorff dimension and the capacity dimension of the empty
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set are equal: dfj = AC =• —oo. In this particular case the topological dimension is not defined. Example. For common sets such as a point, a segment, a surface, the Hausdorff dimension and the capacity are equal to the topological dimension which is respectively 0, 1 and 2. Actually, one can prove that these 3 dimensions are equal for any differentiable manifold. Example. Let us consider the triadic Cantor set. This set is constructed in the following way: the segment [0,1] is divided into 3 parts of equal length and the middle part is taken out. Then, we repeat the same process on the two remaining subsegments and so on recursively. The limit set so obtained is called the triadic Cantor set (Falconer 1985). An illustration of its construction process is shown in Figure 1. Let us note that it is a closed set whose interior is empty and which is made up of nonisolated points. At the nth step of the construction process, the set is made of 2n intervals of equal length 3~ n : its total length is (2/3) n . Thus, the Lebesgue measure of the triadic Cantor set obtained when n —*• +00 is 0; therefore its topological dimension is dj = 0. Let us consider the covering corresponding to the nth step of the construction process. As we already pointed out, it is made of N(c) = 2n intervals of size e = 3"™. When € goes to 0 (i.e., n —+ +00), one thus deduces that the capacity of the triadic Cantor set is dc = In 2/ In 3. The Hausdorff dimension can be obtained in a similar way (Falconer 1985) and is found to be the same as dc- The triadic Cantor set is then characterized by the following dimensions:
Let us note that in the former examples the different dimensions verified: Actually, one can prove (Farmer et al. 1983) that:
A classical example for which dfj < dc is the set of all the rational numbers in [0,1]. As it is dense in [0,1], the capacity is dc = 1. However the Hausdorff dimension is equal to the topological dimension which is 0.
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Figure 1: The construction process of the triadic Cantor set.
A definition of a fractal set. Originally Mandelbrot (1977, 1982) suggested that a fractal set should be defined as a set whose Hausdorff dimension is strictly greater than its topological dimension. Even though this definition is adequate for a lot of sets, there exists a whole class of sets (of the same type as the triadic Cantor set) whose Lebesgue measure is finite and thus whose Hausdorff dimension is dfj = 1. These sets, which are generally referred to as fat fractals (Farmer 1984), are not fractal sets according to the original definition. Thus, along with Mandelbrot (Mandelbrot 1988, 1989b; Evertsz & Mandelbrot 1992), we will rather say that a fractal set is a set which has some self-similar properties in the sense that its singular structure is the same at any scale. In other words, the set is invariant (either in a deterministic or in a statistical sense) by the iterative action of some elementary similitudes. This definition is much wider than the one based on the Hausdorff dimension and, as we will see later on, can be applied to a larger class of objects such as measures and functions.
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Fractal Measures
2.2.1
Spectrum of singularities
The concept of fractal measure first appeared in the works by Mandelbrot (1974) on the spatial distribution of dissipative regions in turbulent flows. Since then, this new concept has proven very fruitful for modeling singular objects arising in a variety of physical situations. It has notably been developed in the framework of dynamical systems by Grassberger (1983), Hentschel & Procaccia (1983) and Grassberger & Procaccia (1984). The term "multifractal" and the concept of singularity spectrum have first been introduced by Frisch & Parisi (1985) in the context of fully developed turbulence, and formalized by Halsey et al. (1986) who made the link with the original works mentioned above. A measure assigns weights to different parts of a set. It could represent, for instance, a distribution of charge or of mass, the distribution of energy in a turbulent flow or of course, any probability distribution such as equilibrium, measures in statistical physics or invariant measures in dynamical systems. In many cases, a measure // on 1R can be described by its density p(x) = lim£_).0+ //([a;, x + e])/e. However, a point charge in electrostatic theory represented as a Dirac distribution 8(x) does not correspond to any density function. A measure which cannot be expressed in terms of either a density function or a sum of Dirac distributions is called a singular measure. An example of such a measure is the Hausdorff measure /[ n 2/i n 3 on the triadic Cantor set. In the same way the Hausdorff (or capacity) dimension is used to describe sets whose Lebesgue measure (in IK] is 0 and which are made of nonisolated points, the /(a) singularity spectrum gives a characterization of singular measures which do not have any density component (Farmer et al. 1983; Halsey et al. 1986; Grassberger et al. 1988; Mandelbrot 1988, 1989b). For the sake of simplicity, we will work exclusively in IR and we will use the words "interval", "ball" or "box" interchangeably. Definition 1. Let n be a measure on IR and sup (//) its support. • We call singularity exponent at the point x0 e sup (p,), the
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where BXo(e) is the ball centered at x0 and of size e. Let us note that when this limit does not exist, one can use either a limsup or a liminf in the above equation. • The /(a) singularity spectrum of a measure fj, associates to any given a, the Hausdorff dimension of the set of all the points XQ such that a(#o) —a:
The support of the /(a) singularity spectrum is the set of all a values for which f(ot) ^ —oo The exponent a(xo) represents the singularity "strength" of the measure fi at XQ. Indeed, if JJL is singular, fj,(BXQ(€)) does not correspond to any density function, therefore it cannot be written as /9(xo)e; according to the definition (8) of a(xo), it can be expressed as
The smaller the exponent a(a;o), the more singular the measure p, around x0 and the "stronger" the singularity. The limit a = 0 corresponds to a behavior similar to a Dirac distribution at XQ. Let us note that C in (10) can be a function of e which varies slower than any power of e. The singularity spectrum describes the statistical distribution of the singularity exponents a(x). If we cover the support of the measure /u with balls of size e, the number of such balls that scale like €a for a given a is (Halsey et al. 1986; Mandelbrot 1988, 1989b): Thus, /(a) describes how the "histogram" Na(e) varies when e goes to 0. Let us illustrate this concept on two examples. Example. Let /.i be a uniformly distributed measure on the triadic Cantor set. One can cover this set by 2ra disjointed intervals of size e = 3~™ (the same ones as the ones obtained at the step n of the construction process in Figure 1). Each of these intervals fix,(f) has a mass
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n(BXi(e}) = 2~", thus we get lnn(BXt(e))/lne = In 2/In3. In the limit e —> 0 + , each point of the cantor set will correspond to the same singularity exponent a(z) = In 2/In 3. Moreover, the HausdorfF dimension of the Cantor set is In 2/In 3. The support of the singularity spectrum of the measure /tt thus reduces to a single point /(a = ln2/ln3) = ln2/ln3. Example. Let us consider a nonsingular measure fi(A) = fAp(x)dx.dddfGenerally, for all x in the support of n, we have a(x) = 1. Since the Lebesgue measure of the support of // is not zero, its Hausdorff dimension is 1. Thus, the singularity spectrum of this nonsingular measure is /(a) = —oo for a / 1 and /(I) = 1. The support of /(a) is a single point. Remark. Let us point out that, as a direct consequence of the example just above, the /(a) singularity spectrum fails to account for singularities of exponent a > 1 as soon as some density component is present in the measure. . The first example corresponds to a singularity spectrum whose support is a single point («0; f(&o))' only one "sort" of singularity is found in the measure. Such measures are called homogeneous measures (Mandelbrot 1977, 1982, 1988; Halsey et al. 1986; Tel 1988). Let us note that for this type of measure, the relation /(cto) = «o is always true. Indeed, if we cover the support of// with N(e) disjointed balls of size e : {JBj(e)}t=i..jv(e)) we nave
Moreover, as n is homogeneous of exponent cio, we get
and thus /(«o) = «oHowever, there exist nonhomogeneous measures, i.e., measures whose singularity spectrum is supported by more than a single point. One can build such a measure on the triadic Cantor set in the following way: at the n = 0 step of the construction process, we associate the weight //Q = 1 to the interval [0,1]. At step n = 1, /.io is distributed on the two subintervals: /*i = Pifio for the interval [0,1/3]
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and // 2 = P2l^o = (1 - Pi)/-*o for the interval [2/3,1]. We then repeat recursively this process: ^ (resp. // 2 ) is distributed in two parts: p\[JL\ (resp. pi^i) and p2Mi (resp. pijJLj.} and so on. The same weights p\ and p^ are used at each step. Clearly the so obtained measure is not homogeneous, i.e., it involves singularities of different strengths a. Indeed, let us consider at each step n the first interval on the left side Bi(f = 3~ n ) = [0,1/3"]. This interval always contains XQ = 0 and is such that (JL(BI) = P'^HQ = p™. Thus, the singularity exponent of the measure fj, at the point x0 = 0 is a(0) = lnpi/ln(l/3). In a similar way, by .considering the last interval on the right side B^^ — 3~ n ), one can prove that the singularity exponent at the point XQ = 1 is a(l) = lnp2/ln(l/3). Clearly, when Pi 7^ P2 then a(0) ^ a(l). Moreover, one can prove that there exist many points corresponding to the singularity exponent a(0) and a(l). Thus the corresponding Hausdorff dimensions /(a(0)) and /(a(l)) are both different from —oo and therefore the support of the singularity spectrum is not reduced to a single point. Actually one can prove that the support of /(«) is a whole interval [ot.min,a.max\ where amin = min{a(0),a(l)} and amax = max{a(0),a(l)}. Moreover f(ctmin} = f(<Xmax) = 0 an(i f(a) ig a single humped curve whose maximum is /(QJM) = da = In 2/In 3. Figure 2a illustrates the construction process of this measure. The singularity spectrum for p\ = 0.6 and p% = 0.4 is displayed in Figure 2b. This spectrum can be computed analytically (Halsey et al. 1986): each point of the triadic Cantor set can be addressed by an infinite sequence of symbols L and R according to the successive choice of the left (L) or the right (R) subinterval at each construction step. The f ( a ) singularity spectrum is then obtained by counting the number of symbolic sequences such that the ratio of the numbers of symbols L and R converges to a given value; these sequences correspond to the same singularity a. But generally, one cannot compute analytically the singularity spectrum of any given measure. One must then define a numerical algorithm for computing singularity spectra. The most natural way would be to use directly (8) and (9): one first scans the support of ;u measuring a(x) at each point x by estimating the slope of the curve log ju(#3;(e)) as a function of log e; then one computes the fractal dimension /(a) by using the box counting method corresponding to (4). However, such a method would lead to dramatic errors since, for any c, JJL(BX(()) takes into account many points with very different singu-
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Figure 2: Binomial measure distributed on the triadic Cantor set with the weights pi = 0.60 and p2 = 0.4. (a) Construction rule, (b) /(a) singularity spectrum. The support, of the singularity spectrum is a whole interval; therefore the measure /j, is not homogeneous.
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larity exponents; the estimate of a(x) on a finite range scales is often extremely unstable. One can use a slightly different method called the histogram method (Arneodo et al. 1987; Badii & Broggi 1988; Grassberger et al. 1988). It consists in covering the support of the measure /i with balls {Bi(c)}i of size e. For each ball B,(e), we define the exponent a;(e) = ln//(.B;(e))/lne. This exponent is like a singularity exponent "seen" at the the scale t. Then if Nf(a~) is the histogram of the values {a,-(e)},-, /(a) can be computed using the following relation:
Even though this method is stable under certain conditions, the convergence when ( goes to 0+ is very slow (Grasseau 1989). In most cases, the range of scales available in the numerical data is too small and the histogram method leads to very approximate results because of scale dependent prefactors. Basically, this is due to the fact that this method is based on the computation of scaling exponents which represent "local" quantities that can vary a lot from one point to another. In the next paragraph, we present the inultifractal formalism which uses the generalized fractal dimensions of a measure as intermediate "global" quantities from which one can compute the /(a) singularity spectrum. 2.2.2
Generalized fractal dimensions
Let fj, be a measure on IR, K(t) a covering of the support of /j, with intervals of size e and JV(c) the number of intervals in Ii'(e), i.e., Ji'(e) = {£,-(e)}!=i..jv(e)- Let ^(c) = / B . (e) rf/x. For all q e JR, we consider the partition function:
We then define the r(q) spectrum from the power law scaling behavior of Z(q, e) when e —>• 0 + :
The spectrum of generalized fractal dimensions l)q is obtained from the spectrum r(q) (Grassberger 1983; Hentschel & Frocaccia 1983;
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Grassberger fe Procaccia 1984):
Let us note that the notion of generalized fractal dimensions was originally introduced in dynamical system theory to characterize an ergodic measure associated with a given dynamical system. The capacity dimension (see (4)) of the support of n corresponds to D0, whereas D\ characterizes the scaling behavior of the information I ( e ) = ^j/i,-(c)ln//j(e): it is called the information dimension (Renyi 1970). Moreover, for q > 2, the Dq's can be related to the g-point correlation integrals (Grassberger 1983; Grassberger & Procaccia 1983, 1984; Hentschel & Procaccia 1983). The spectrum r(q) is a "global" quantity which describes the behavior of the "mean" value Z(q,e). Thus, it is likely that a r(q)based method for computing /(a) will be much more stable than any a-based method such as the histogram method. Let us see how one can relate the f ( a ) singularity spectrum to the r(q) spectrum. At the scale c, if we consider that the distribution of the a's is of the form p(a)e~^a^ and if we use this expression in (15), it follows (Frisch & Parisi 1985; Halsey et al. 1986):
In the limit e -*• 0 + , this sum is dominated by the term €™™<*(
Thus the r(q) spectrum is obtained by Legendre transforming the /(a) singularity spectrum. By inverting this transformation one.gets:
One can thus compute the /(a) singularity spectrum from the scaling exponents r(q) of the partition functions Z(q,c). When r(q) is a continuously differentiable function, the equation (19) can be rewritten in the following way:
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In the same way, (20) becomes:
Let us study these last equations for some particular values of q (Figure 3). « For q = 0, by using both the definitions of r(q) (see (16)) and of Dq (see (17)), one gets from (21):
Thus the fractal dimension of the support of the measure corresponds to the maximum of the /(a) curve. By definition of /, one can deduce that the value a(q = 0) corresponding to f ( a ( q = 0)) — maxa /(a) = DQ is the most "frequent" singularity. » For q = 1, (19) becomes r(l) = mm a (a - /(a)) = <x(q — 1) — f ( a ( q = !))• Moreover, we know from the definition of r (see (16)) that r(l) = 0 (£Mt( c ) = 1). We thus get /(a) = a when q = 1. The singularity spectrum /(a) stands below the diagonal /(«) = a and reaches it for a = a(q = 1). Let us note that if / is differentiable at a ~ a(q = 1), then df/da = 1 (see (21)) and thus /(a) is tangent to the diagonal at a(l). • For q — ±00, one can deduce from (17) and (19) that:
Thus, in these limits, the Dg spectrum directly provides the strongest and the weakest singularity exponents that respectively characterize the densest and the most rarefied regions of the support of fj,. Moreover, (21) shows that the tangent of /(a) at amin and amax are vertical. If /(a) is C 2 , (19) implies that cf 2 //cPa < 0 for all values of a, thus / is convex.
2.2.3
The multifractal formalism
The multifractal formalism is the framework in which we consider the /(a) singularity spectrum as the Legendre transform of the r(q)
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Figure 3: Generic shape of the /(a) singularity spectrum considered as the Legendre transform of r(q) (Eq. (20)).
spectrum. It is represented by (15), (16), (17), and (20). In this context, it has been proved that /(a) has a generic single humped shape as shown in Figure 3. Let us illustrate this formalism on two specific examples. Example. Let n be the homogeneous measure lying on the triadic Cantor set. We saw in the last section that the singularity spectrum of this measure is /(a) = a = In 2/ In 3. The spectrum r(q) can be easily obtained by considering the covering K(e) with intervals of size € = 3~ n , corresponding to step n of the construction process of the measure H. From (15) and (16), we get, in the limit n -> +00:
which yields r(q) = (q - I)ln2/ln3. The spectrum r(q) is a linear function whose slope is the singularity exponent characterizing the homogeneous measure fj,. Let us note that we could have obtained this result directly by Legendre transforming /(a). The spectrum of the generalized fractal dimensions is independent of q:
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The fact that the generalized fractal dimensions are all equal to the box (or Hausdorff) dimension (i.e., the r(g) spectrum is linear) characterizes the homogeneity of the measure. Both T(q) and Dq spectra are displayed in Figures 4a and 4b respectively. Example.
Let us now consider a nonhomogeneous measure lying on the triadic Cantor set such as the binomial measure described in Section 2.2.1 (Figure 2). It is characterized by two parameters p\ and p2 which represent the distribution factors of the weight at each step of the construction process. By considering again the same covering K(e) made of intervals of size e = 3""™, one gets
The behavior of Z(q,e) when e goes to 0+ (i.e., n —»• +00) leads to the following expression for the exponents r(q):
The r(q] and Dq spectra are displayed in Figures 4a and 4b respectively. Let us note that r(q) is no longer linear and that Dq is a decreasing function from -D-oo = otmax = — Inp 2 /ln3 to D+ca = <*min — — I n pi/ In 3. The Legendre transform relation (20) between r ( q ) and /(a) can be checked analytically for this particular example (Halsey et al. 1986; Mandelbrot 1988; Tel 1988). A definition of a fractal measure. Actually, there exists a deep analogy that links the multifractal formalism with that of thermodynamics (Badii 1987; Collet et al. 1987; Feigenbaum 1987; Bohr & Tel 1988; Rand 1989). The variables q and r(q) play the same role as the inverse of temperature and the free energy in thermodynamics, while the Legendre transform (20) indicates that instead of energy and entropy, we have a and /(a) as the thermodynamic variables conjugate to q and r(q). This thermodynamic formalism has been worked out in the context of dynamical system theory (Sinai 1972; Bowen 1975; Ruelle 1978). But rigorous proofs of the above connection have been limited to some restricted classes of singular measures, e.g., the invariant measures of some expanding Markov maps ("cookie-cutter" Cantor sets) on an interval or a circle and the invariant measures associated with the dynamical systems for period
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doubling and with critical circle mappings with golden rotation number (Collet et al. 1987; Rand 1989). Nevertheless, successful applications of this thermodynamic formalism have been reported for singular measures which appear beyond the scope of dynamical systems (Paladin & Vulpiani 1987; Feder 1988; Vicsek 1989). Therefore, the generalized fractal dimensions Dq and the /(a) singularity spectrum are thermodynamic functions, i.e., statistical averages that provide macroscopic (global) information about the scaling properties of singular measures that are self-similar. In the same spirit as for fractal sets, we will thus define a fractal measure as a measure which is invariant (either in a deterministic or in a statistical sense) by the iterative action of some elementary similitudes. We will call a multifracial measure a fractal measure which is not homogeneous. Let us note that this analogy with thermodynamics provides a very attractive understanding of the phase transition phenomena sometimes observed in the scaling properties of fractal measures (Badii 1987; Cvitanovic 1987; Grassberger et al. 1988).
2.3
2.3.1
Fractal Functions
The concept of self-affinity
Generally, the self-similarity properties of fractal sets and fractal measures induce some very intricate singular behavior. The spatial distribution of these singularities has been studied via the multifractal formalism. One could think of using the same kind of methods in order to study very irregular functions such as rough surfaces, stock exchange data, ID component of the velocity in a turbulent flow, signals obtained from spectroscopy, I// noise... (Mandelbrot 1977, 1982; Pietronero & Tosatti 1986; Stanley & Ostrowski 1986, 1988; Feder 1988; Aharony & Feder 1989; Vicsek 1989; Family & Vicsek 1991), i.e., functions that do not have any characteristic scale and which are highly singular. These functions can be qualified as fractal functions in the sense that their graphs are fractal sets in fft2 (we will only consider functions from ffi to JK). The self-affinity property characterizes the (fractal) sets which are invariant under affine transformations (Mandelbrot 1982; Peitgen &
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Figure 4: T(qJ and Dq spectra for a homogeneous (- -) and a binomial nonhomogeneous ( ) measure lying on the triadic Cantor set. (a) r(q) spectrum, (b) Generalized fractal dimensions Dq.
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Figure 5: Self affinity of the graph of a Brownian realization, (a) Graph of a realization of the Brownian motion Bi/^(t}. (b) Isotropic zoom of a part of the graph of Bi/2(t). (c) Zoom with a factor A in the t direction and with A""1/2 in the direction of B^/^i], Saupe 1987; Voss,1987, 1989; Dubuc et al 1989; Edgar 1990). In 1R2, it means that the considered set is similar to itself when transformed by anisotropic dilations. A particular case of self-affinity is self-similarity which implies isotropic transformations. We will say that a function is self-affine when its graph is a self-affine set. We have displayed in Figure 5a the graph of a Brownian motion Bi/^t) which is a typical self-affine function (in a statistical sense though) (Levy 1965; Mandelbrot 1982; Voss 1987, 1989). Figure 5b repre-
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sents an isotropic zoom of a small part of Figure 5a; it is clearly not "similar" to the initial figure. On the other hand, Figure 5c represents an anisotropic zoom by a factor of A in the abscissa direction and of A" 1 ' 2 in the ordinate direction; the similarity with Figure 5a appears clearly. If f ( x ) is a, self-affine function then, Vx 0 G IR, 3H £ 1R such that for any A > 0, one has
If / is a stochastic process, the identity holds in law for fixed A and XQ. The exponent H is called the Hurst exponent (Mandelbrot 1977, 1982; Feder 1988). The graph of the function is self-similar only if the Hurst exponent is H = I (the graph is then invariant under some isotropic dilations). Let us note that if H < 1, then / is not differentiable and that the smaller the exponent H, the more singular /. The Hurst exponent indicates how irregular the function / is. Let us present some "classical" examples of self-affine functions. Example. Let us consider the triadic Cantor set (introduced in Section 2.1) and ft the homogeneous measure on this set. We define / = [0,1] -+ [0,1]
as the distribution function of ft, i.e.,
This function is displayed in Figure 6a. It is almost everywhere constant on [0, 1] and looks like a staircase whose "steps" are uncountable and infinitely small. The so denned function / is called the devil staircase (Mandelbrot 1977, 1982). By definition, fj, is invariant under the map T(x) defined by T [o,i/3](^) = TI(X) = 3x and T \ ^ / 3 r i ] ( x ) — TI(X] = 3x-2, i.e., for all intervals A, fjL(T~1(A)) = A*(A). Moreover fi(A) = 2/i(Tf 1 (A)) = 2 / u(T 2 ~ 1 (A)). Thus if A = [0, a;], we get f ( x ) = fjt(A) = 2/z(Tf 1 (A)) = 2/(x/3) and then f(\x) = A ln2 / ln3 /(a:). The Hurst exponent of / is the fractal dimension of the triadic Cantor set // = l n 2 / l n 3 . Example. One can generalize the above construction to a norihomogeiieous
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measure lying on a "generalized" Cantor set. For instance, at the first step of construction, we divide the interval [0,1] into four subintervals of equal length but of different weights p\, P2-, P3 and p4. We then repeat the same operation recursively ad infinitum: at each step we distribute the former weights on 4 subintervals of the same size and using the same distribution weights p± (i=l, 2, 3 and 4). We can choose any real value for each pi (positive or negative); the only constraint for convergence is that ^2^pi = 1. The soobtained object is a "signed" measure fj, whose distribution function /(x) = //([0,x]) is a self-affine function. This function is commonly referred to as generalized devil staircase. We illustrate its construction for pi = p2 = P4 = 1/2 and p$ = —1/2. The corresponding everywhere continuous but nowhere differentiable function is displayed in Figure 6b. One can prove easily that for all x, /(z/4) = /(x)/2. The Hurst exponent is H = In 2/ In 4 = 1/2. When the \pk 's are not equal, the Hurst exponent is much harder to get analytically. One can show that it is not unique; it actually depends on the point x0 (29). We will study this case in Section 2.3.2. Let us note that there exists a lot of different methods for constructing self-affine functions with deterministic rules. However, they are all based on the same process: the iteration of a multiplicative rule for distributing the weights at smaller and smaller scales (Peitgen & Saupe 1987; Voss 1987, 1989; Dubuc et al. 1989; Edgar 1990; Barabasi & Vicsek 1991). Example. Originally introduced by Mandelbrot & Van Ness (1968), the fractional Brownian motions generalize the classical Brownian motion. They have been extensively used to model various physical phenomena (Mandelbrot 1977, 1982; Feder 1988; Stanley & Ostrowski 1988; Vicsek 1989; Family & Vicsek 1991). A fractional Brownian motion Bff(t), indexed by H G]0,1[, is a Gaussian process of mean value 0 and whose correlation function is
where < > represents the mean value. The variance of such a process is The classical Brownian motion corresponds to H = 1/2 and to a variance var(Bi/'2(t)} = cr'2\t\. One can easily show that the increments
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of a fractional Brownian motion, i.e., 8Bf{,i(t) = Bfj(t + /) — Bf{(t) (I € IR+* fixed), are stationary. Indeed, the correlation function < 6BH,i(t),6BH,i(8» >=£(\t-s + l\2H + \t-s-l2H-'2\t- s 2H) depends only on t — s. For // = 1/2, we recover the fact that the increments of the classical Brownian motion are independent. For any other value of H, the increments are correlated. From (31), one gets where ~ stands for the equality in law (for fixed t and A). This means that fractional Brownian motions are self-affine processes and that the Hurst exponent is H. The higher the exponent H, the more regular the motion. A definition of a fractal function. Along the same lines as for sets and measures, we will define a fractal function as a self-similar function which is invariant (either in a deterministic or in a statistical sense) by the iterative action of some elementary similitudes. Let us anticipate that this definition extends quite naturally to fractal distributions.
2.3.2
A first step towards a multifractal formalism for fractal functions: the structure function method
In order to study fractal functions, one could just study their graphs as fractal sets in ZR 2 . Basically, it would lead to computing the fractal dimension of a given graph in ffi2. It would be, however, a very poor characterization as compared to the multifractal formalism for fractal measures which instead provides a continuous spectrum of fractal dimensions to account for the statistical contributions of singularities of various strengths. Could not we apply this formalism to study self-affine functions? For that purpose, we have to consider a self-afnne function no longer as a geometric graph in M2 but as a function with local singular behavior. Let us change slightly the definition of the Hurst exponent of / so that it becomes a local quantity: This "local Hurst exponent" h(x) is generally called the Holder exponent of / at the point x. In the next section, we will give a rigorous definition of this exponent; for now we will keep calling it the local
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Figure 6: Generalized devil staircases, (a) Devil staircase associated to the triadic Cantor set. (b) Distribution function of the signed measure recursively distributed on [0,1] with the weights p\ — p2 = p4 = -P3 = 1/2.
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Hurst exponent. This exponent characterizes the regularity of / at x\ the closer this exponent to zero, the less regular the function. The way of adapting the multifractal formalism from fractal measures to fractal functions is clear when one compares (34) with (10). We just need to substitute the singularity exponent a(x) by the exponent h(x). The balls of size e that we used for covering the support of a measure are replaced by "increments" of the function over a distance /. In the same way we have defined the /(a) singularity spectrum of a measure ft, we will then define the D(h) spectrum of local Hurst exponents of the function f ( x } by the function which associates to any h, the Hausdorff dimension of the set of points x which verify h(x) = /i, i.e. :
In the simple case where / is a devil staircase, i.e., f ( x ) = H([Q, a;]), let us note that the notions of local Hurst exponent h and of D(h] spectrum correspond respectively to the singularity exponent a(x) and the /(a) spectrum associated with the measure /,«. The increments of / over a distance I can be computed as the measure in balls of size /. These new concepts have been initially introduced by Frisch fe Parisi (1985) for studying fully developed turbulent flows. In order to estimate the D(h) singularity spectrum they have suggested considering the structure functions Sp(l) (Monin & Yaglom 1971) to replace the partition functions Z(q,e] (see (15)). Their power law scaling behavior (when the scale / varies) defines the exponents (p (to be compared to the exponents r(q)):
In the same way as we did for measures, a steepest descent argument shows that (p and D(h) are related (as r(q) and /(a)) by a Legendre transform (Frisch & Parisi 1985):
This last equation completes the analogy with the multifractal formalism for fractal measures. It is clear that this new description gives a much richer characterization of self-affine functions than the one given by the fractal dimension of the graph (Frisch & Parisi 1985;
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Barabasi & Vicsek 1991). Indeed, it tells whether the function is an homogeneous fractal function, i.e., h(xg) = H, V£Q, or a multifractal function characterized by a nontrivial D(h) spectrum. Example. Let us consider the recursive fractal function which is the distribution function of a "generalized" Cantor measure, with the weights pi = P2 = —pz = p4 = 1/2 (same as in Figure 6b). If we choose I = 4" n , then the structure functions are 5 p (4~ ri ) = 4~ 1 2~ np . Thus the (p spectrum is a linear function of p: (p = p i n 2 / I n 4 = p/2 = pH. The function is an homogeneous fractal function. The slope of £p is the Hurst exponent which characterizes the function. We find h(x) = H = 1/2 and D(H) = 1. Example. One can generalize the latter construction to any value for the weights n 1 p Pi. One can prove that 5 p (4- ) = 4- (|p1 P + \p2\P + \p3 + NT and thus (p = Iog4(|pi p + P2\p + Ps\p + \P4\P)- This function is generally nonlinear; the support of the D(h) spectrum is the interval [hmin,hmax] whose limits are fomin = minj(- Iog4p;) and hmax = max,-(—Iog 4 pj). The function is multifractal. Drawback of the structure function method. The structure function method is a first step to characterize the singular structure of selfaffine functions through a statistical description like the one used in the multifractal formalism for singular measures. However, this method has some main drawbacks which can basically be summarized as follows: • (36) involves a sum over the whole space. Generally, fractal functions are likely to have, at any scale, increments as close to zero as we want. It then appears clearly that Sp(l) will diverge for p < 0. This means that the exponents (p are only denned for positive values of p. Let us note also that Co is generically always equal to 0 and thus does not carry any information on the dimension of the set of all the singular points of /. • If we analyze a signal which has some very strong singular behavior (e.g., Dirac distributions), the structure function method is extremely unstable. Indeed, the computation of the increments is very unstable. On the other hand, if the function is very smooth (i.e., more than C 1 ), then the (p spectrum is
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A. Arneodo trivially (p -~ p. In other words, it does not give any characterization of the singular behavior that could appear in the successive derivatives of /.
These drawbacks show that the structure function method does not provide a reliable generalization of the multifractal formalism to fractal functions and more generally to fractal distributions. In the next section, we try to analyze these problems, and the associated remedies, in more depth. We will define a new method that will be a general unified multifractal formalism for fractal distributions (including measures and functions). This method is based on a very powerful tool for analyzing singular behavior and was introduced in the early eighties: the wavelet transform (Combes et al. 1989; Meyer 1990, 1992; Lemarie 1990; Daubechies 1992; Ruskai et al. 1992; Meyer & Roques 1993).
3
Singularity Detection and Processing with Wavelets
The wavelet transform has been introduced about ten years ago by the geophysicist Morlet (1983). While studying seismic signals for petroleum research, Morlet realized that both the regular and the short term Fourier transforms are not well-adapted to analyzing signals with a wide range of scales. To circumvent these difficulties, he built a new transformation which provides a space-scale representation of a signal. This transformation basically consists in decomposing the signal in terms of some elementary functions ^>)0 obtained from a "mother" function V by dilations and translations: V'fe,a( a; ) = ffl~1'2V'((a; ~~ fr)/a)- In 1984, Grossmann, Morlet and their collaborators formalized this new transformation and showed that it can be inverted if the "mother" function ^ is a wavelet, i.e., a zero-mean function with some oscillations (Goupillaud et al. 1984; Grossman fe Morlet 1984, 1985; Grossmann et al. 1985, 1986). Later on, Daubechies et al. (1986) found a way to discretize the wavelet transform on "frames". Frame theory is the foundation of all subsequent work on discrete wavelets. Lemarie & Meyer (1986), Meyer (1987) and Jaffard & Meyer (1989a, 1989b) construct some orthogonal wavelet bases. The multiresolution analysis discovered by Mall at (1989a, 1989b) and Meyer (1990) describes any orthogonal wavelet
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basis in terms of a common multiscale structure. It allows us to build a fast algorithm for computing the wavelet transform. This fast algorithm combined with the compactly supported wavelet bases built by Daubechies (1988,1992) makes the wavelet transform a very powerful tool for numerical applications. The wavelet theory has been applied recently to various domains such as functional analysis, theoretical physics, signal and image processing, partial differential equations and fractal theory (Combes et al. 1989; Lemarie 1990; Meyer 1990, 1992; Daubechies 1992; Ruskai et al. 1992; Meyer & Roques 1993). 3.1
3.1.1
The Continuous Wavelet
Transform
Definitions
In this section, we introduce the 1-D continuous wavelet transform and some of the basic mathematical results. We consider a function s(x~) in the Hilbert space L2(IR.dx). We are going to decompose this function s in terms of elementary functions obtained by dilations and translations of the real-valued mother function ^(x). Let us define ^6 >a (x) = a~ 1 / 2 ^((x — b)ja). We then define the wavelet transform (WT) of s(x) by (Grossrnann & Morlet 1984, 1985):
where < .). >£2(,H,ete) '1S the scalar product in L2(IR,dx). The parameter 6 is a real valued space (or time) parameter whereas a is a scale parameter (a 6 IR+*). Thus the wavelet transform is basically the scalar product of the function with the mother function dilated by a and translated by b (Figure 7). In the Fourier domain, (38) can be rewritten in the following way:
where s(u) = / e~lljixs(x)dx represents the Fourier transform of s ( x ) . Thus at a fixed scale a, the wavelet transform corresponds to filtering the function s with the band-pass filter ^(au;).
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figure 7: The wavelet coefficients correspond to the scalar product of the given signal (- -) with the wavelets ^6,0(2) obtained by dilating and translating the mother function ^(x) ( ).
3.1.2
Inversion formula and reproducing kernel
In order to be able to reconstruct the original function s ( x ) from its wavelet transform T^,[s](6, a), we would like T^ to be an isometry, i.e., to verify for any function s a relation of the type
where C^ is a constant and d/j,(b, a) a measure in the space-scale half-plane. The wavelet transform is invariant under dilations and translations in the sense that (Goupillaud et al. 1984; Grossmann & Morlet 1984, 1985; Grossmann et al. 1985):
Thus, according to (40),rf/i(6,a) must be invariant under the transformation (6, a) —» (6 — xo,a) and (6, a) —> ( 6 / A , a / A ) . It is then
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natural to choose d[i(b,a) = dbda/a2. By using this expression of dp one can prove, using the Plancherel formula, that the relation (40) holds if 0 satisfies
This relation is generally referred to as the admissibility condition. Let us note that it implies that V>(0) — 0 and therefore, if tj) G Ll(IR),
A mother function if) which satisfies (42) will be called an analyzing wavelet. For such a mother function, the wavelet transform is an isometry (modulo the factor C^) and we have the following reconstruction formula (Grossmann & Morlet 1984, 1985; Grossmann et at. 1985, 1986; Meyer 1990; Daubechies 1992):
Let us note that relation (42) is not a necessary condition and that one could choose a slightly different admissibility condition (Grossmann fe Morlet 1985; Daubechies 1992). Actually, one can obtain a much more general reconstruction formula. Indeed, it is possible to reconstruct s by using a mother function if}r different from the function ij> used to compute the wavelet transform (Holschneider & Tchamitchian 1990). The reconstruction formula then becomes
where ipr and tjj are two mother functions satisfying the new admissibility condition
Let us note that if ^>r — if}, then we recover (42) and (44). The new admissibility condition (46) is very "weak" and allows a very wide range of the reconstructing wavelets ^ r . One interesting choice is the
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Dirac distribution 4'T(X) — 8 ( x ) . It leads to a very simple inversion formula (Holschneider 1990)
By taking the scalar product of the inversion formula (44) with the wavelet Vv,a' one can easily get the interpolation formula (Grossmann & Morlet 1984, 1985; Grossmann et al. 1985; Daubechies et al, 1986)
where K^,(b,a) is the scalar product < ^t, ia ]i/> >. The function K^(b,a) is called the reproducing kernel associated with the analyzing wavelet ij). Equation (48) means that the wavelet coefficients are correlated and thus that a given function S(b, a) is the wavelet transform of a function s iff it satisfies some correlation properties. For a given analyzing wavelet, the correlation "length" depends linearly on the scale a. That means that the value of the function s at a point a:0 will "influence" the values of the wavelet transform T^,[.s](6, a) for (6, a) in a domain of the form
where cr'j, is the variance of the analyzing wavelet ^). This domain is a cone which "points" to XQ on the space axis. This cone of influence is illustrated in Figure 8. 3.1.3
Some examples of analyzing wavelets
There are almost as many analyzing wavelets as applications of the wavelet transform. The first one that Morlet (1983) used when he first developed the wavelet transform is a complex valued function referred to as the Morlet wavelet. It is a Gaussian function which is modulated so that its mean value is close to zero (see (43)). A very important class of wavelets are the ones that can be associated with orlhonormal bases of L 2 (/R). It is the case, for instance, of the Meyer wavelet (Meyer 1987; Meyer 1990), the spline wavelets built by Lemarie & Meyer (1986) and the compactly supported wavelets of Daubechies (1988, 1992). When one uses these wavelets, the wavelet
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Figure 8: The cone of influence associated to the point x0. -ip is the analyzing wavelet of variance a^.
transform is performed only on a dyadic grid of the space-scale halfplane corresponding to a wavelet basis of L'2(1R). One generally refers to this method as the orthogonal (discrete) wavelet transform. However, for the study of fractals, we will need to compute the wavelet transform on a continuous (6, a) grid, i.e., we will perform a continuous wavelet transform. Since there is no advantage to using orthogonal wavelets, we will use the wavelets corresponding to the successive derivatives of the Gaussian function. Let us define ^'dsffd
Its Fourier transform is
These functions are C°° and are well-localized both in space and frequency. The wavelet ip^ is a very "classical" wavelet and it is
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generally referred to as the "Mexican hat" (Figure 9). Let us note that the first TV moments of t/i^) are vanishing:
As we will see in the next section, this property is fundamental for analyzing singular behavior. Figure 9 displays t/}(N)(x)sjdfhdsddfdfdsfdfdf for N = 1, 2 and 4.
3.2
3.2.1
Singularity Analysis with the Continuous Wavelet Transform
Local regularity and Holder exponents of a distribution
In Section 2.3.2, we have seen how the "strength" of a singularity can be described by an exponent called local Hurst exponent or Holder exponent (see (34)). Let us give a more rigorous definition of a Holder exponent. Definition 2. The Holder exponent h(x0) of a distribution f at the point XQ is the greatest h so that f is Lipschitz h at XQ, i.e., there exists a constant C and a polynomial Pn(x) of order n so that for all x in a neighborhood of XQ we have
If h(xo) 6 ]n, n + 1[ one can easily prove that / is n times but not n + 1 times differentiable at the point XQ. The polynomial Pn(x) in (53) corresponds to the Taylor series of / around x — x0 up to the order n. Thus //-(.TO) measures how irregular the distribution / is at the point XQ. The higher the exponent /I(XQ), the more regular the distribution /. Let us note that if / has a Holder exponent h(xo] — h at the point XQ, then the primitive of / corresponds to a Holder exponent h(xQ~) > h + 1. Actually, in most cases, h(x0) = h + l for the primitive of /. This is not true however if the singularity of / at x0 is an oscillating singularity (Mallat & Hwang 1992). Indeed, for instance, the Holder exponent of f ( x ] = sin(l/,-c) at XQ = 0 is h(0) = 0,
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Figure 9: Some analyzing wavelets of the class of the successive derivatives of the Gaussian function: (a) ^M(x), (b) ^(1)(o;), (c) V'(2)(.'0, (d) V' ( 2 ) (w), (e) V ( 4 ) («), (0 V; (4) H-
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whereas its primitive f\x) verifies \F(x) — -FXO)| = O(x2) in the neighborhood of 0, thus its Holder exponent at .TO = 0 is greater than 2 ! Henceforth, we will only consider distributions whose singularities are not oscillating. Basically, when one differentiates (integrates) such a distribution, the Holder exponent is decreased (increased) by 1. Thus, for instance, the Heaviside function H ( x ) = X[o,+co[(x) corresponds to /i(0) = 0 and its derivative, the Dirac distribution 6(x), corresponds to /i(0) = — 1. The function -\fx has a Holder exponent /t(0) = 1/2 and its derivative, the distribution l/2i/x, a Holder exponent ft(0) = -1/2. Remark. In order to characterize the regularity of a distribution /, one generally studies the behavior of its Fourier transform / at infinity. However, this method characterizes the global regularity of / and does not give any information on how regular / is around a given point XQ. We will see that, the wavelet transform is able to do so. 3.2.2
Wavelet analysis of local Holder regularity
Let us suppose that the Holder exponent of a distribution f ( x ) around the point x — XQ is h(xo) € ]n,n + l[ and that the behavior of f ( x ) around x = XQ is given by: f ( x ) = c0 + ci(a; - x0) + ... + cn(x - x0)n + C x - xQ\h(-x°\ (54) The wavelet transform of / is the scalar product of / with the wavelets V ; 6,a(^) = a~"1//2-^((?:> — x ) / a ) (see (38)). Thus, if we suppose that the analyzing wavelet ^(x) has n^ > n vanishing moments (see (52)), then ip is orthogonal to polynomials up to order n (included). One can derive from (54) the expression of the wavelet transform of / at ,TQ, when the scale a goes to 0:
For the sake of simplicity, we slightly change the definition of the wavelet transform by multiplying its expression (see 38)) by the factor a^" 1 / 2 :
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In this definition, we no longer restrict / to i 2 (jR, dx\ it can be any distribution. The relation (55) then becomes: Thus, the local singular behavior C\x — a;o| •kk of f around x = XQ is characterized by a power law scaling exponent h(xo) of the wavelet transform of / at the point XQ when the scale a goes to 0. On the other hand if / were C°° at XQ, one could prove that we would get a power law scaling exponent n^, i.e.,
Thus, around a given point XQ. the faster the wavelet decreases when the scale a goes to 0, the more regular / is around that point. This result is summed up in a single theorem which was proved independently by Jaffard (1989) and by Holschneider fe Tchamitchian (1990). Theorem 1. Let i/) an analyzing wavelet ments and let / a bounded function.
with n^ vanishing mo-
a
) V f is Lipschitz 7 < n$ at XQ, then the wavelet transform of f satisfies:
b) Reciprocally, let 7 < n^ and let us suppose that
then f is Lipschitz 7 at XQ. This theorem is the "rigorous" version of (57) and (58). The greatest value of 7 corresponds to the Holder exponent h(xo). Let us note that the necessary condition (59) is not sufficient: one has to add a logarithmic correction by replacing x — XQ\"* in a) by \x - xo| 7 /|ln x - a;0||. However, from a numerical point of view, a logarithmic correction does not make any difference, and the condition a) can be considered as a necessary and sufficient condition. For nonoscillating singularities, one can prove that the behavior of the wavelet transform inside the cone x — XQ < Ca is enough to recover the Holder exponent /i(xo) (Mallat & Hwang 1992). The Holder exponent h(xo) (< n^) is the greatest exponent such that
390 3.2.3
A. Arneodo Detection and identification of singularities with the wavelet transform. Wavelet transform modulus maxima and maxima lines
As just explained, in order to recover the Holder exponent II(XQ) of a distribution / at the point XQ, we need to study the behavior, when a goes to 0, of its wavelet transform inside a cone x — XQ\ < Ca. For that purpose, we just need to look (at each scale a) at the "maximum" values of the wavelet transform inside the cone, and to study how these "maximum" values vary when a goes to 0 (see (60)). lit order to do so Mallat & Hwang (1992) introduced the notions of modulus maxima and of maxima lines of the wavelet transform (Mallat & Hwang 1992; Mallat & Zhong 1992).
Definition 3. • We call modulus maximum of the wavelet transform T^[f], any point («o,«o) of the space-scale half-plane which corresponds to a. local maximum of the modulus of T^[f](x, OQ) considered as a function of x, i.e., |T^,[/](xo,ao)| > T^[f](x, O,Q)\ for all x in a right neighborhood of XQ and \ r r ^ [ f ] ( x 0 , a 0 ) \ < I^W/K^ ao)| for all x in a left neighborhood of XQ. Thus, d \ T ^ [ f } \ / d x ( x Q , a0) = 0. e We call maxima line, any connected curve in the space-scale half plane made of modulus maxima. We illustrate these definitions in Figure 10. The considered function is f ( x ) = k0\x - XQ\OA + kie-<x-X1?/2. This function is C°° everywhere but at x = XQ where / is singular with h(xo) — 0.4. Its wavelet transform is shown in Figure lOb; it is coded at each scale a from black (bottom of color bar, T^[f] < 0) to the medium gray at the top of the color bar (max x r^,[/](a;, a)). Figure lOc represents a horizontal section of the wavelet transform at some given coarse scale do- The modulus maxima at this scale are marked on the same figure using the symbol ( x ) . The set of all the modulus maxima lies on maxima lines in the space-scale half-plane (6, a); they are displayed in Figure lOd. The analyzing wavelet is the Mexican hat V/ 2 ' (n^ = 2). Mallat & Hwang (1992) have shown that a singular behavior of a distribution around a point x,0 implies that there exist maxima lines
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Figure 10: Modulus maxima of the wavelet transform of the function f ( x ) = k0\x - XO\OA + jfeje-f*-*') 2 /*. ( a ) Graph of f ( x ) . (b) The wavelet transform of f ( x ) coded, independently at each scale a, from black (bottom of color bar, T,/, < 0) to the medium graph at the top of the color bar (max^ T^ > 0); the small scales are at the top. (c) Horizontal section of the wavelet transform 70[/](a;, a) at a = CQ; the symbols ( x ) represent the modulus maxima, (d) Maxima lines of the wavelet transform in the (x,o) half-plane. The analyzing wavelet is the Mexican hat ip = V> (2) (Eq. (50)).
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converging (when a goes to 0) towards the point x0 on. the space axis. Then, if there are no such lines, the distribution is uniformly Lipschitz n^ in a neighborhood of x0 (i.e., h ( x ) > n^ in a neighborhood of ,TQ). Thus the modulus maxima lines of the wavelet transform pro-vide information on the location of the singularities of a distribution /. One can check in Figure lOd, that on any interval which does not contain, at small scales, any modulus maximum, the function / is uniformly Lipschitz n^, = 2, whereas some maxima lines are converging towards the points x = x0 (which is singular). Let us note that it does not exclude that there could be maxima lines on an interval where / is not singular. Indeed, in Figure lOd, some maxima lines are converging towards the point x = x\ where / is not singular. From (60) one can expect that by looking at the value of the wavelet transform along a maxima line II(XQ) converging to a given point £0, one can estimate h(xo) (< ra,/,) as the greatest exponent such that
in the limit a —^ 0 + . Thus the maxima line allow us to locate and to estimate local singular behavior. In figure lOd, we see that the maxima lines converge towards 4 different points. By looking at the power law scaling of |T,/,[/]| along these lines, one can recover that only the point x = XQ corresponds to a singular behavior. Indeed, the exponent which characterizes the power law scaling of the 3 other points is h = n,/,, whereas the exponent corresponding to XQ is II(XQ) < n^. In Figure 11, we show log-log plots of the wavelet transform respectively along a maxima line which converges towards XQ and along a maxima line which converges towards x = x\. According to (61), the slope of these curves gives an estimate of h(x} = imn(h(x),n^ = 2). By a linear regression fit we obtain h(xo) — 0.4 and h(xi) ~ 2; thus / is singular at x = XQ with Holder exponent h(x0) = h(x0) = 0,4 but it is not singular at x = x\ (it is at least once differentiable and the Holder exponent is h(x1)>h(xl) = n^ = 2). Thus the modulus maxima of the wavelet transform allows us to study in a very efficient way the isolated singularities of a distribution /. In the case of fractal distributions, these singularities are not isolated. However, as we will see in the next sections, in most cases, the same analysis can be carried out.
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Figure 11: Estimating the Holder exponent from the behavior of the wavelet transform along the maxima lines. The slope of the curve log 2 (|T,/,[/](a:,a)|) versus Iog 2 (a) along a maxima line which converges towards a given point provides an estimate of the Holder exponent at that point (Eq. (61)). (a) Measurement for a line converging towards x0 in Fig. lOd. (b) Measurement for a maxima line converging towards the point x\ of the same figure. The analyzing wavelet is the Mexican hat -0 = ip2'.
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4.1
4.1.1
A. Arneodo
Multifractal Formalism for Distributions Based on Wavelets
Wavelet Transform of Multifractal Measures
Wavelet analysis of the local scaling properties of singular measures
The singularity-behavior of a measure at a given point x0 is generally characterized by the exponent a(xo) which satisfies the relation (10) (Farmer et al. 1983; Halsey et al. 1986; Grassberger et al. 1988):
where BXo(t) is an interval of size e, centered at x0. Let us note that this definition is very close to the definition (34) of the local Hurst exponent /I(.TO) introduced in Section 2.3.2. Indeed, if (62) holds and if s ( x ) is the distribution function associated with /j, ( s ( x ) = /.t([0, a;])), then one can prove that h(xo) = a(a-'o); where h(xo) is the local Hurst exponent of s at XQ. Thus, in the same spirit we derived (57) from (54) in the case of Holder exponents, one can derive from (62) a formula which shows that the wavelet transform of [j, behaves like a"^0"1 in the limit of the scale a going to 0 (Arneodo et al. 1988, 1989a, 1989b, 1990; Grasseau 1989):
where the wavelet transform of the measure ft is defined by (Arneodo et al. 1988; Holschneider 1988b):
Let us note that this latter definition is nothing but the definition (56) of the wavelet transform of a distribution except for the I/a factor that we have removed so that 7!0 scales exactly like a 0 ^ 0 ) (and not like a"^ 0 )- 1 ).
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Remark. In Section 2, we have pointed out that the definition of the scaling exponent a in (62) is not appropriate to account for singularities of strength a > I in the presence of some density component. This is the main reason why we have introduced in Section 3 the wavelet transform. Actually, this transformation is a mathematical tool which is well-suited to detect and measure local Holder exponents which are in fact a generalization of both the a scaling exponents of measures and the local Hurst exponents of functions. Indeed the wavelet transform can detect Holder exponents up to h < n^,, where n^ is the number of vanishing moments of the analyzing wavelet ^. However, the multifractal measures we will consider in this section are "purely" singular, i.e., they don't have any density component. Thus, there is a priori no point to use a function i/} with some vanishing moments (i.e., n,/, > 1). Using a Gaussian function if) or any function localized in space, in (63) and (64), is sufficient since, in this case, measuring a is equivalent to measuring the Holder exponent h (Bacry 1992; Ghez & Vaienti 1992; Muzy 1993). Indeed, if we use the "box function" t/j = X[o,i] (i- e -) the characteristic function of the interval [0,1]) as the analyzing wavelet, then T^[fi](xo,f) gg = fi(BXo(e}) and (63) is nothing but the definition (62) of the exponent a(xo). Figure 12 displays Iog2 \T^i}(x0,a) as a function of Iog2 a at 3 different points x of the triadic Cantor set on which a nonuniform measure has been distributed with the weights pi = 0.6 and p2 = 0.4 (Figure 2). One can see in Figure 12a that when x corresponds either to the sequence LLL...LL... or to the sequence RRR...RR..., the loglog plot displays a clear linear behavior (the oscillations are due to the fact that n is invariant under discrete dilations of a factor 3", n > 0) whose slope gives, with a very good precision, the expected values a(x) = amin = ln0.6/lnl/3 and amax = ln0.4/lnl/3 respectively. But when x corresponds to the sequence LLLLRRRR, Figure 12b displays a cross-over from amin to amax and thus, measuring the exponent a(x) becomes inaccurate and somehow inappropriate. In fact, the exponent a obtained from a linear regression fit on the entire available range of scales, corresponds to the strength of the singularity with periodic symbolic sequence of period LLLLRRRR. The singularities associated with nonperiodic sequences are not amenable to such a local scaling behavior analysis (Arneodo et al. 1989b, 1990). We understand now why (63), in the same way as (62), cannot be used for systematically "counting" the number of points x corre-
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Figure 12: Iog2 |T^[ju](a;,a)| as a function of log2 a at three different points x of the triadic Cantor which supports a Bernoulli measure associated to the weights p\ — 0.6 and p-^ = 0.4. (a) x = x\ and x = x-2 where x\ and x^ correspond respectively to the kneading sequences LLLLLLLL (a = amin = - I n p i / l n 3 ) and RRRRRRRR (a = amax = - In pa/In 3). (b) x = x3 which corresponds to the kneading sequence LLLLRRRR. The analyzing wavelet is the Mexican hat ^ = ^ 2 ).
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spending to the same value a(x) = a and thus for computing the singularity spectrum of a measure. For that purpose, we need to use a "global" method (which goes beyond local estimates) and which is based on the computation of partition functions from the continuous wavelet analysis. Furthermore, there is a strong argument in favor of using the wavelet transform for studying fractal objects. The wavelet transform is a space-scale analysis; it actually plays the role of a "mathematical microscope" whose enlargement factor is I/a and whose optics is defined by the shape of the analyzing wavelet ip. As we will illustrate in the next section, this microscope provides some unfolding, in the space-scale half-plane, of the hierarchical structure of the analyzed fractal object; this unfolding can eventually "reveal" the underlying construction process (Arneodo et al. 1988, 1989a, 1989b, 1990; Holschneider 1988a, 1988b; Grasseau 1989). It is thus tempting to develop a multifractal analysis relying on such a tool which is likely to capture the main structural ingredients to make this analysis optimal in the sense that it will converge faster than any other analysis. 4.1.2
Wavelet transform and renormalization
In this section, we will illustrate with three examples how the wavelet transform unfolds the hierarchical structure of a fractal measure by revealing the corresponding "renormalization operations" (Arneodo et al. 1990). Uniform measure on the triadic Cantor set. Let /j, be the uniform measure on the triadic Cantor set (Figure 1). The self-similarity properties of this measure are mathematically described by saying that it is invariant under the application T, (i.e., n(T~lA) = ii(A), VA) denned as follows:
Let us recall that the wavelet transform of \i is defined by (64):
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Letting x' = Tj~ (x) where k = 1 or 2, we thus obtain
Then, by using the fact that Tk is linear, we get
However, if we suppose that -0 has a compact support, then for a small enough
and thus This relation shows how the self-similarity of the measure /.t is reproduced in the space-scale representation given by its wavelet transform. The gray-shaded pictures in Figures 13a and 14a represent the wavelet transform of JJL using the analyzing wavelets -i/^0' and ?/>( 2 ) respectively. The amplitude is coded using 256 gray shades from black (top of diagram, T$ < 0) to medium gray (bottom of diagram, maxxT^,(x,a)). The coding is denned, independently at each scale, according to the maximum value of the wavelet transform at this scale. The construction process is clearly revealed through the relation (70) for both k = 1 and k = 2 (Arneodo et al. 1988, 1989a, 1989b, 1990). Indeed, the "structure" at any scale a* is exactly reproduced at the scale a — a*/3 in two similar structures reduced by a factor of 3. Thus, in both Figures 13a and 14a (independently of the analyzing wavelet) the original portion of the space-scale half-plane can be uncovered in the two symmetric dashed rectangles obtained from the original (large scale) picture by applying the two renorrnalization operations:
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Figure 13: Continuous wavelet transform of (a) Top picture: a multifractal measure distributed nonuniformly on the triadic Cantor set with the weights p\ ~ 0.6 and pi = 0.4 and (b) bottom picture: the natural measure of the period-doubling quadratic map $*(a;) (Eq. (75)). The gray-shaded coding is the same as in Fig. 13. The left (right) dashed rectangle corresponds to the renormalization operation RI (R?) discussed in the text. The analyzing wavelet is the Gaussian function ^M°'.
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Figure 14: Continuous wavelet transform of the uniform measure on the triadic Cantor set. Same calculation as in Fig. 13 except that the analyzing wavelet is the Mexican hat ^>'2) (Eq. (50)).
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Figure 15 illustrates the wavelet transform computed with the Mexican hat (Figure 14a) using a three-dimensional representation. To make the singularities easy to visualize, T^ is normalized by the factor a~2 so that, when a —* 0, it diverges (~ a1"2/1"3"2), at each point of the triadic Cantor set. Figure 15 illustrates the fact that the singularities can be tracked in a very efficient way with the wavelet transform microscope. Indeed, the lines formed by the higher wavelet coefficients converge towards the support of //. These lines correspond to the maxima lines made of local maxima of the function |T^,[/tt](x,a)| considered as a function o f x . They are pointing towards the singularities of the measure. Moreover they follow the same renormalization rules RI and RI as the wavelet transform itself. Thus the behavior of the wavelet transform along these lines is of the same type as the one described by (63). According to (71) (which transforms a maxima line into another maxima line), dilating the scales a by a factor 3"1 corresponds to a multiplication of the wavelet coefficients by a factor 2"1; thus, along these lines T^(b,a) ~ a l n 2 / l n 3 (Arneodo et al. 1991; Bacry et al. 1993). As illustrated in Figures 13b and 14b, the maxima lines of the wavelet transform define a tree whose branching process contains all the information on the hierarchical structure of the triadic Cantor set which justifies its construction. This explains why such a skeleton is a "key" tool in processing fractal measures and detecting singularities. As we shall see in the next section, this wavelet transform skeleton will play a very important role in the definition of the new multifractal formalism based on wavelets. Nonuniform measure on the triadic Cantor set. We can perform the same kind of analysis on a Bernoulli measure distributed nonuniformly (e.g., p\ = 0.6 and p-2 = 0.4) on the triadic Cantor set (Arneodo et al 1988, 1989a, 1989b, 1990). One can easily prove that the self-similarity of // is reproduced on its wavelet transform according to the following rule (corresponding to (70)):
where R^ for k = I or 2 are the renormalization operations defined in (71). The color picture in Figure 16a displays the wavelet transform, of /7,; the coding is the same as the one used for the former color figures. The analyzing wavelet is the Gaussian function t/>(°'. The two renormalization operations are once again illustrated by two small dashed rectangles reproducing the structure of the original picture
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Figure 15: 3D representation of the continuous wavelet transform (a~ 2 T^,[/i](6, a)) of the uniform measure on the triadic Cantor set. The analyzing wavelet is the Mexican hat 0( 2 '.
at smaller scales. But this time the reproduction is "attenuated" by 2 different amplitude factors (p\ and p^) depending upon which rectangle is considered. A 3D representation of this image is displayed in Figure 17. Again the maxima lines are converging towards the singularities of the measure, i.e., towards the triadic Cantor set, and they reproduce its hierarchical structure. Moreover, the behavior of the wavelet transform along these lines characterizes the strength a of the singularity it is pointing to (even though, as noted before, this value cannot reliably be computed by this local method). The fact that the wavelet transform displays very different slopes in Figure 17 indicates that the measure is not homogeneous. Let us note that the symmetric "forking" process in Figure 15 has disappeared; this comes from the fact that the symmetry of the uniform Cantor set is broken by the nonuniform weights pi ^ p%. This measure is
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Figure 16: Continuous wavelet transform of (a) Top picture: a multifractal measure distributed nonuniformly on the triadic Cantor set with the weights p\ = 0.6 and p2 - 0.4 and (b) bottom picture: the natural measure of the period-doubling quadratic map $*(x) (Eq. (75)). The color coding is the same as in Fig. 13. The left (right) dashed rectangle corresponds to the rcnormalization operation 7?i (_ff 2 ) discussed in the text. The analyzing wavelet is the Gaussia.n function w(°>.
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multifractal; the scaling exponent a fluctuates from point to point on the triadic Cantor set.
Figure 17: 3D representation of the continuous wavelet transform (a"2T^,[/f](6, a)) of a multifractal measure distributed nonuniformly on the triadic Cantor set with the weights p\ = 0.6 and pi — 0.4. The analyzing wavelet is the Mexican hat -tp^.
The period-doubling Cantor set. The transition to chaos in dissipative systems (Eckmann 1981; Ott 1981; Cvitanovic 1984; Guckenheimer fe Holmes 1984) presents a strong analogy with second-order phase transitions (Hu 1982; Crutchneld et al. 1982; Coullet 1984; Argoul & Arneodo 1986). Among the different scenarios which explain the passage from ordered to disordered temporal patterns, the most popular is undoubtedly the cascade of period-doubling bifurcations (Coullet & Tresscr 1978; Feigenbaurn 1978, 1979; Tresser & Coullet 1978) and the transition to chaos from quasi-periodicity
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with irrational winding numbers (Feigenbaum et al. 1982; Ostlund et al. 1982, 1983; Shenker 1982). In this section, we focus on the period-doubling scenario and refer the reader to our original work for a similar analysis of the evolution from quasi-periodicity with golden mean winding number (Arneodo et al. 1988, 1989a, 1989b, 1990).
Figure 18: The construction rule of the period-doubling Cantor set; the indices refer to the number n of the iterate $^c (0) of the critical point xc — 0 of the map defined in Eq. (75). Dissipative dynamical systems that exhibit the cascade of perioddoubling bifurcations are in practice well-modeled by one-dimensional maps with a single quadratic extremum such as the map (Coullet & Tresser 1978; Feigenbaum 1978, 1979; Tresser & Coullet 1978; Collet & Eckmann 1980):
or quadratic maps of the form $R(X) = Rx(l — x), Rsimrx... As one increases the parameter R which determines the height of the maximum of $/} at x = xc = 0, one observes an infinite sequence of subharmonic bifurcations at each stage corresponding to a doubling of the limit cycle period. This period-doubling cascade accumulates at Rc = 1.40115... where the system possesses a 2°°-orbit that displays scale invariance (Figure 18). Beyond this critical value, the attractor becomes chaotic, even though there still exist parameter
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windows of periodic behavior. As originally emphasized by Coullet & Tresser (1978), Feigenbaum (1978, 1979) and Tresser & Coullet (1978), this scenario presents a strong analogy with second-order phase transition in critical phenomena. Above criticality (R > Rc), the envelope of the Lyapunov characteristic exponent (which provides a quantitative estimate of chaos) displays a universal "order parameter" like behavior L(R) ~ (R — Rc)v, where v is a universal exponent in the sense that it does not depend on the explicit form of the map but only on the quadratic nature of its maximum. Below criticality (R < Rc), the period of the bifurcating cycles is a "characteristic time" which diverges at the transition according to the scaling law P(R) ~ (Rc — R)"", with the same critical exponent v = In 2/ In A as for the Lyapunov exponent. This universal behavior results from the observation that the bifurcation parameter values Rn (from an orbit of period 2" to an orbit of period 2 n+1 ), converge to Rc = #00 according to the geometric law (Rc — Rn) ~ A~ r a , where A = 4.669... for quadratic maps. Very much like in critical phenomena, Coullet & Tresser (1978), Feigenbaum (1978, 1979) and Tresser & Coullet (1978) have demonstrated that these universal properties can be understood using renormalization group techniques. Indeed, at criticality R = Rc, the attractor of the quadratic map (73) exhibits scale invariance: the adherence of the asymptotic orbit of almost all initial conditions in the invariant interval is a Cantor set. As sketched in Figure 18, the iterates of the critical point xc = 0 form this Cantor set, with half of the iterates falling between 3># (0) and $H C (0), and the other half between $^(0) and $^(0). At the next stage of the construction process, each subinterval is again divided into two subintervals with equal probability and so on. Consequently, the visiting probability measure is symmetrically distributed with the weights p\ = p% = 1/2. Actually, the critical map $RC belongs to the stable manifold of the fixed point $* of the renormalization operation (Coullet & Tresser 1978, 1981; Feigenbaum 1978, 1979; Tresser & Coullet 1978; Collet & Eckmann 1980):
where a = !/$(!). Up to some dilation by a scale factor a, $* is identical to its second iterate. In the generic case of quadratic maps, the functional equation $* = 7£($*) was solved using a truncated recursion formula (Derrida et al. 1979) and numerical algorithms (Coullet
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& Tresser 1978; Feigenbaum 1978, 1979; Tresser & Coullet 1978):
with a = —2.5029... Later on, rigorous mathematical proofs were carried out by Campanino & Epstein (1980), Epstein & Lascoux (1981), Campanino et al (1982), Lanford (1982, 1984) and Epstein (1986). Since $RC belongs to the stable manifold of $*, their respective invariant measures display similar multifractal characteristics. In particular, it was shown that the /(a) singularity spectrum of the invariant measure of $* is universal (Halsey et al. 1986; Collet et al. 1987). Moreover Ledrappier & Misiurewicz (1985) succeeded in proving that this measure can be considered as the invariant measure obtained by iterating backward the following dynamical system denned on the interval A = [$*(!),!]:
where x* is the fixed point of $, in A ($*(»*) = x*). The construction rule of the period-doubling Cantor set J* = n^0T~n(A) is illustrated in Figure 19. Let us point out that, as compared to the Bernoulli measures distributed on generalized Cantor sets, the self-similarity properties of the invariant measure of critical period-doubling dynamical systems depend dramatically on the fact that one branch of T(#), i.e., T2(a:) = $*(x)/$*(l) is nonlinear. Nevertheless, if one supposes again that the analyzing wavelet has a compact support, one can investigate the self-similarity properties of the wavelet transform in the same way as for invariant measures of piece-wise linear maps:
where AI = A n ^(A) and A2 = A n T^l(A) (Figure 19). Now using the fact that \T'\ is bounded on A by two constants larger than 1, for a small enough the above integral can be restricted to a subinterval of the form [b1 - c\a, b' + c 2 a], where b' = T^{(b). Then,
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Figure 19: The natural measure of the critical quadratic map $*(a;) (Eq. (75)) as "seen" as the invariant measure of the hyperbolic mapping T ( x ) defined in Eq. (76).
after expanding T(a:) — T(b') to first order, one can rewrite (77) in the form
where we have implicitly assumed that T^[/j,](b, a) = O(T^i[n](b,a}). This relation accounts for the self-similarity properties of the wavelet transform of the natural measure associated with the critical quadratic map $* (see (75)) in the limit a —y 0 + . Let us note that the main difference between this new relation and (70), is the fact that the dilation factor in the right-hand side of (78) depends explicitly on the point 6 as a direct consequence of the nonlinearity of T. A color coding of T^[p](b,a) is shown in Figure 16b, in comparison with the
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wavelet transform of a nonuniform measure distributed on the triadic Cantor set (Figure 16a) (Arneodo et al. 1988, 1989a, 1989b, 1990, 1992a). The analyzing wavelet is the Gaussian function ^>(°). The two dashed rectangles illustrate the two renormalization operations RI and R-2 associated respectively with the two branches Tf1 and T^1 of the inverse mapping T~l (Muzy 1993):
The self-similarity of the wavelet transform in the space-scale halfplane observed in Figure 16b is thus contained in (78). The multifractality of the invariant measure of the critical quadratic map $*(cc) is conspicuous in the 3D representation of the wavelet transform shown in Figure 20. The singularity exponent a(x) clearly fluctuates from point to point on the period-doubling Cantor set. In the concluding section, we will elaborate on a very promising attempt to solve the inverse fractal problem, taking advantage of the self-similarity properties of the wavelet transform in the space-scale half-plane (see (70) and (78)).
4.2
The Multifractal Formalism for Singular Measures Revisited with Wavelets
In Section 4.1, we already pointed out that if one takes an analyzing wavelet tj) which is the box function X[o,i] (i- e -j the characteristic function of the interval [0,1]), then (62) and (63) become the same. Indeed, n(BXfl(t}) is nothing but the wavelet transform of// using x as the analyzing wavelet. In this way, (63) can be seen as a "generalization" of (62). However, it is a generalization not only because it holds for any wavelet 4> but also because it still holds when "perturbing" fj, with regular behavior (provided n^ is large enough). As we have emphasized at the end of Section 4.1.1, this last point is the main reason why we advocate the use of wavelets instead of "boxes": our goal is to get rid of possible smooth behavior that could mask the singularities or perturb the estimate of their strength a and thus would lead to dramatic errors in the computation of the singularity
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Figure 20: 3D representation of the continuous wavelet transform (a~lT^,[n}(bi a)) of the invariant measure of the critical quadratic map $*(x) (Eq. (75)). The analyzing wavelet is the Mexican hat ^( 2 ).
spectrum (as an exercise just consider that /.i is "perturbed" by a uniform measure on [0,1] and try to apply (62)...). Our aim is to revisit the multifractal formalism biit substitute the box functions by wavelets. Let /j, be a "purely" singular measure. A "naive" way to proceed would be to define the new following partition functions (Holschneider 1988b; Arneodo et al. 1991, 1992b; Muzy et al. 1991, 1992; Bacry et al. 1993):
where q 6 JR. One can prove that this definition would lead to a consistent multifractal formalism for positive values of q only (Bacry et al. 1993; Muzy 1993). Indeed, nothing prevents T^,[/,i](b, a) from vanishing at some points (6, a) of the space-scale half-plane (even for
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b in the support of fj,). The function K(q, a) would then diverge for q < 0; thus, (80) is not a good definition of a partition function. In order to avoid these divergences, we could change the continuous sum over space in (80) into a discrete sum over the maxima of |T.0[/4](£,a)| (considered as a function of a;). As seen in Section 4.1.2, the maxima lines satisfy two main properties (Muzy et al. 1991, 1993a, 1993b; Mallat & Hwang 1992; Bacry et al. 1993): (i) They follow the same renormalization rules as the wavelet transform itself and thus, reproduce the hierarchical structure of the fractal measure fj, (see Figures 13 and 14). (ii) Each line / = {6|(a),a} is pointing (when a goes to 0) towards a point 6;(0) which corresponds to a singularity of yu; moreover, along such a line the wavelet transform behaves like
(The maxima line / is then said to be associated to the exponent a(6/(0))). Let Na(a) be the number of maxima lines, at the scale a, associated with a given exponent a. From (i) and (ii), one can prove that Na(a) scales like This result has been proved rigorously by Bacry et al. (1993) in the case of Bernoulli measures. It can be intuitively understood by saying that the maxima lines reproduce the hierarchical structure of yu, i.e., when a goes to 0, the rate of multiplication (when constructing H~) of singularities corresponding with the exponent a (which is by definition a~-^ a ') is given by the rate of multiplication of maxima lines (in the wavelet transform skeleton) associated to the exponent a. Example. Let us illustrate this result on a very simple case: the uniform measure lying on the triadic Cantor set. From Section 2.2.1, we know that this measure is homogeneous and that its singularity spectrum is represented by a single point f(Djf) = DH where Dfj = In 2/In 3 is the Hausdorff dimension of the support of fj,. Then, according to (82), N(a) = NDH(O) ~ a~D". Figures. 13b and 14b display the maxima lines (in the space-scale half-plane) of /i computed with the
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analyzing wavelet t[>(°') (Gaussian function) and V^ 2 ^ (Mexican hat) respectively. Both figures reveal clearly the hierarchical structure of fj,. One can "read" the self- similarity of n in these perfect symmetric "forkings" which look exactly the same at any scale. One can estimate easily the multiplication rate of the maxima lines: each of the N(a) maxima lines at the scale a is going to give birth to 2 maxima lines at the scale a/3. Thus we find the expected result N(a) — 7Voa~ ln2 / ln3 . Let us notice that the constant N0 depends basically on the shape of the analyzing wavelet we use; it is generally proportional to the number of vanishing moments n^ of V; (Muzy et al. 1991, 1993a, 1993b; Bacry et al. 1993). Let us come back to the original purpose: the definition of waveletbased partition functions. As suggested earlier, in order to avoid the divergence of the partition functions for q < 0, we could change the continuous sum in (80) into a discrete sum over the maxima of |T^[yw](a:, a) | (considered as a function of a;) only. We then obtain the following partition function (Arneodo et al. 1991; Muzy et al. 1991, 1993a, 1993b; Bacry et al. 1993):
where £(a) is the set of all the maxima lines / existing at the scale a, and b[(a) the position, at the scale a, of the maximum belonging to the line I. Then by using (81), the partition function Z(q, a) becomes
and by substituting the expression of Na given in (82), one gets
Thus, if T(5) is the scaling exponent characterizing the power-law behavior of Z(q,a): we then obtain
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where Ts(q) is the exponent defined in (16) corresponding to the "standard box-counting" multifractal formalism. Consequently, the newly denned multifractal formalism is consistent with the classical formalism originally introduced for singular measures which allows us to obtain the f ( a ) singularity spectrum of a multifractal measure fj, as the Legendre transform of the r(q) function. However, the definition (83) is not yet general enough since it is unstable for negative values of q, very much like fixed-size box-counting algorithms. Indeed, let us suppose that along a maxima line the value of the wavelet transform goes from positive to negative values; then there is a scale where the maximum of |T^,[/u]| is 0! This corresponds to an inflection point of the wavelet transform (considered at a fixed scale a). So we need to change slightly the definition of the partition function by replacing the value of the wavelet transform modulus at each maximum by the supremum value along the corresponding maxima line at scales smaller than a (Arneodo et al 1991; Muzy et al 1991, 1993a, 1993b; Bacry et al. 1993):
We have proved rigorously, that for any Bernoulli measure // and for a large class of analyzing wavelets ip, the Legendre transform of the so defined r(q) function is the /(a) singularity spectrum of // (Bacry et al. 1993). Numerical examples are given in Section 5. Let us note that this new definition of Z(q,a) corresponds to a "scale-adapted" partition with wavelets at different sizes (smaller than a) whereas the former definition (83) uses a uniform partition (Arneodo et al 1991; Muzy et al. 1991, 1993a, 1993b; Bacry et al. 1993). This is illustrated in Figure 21 where the support of the measure is covered with wavelets centered at the maxima positions (instead of covering the support with boxes along the line of the "standard" multifractal formalism").
4.3
Generalization of the Multifractal Formalism to Fractal Distributions: the Wavelet Transform Modulus Maxima Method
Before describing how this new multifractal formalism can be used to analyze any fractal distribution, we need to define the notion of
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Figure 21: Representation of the uniform (Eq. (83)) and scaleadapted (Eq. (88)) partitions, (a) Uniform partition : Z(q,a) involves wavelets of the same size a. (b) Scale-adapted partition: Z(q,a) involves wavelets of different sizes a' < a. singularity spectrum of a distribution. The Holder exponent h(xo) (see (53)) provides a quantitative characterization of the degree of regularity of a distribution s at the point XQ. It is thus natural to use it in order to define the singularity spectrum of a distribution. Definition 4. Let s be a distribution and Sh the set of all the points XQ such that the Holder exponent of s at XQ is h. The singularity spectrum D(h) of s is the function which associates to any h the Hausdorff dimension of Sh'-
Let us note that when the distribution s is a singular measure with no density component, the exponent a(xo) defined in (62) is related to the Holder exponent h(xo) according to
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Therefore the D(h) spectrum of Holder exponents is obtained by just translating /(a) by 1 (Muzy 1993):
This comes from the fact that instead of considering // as a measure, we consider it as a distribution. In the following, we will only talk about distributions and the term singularity spectrum will always refer to D(K). Similar to (88), let us define the partition function for any q 6 1R (Bacry et al. 1993; Muzy et al. 1993a, 1993b):
The exponent r(q) is then given by
in the limit a —*• 0+. We then suggest the following theorem: Theorem 2. The D(h) singularity spectrum, of the distribution s is obtained by Legendre transforming the function r(q) defined in (93):
Both the D(h) and r(q) spectra involved in this new multifractal formalism are much more general that the ones defined in the "standard" multifractal formalism described in Section 2.2. The above theorem is likely to hold for a very large class of fractal objects. For rigorous results concerning this new formalism we refer the reader to the work of Bacry et al. (1993). Remark. In Section 5, we present some numerical applications of this wavelet-based multifractal formalism. From a numerical point of view, in order to compute D(h) we either directly perform the Legendre transform of r(q) (see (94)) or calculate two intermediate partition functions:
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and
from which we extract the scaling exponents
and
and in turn D(h). This alternative method is inspired from the socalled canonical method (Badii 1987; Jensenfdggfgfgfgfdfdgfgfdfgg fe Jensen 1989; Chhabra et al. 1989), which requires first the computation of the "Boltzmann probability measure" from the wavelet transform modulus maxima:
where Z(q,a) is the partition function defined in (88). In the following, both methods will be referred to as the wavelet transform modulus maxima (WTMM) method (Arneodo et al. 1991; Muzy et al. 1991, 1993a, 1993b; Bacry et al. 1993). Generally they give consistent results and robust estimates of the D(h) singularity spectrum.
Interpretation of r(q) for select values of q. • For q = 0, one can see from (92) and (93) that the exponent r(0) accounts for the divergence of the number of maxima lines in the limit a —* 0 + . This number basically corresponds to the number of wavelets of size a required to cover the set of singularities of the distribution. In full analogy with standard box-counting arguments. —r(0) can be identified with the fractal dimension (capacity) of this set:
« For 9 = 1, one can prove that the value of the exponent r(l) is related to the capacity of the graph Q of the considered distribution (provided Q is well-defined). More precisely:
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• For g = 2, if /? is the scaling exponent of the spectral density S(k) =| s ( k ) | 2 ~ k"13, then on can show that
5
5.1
5.1.1
Numerical Applications of the Wavelet Transform Modulus Maxima Method
Generalized Devil Staircases
Deterministic and stochastic signals
Before applying the WTMM method to analyze experimental signals, it is important to test it on "simple" functions for which the singularity spectra can be computed analytically. In the former sections, we introduced the class of self-similar measures lying on "generalized" Cantor sets. These measures are constructed using some very basic recursive rules which make the analytic computation of their singularity spectra particularly easy. The corresponding distribution functions(i.e., f ( x ) = //,([0,£'])) are self-affine and thus are particularly "good" functions for testing numerically the WTMM method. Moreover, their singular behavior is exactly given by the singularities of their associated fractal measures. Therefore one can prove easily that the D(h) singularity spectra of these functions are equal to the /(a) singularity spectra of n (Arneodo et al. 1991; Muzy et al. 1991):
In this section, we illustrate the WTMM method on three such distribution functions and we compare the so-obtained singularity spectra to their theoretical counterparts. First note that it has been proved that for any distribution function of a self-similar measure [j, lying on a generalized Cantor set and for a large class of analyzing wavelets, the WTMM method converges to the singularity spectrum of//. This is equivalent to saying t h a t for these particular functions both (93)
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Figure 22: Continuous wavelet transform of the devil staircase corresponding to the uniform triadic Cantor set. (a) Graph of the function. (b) Wavelet transform computed with the analyzing wavelet V^1); the amplitude is coded, independently at each scale a, using 32 grey levels from white (T^,[/](,r,a) < 0) to black (max^T^f/^a;, a)). (c) Definition of the modulus maxima at a given scale a0 corresponding to the dashed line in figure (b). (d) The skeleton of the wavelet transform, i.e., the set of all the maxima lines. In (b) and (d) the large scales are at the top. and (94) hold. For the theoretical proofs of this theorem we refer the reader to Bacry et al. (1993). Example. Let n be the uniform measure lying on the triadic Cantor set (p\ — pi = 1/2), and f ( x ) — /;,([(),. T]) the so-called devil staircase. The function / is shown in Figure 22a. Figure 22b illustrates its wavelet transform coded, independently at each scale, using 32 grey levels. The analyzing wavelet is i/^1' the first derivative of the Gaussian function. Figure 22c represents a horizontal cross section of |^'i/.[/](-x, ao)|
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Figure 23: Determination of the multifractal spectra of the devil staircase associated to the uniform triadic Cantor set using the WTMM method, (a) Iog2 Z(q,a}/(q - I) versus Iog2 a. (b) r(q) versus g; the solid line corresponds to the theoretical curve r(q) — (q — I ) l n 2 / l n 3 . (c) Determination of the exponents h(q)] h(q,a) is plotted versus Iog2 a according to Eq. (97). (d) D(h) versus h. The analyzing wavelet is ^'l). at a fixed scale a0 corresponding to the dashed line in Figure 22b; the modulus maxima are marked by the symbols ( x ) . The skeleton of the wavelet transform is shown in Figure 22d. It is on these maxima lines that the partition function Z(q,a) is computed according to (92). Let us note that the perfect similitude of the skeletons in Figure 22d (the maxima lines of T^jf/]) and Figure 13b (the maxima lines of 7'^(o) [//,]) comes from the fact that the two wavelet transforms
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are proportional at every scale:
this relation is simply derived by integrating (56) by parts. Figure 23a displays some plots of Iog2 •£(
Let / be the generalized devil staircase associated with the selfsimilar measure JJL constructed recursively as follows: each interval at each step of the construction is divided into 4 sub-intervals of same length on which, we distribute respectively the weights p\ = 0.69, p2 = —p3 = 0.46 and p4 = 0.31. One of the weights has been chosen negative so that the so-obtained measure is signed. This definition is consistent as far as the relation pi + p2 + jj;; + p4 = 1 holds. Let us note that in the case of a distribution function on a signed measure, the relation r ( l ) = 0 does not bold a priori since the "norm" T^c(a) '/V(M a ); a ) l '1S 110 l°ngcr conserved through the scales; it di-
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Figure 24: Graphs of the generalized devil staircases, (a) Deterministic signal f ( x ) = //([0,a:]). (b) Random signal fr(x) = fj,r([0,x]). /J, and \ir are signed measures. verges when a goes to 0. Actually, one can prove that in this particular case, r(q) is given by the relation (Arneodo et al. 1991; Muzy et al. 1991):
Figure 24a displays the distribution function f ( x ) = /u([0, x]), whereas Figure 24b shows the distribution function fr(x) = // r ([0, a;]) which is constructed exactly in the same way as / except that, at each step of the construction, the order of the weights is chosen randomly. Their wavelet transforms are illustrated in Figures 25a and 25b respectively. In the case of the random distribution function f r , the partition function is averaged over the realizations of the random process, i.e., Clearly, the analytical expression (105) of r ( q ) does not depend on the specific order of pi, p 2 , p3 and p4, so one deduces easily that
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Figure 25: Continuous wavelet transforms of (a) top picture: the deterministic devil staircase / shown in Fig. 24a, and (b) bottom picture: the random devil staircase fT shown in Fig. 24b. The analyzing wavelet is V 7 ' 1 '- Same color coding as in Fig. 13. The small scales are at the top.
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Figure 26: Determination of the multifractal spectra of the devil staircases displayed in Fig. 24 using the WTMM method, (a) Iog2 a~r(l^Z(q, a)/(q — 1) versus Iog2 a. (b) r(q) versus q; the solid line corresponds to the theoretical curve given by Eq. (105). (c) Determination of the exponents h(q); h(q,a) is plotted versus Iog2 a according to Eq. (97). (d) D(h] versus /i; the solid line corresponds to the theoretical spectrum. The analyzing wavelet is tM 2 ). In (b) and (d) the symbols correspond to the data obtained for the deterministic (•) and the random (A) signal.
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Tr(q) = r(q). The results of the multifractal analysis o f / and /,. using the WTMM method are reported in Figure 26 (Arneodo et al. 1991; Muzy et al. 1991). As shown in Figure 26b, the functions r(q) and Tr(q) are nonlinear convex increasing functions. The numerical data both for the deterministic signal (•) and the random, signal (A) match perfectly the theoretical prediction given by (105). The corresponding D(h) singularity spectra obtained by Legendre transforming r ( q ) and TT(q) are displayed in Figure 26d; their single humped shapes are characteristic of multifractal signals. The support of D(h) extends over a finite interval hmin < h < hmax. This nonuniqueness of the Holder exponent is confirmed in Figure 26c, where the exponent h(q), computed directly from (97), clearly evolves from the value hmin — 0.28 to hmax ~ 0.82 when q varies from q = 10 to q — —10. The maximum of the D(h} curve is obtained for q = 0: D(h(q = 0)) = — r(0) = Dp = 1. The generalized devil staircases in Figure 24 are thus everywhere singular signals that display multifractal properties; the fractal dimension of the support of the set of singularities of these distribution functions is Dp — 1. 5.1,2
The structure function approach versus the wavelet transform modulus maxima method
The efficiency of the WTMM method comes from two basic factors (Arneodo et al. 1991; Muzy et al. 1991). First, it is based on the wavelet transform of the signal, thus it can be adjusted easily (mainly by changing the analyzing wavelet) to the particular singular behavior of the analyzed signal. Furthermore, it uses the modulus maxima of the wavelet transform so that the partition automatically covers only the points where the signal is singular. These two very important features are clearly missing from the structure function (SF) method introduced by Frisch & Parisi (1985). However, this method has been playing a very important role in the context of fully developed turbulence and is a first step towards a multifractal formalism for fractal signals (Frisch fe Orszag 1990; Barabasi & Vicsek 1991; Family & Vicsek 1991). It has been applied and is still very often used to analyze experimental fractal signals. A direct comparison of the SF and WTMM methods when applied to specific examples is thus of great interest for future analysis. In this subsection we review the main cases where the SF method
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leads to drastic bias whereas the WTMM method provides reliable results. We illustrate this bias with simple numerical examples. The very interesting case of the influence of C°° behavior on the computation of the D(h) singularity spectrum will be studied in Section 5.2. Divergences for negative values of q. Let us recall that in the definition initially proposed by Frisch & Parisi (1985), the structure functions S p ( l ) in (36) are not defined for negative values of p. Indeed, there is no reason a priori that the probability density of the increments Sfi of a function / should vanish around 0. Consequently, the Legendre transform (37)
is valid only for q > 0. Therefore, if the singularity spectrum is a single humped function (Figure 26d), only the increasing part of D(h) corresponding to the strongest singularities is accessible to the SF method. Let us note that in order to circumvent this difficulty, Barabasi & Vicsek (1991) have used a slightly different SF method for studying the multifractal properties of some rough interfaces in nonequilibrium growth phenomena. They defined the following "correlation functions of order g":
where the spatial average is performed only on terms for which 6f,(x) = \f(x + /) - f ( x ) \ ^ 0. Then Cg(l) can be defined for any value of q. However, this definition is totally artificial and does not cure at all the divergence problems encountered for q < 0. Practically, this method consists in not taking into account the increments which are smaller than a given threshold. Since this artificial "cutoff" is scale independent, the self-similarity is broken locally where the increments of the signal are small; note that the small increments dominate in (108) for q < 0. This will lead in most cases to a phase transition phenomenon in the (q spectrum. This is exactly what Barabasi & Vicsek (1991) did observe when studying a fractional Brownian motion. Indeed, let Bfj a fractional Brownian motion ofparameter H defined by (31). The corresponding increments 6Bu are Gaussian and stationary. Thus their spatial averages can be computed using an ergodic formula (Azencott & Dacunha-Castelle 1984).
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The spatial average of Cq(l) is given by (Muzy et al. 1993a):
where c > 0 is the cut-off. When / is large enough one gets:
One thus obtains the following expression for the exponent
which differs from the theoretical prediction for an homogeneous fractal signal which is almost everywhere singular with a unique Holder exponent h = H: The nonanalyticity of (q for q = —1 in (111) can be understood as a phase transition in the scaling properties of the signal which is an artifact of the method proposed by Barabasi & Vicsek (1991). This example clearly demonstrates that the divergences encountered in the structure function method for negative values of q are really intrinsic to the method and generally cannot be avoided by using some numerical "tricks". Therefore, only the increasing part of the singularity spectrum (corresponding to the strongest singularities) is potentially accessible to the SF method, whereas, in the case of the WTMM method, the use of the modulus maxima allows a consistent definition of the exponent r(q) for any value of q. This will be illustrated in Section 5.3.1 where a detailed analysis of the D(h) singularity spectrum of fractional Brownian motion using the WTMM method will be reported. Limitations on the range of accessible Holder exponents a) Singularities with negative Holder exponents The increments used in the SF method can be seen as wavelet coefficients using the analyzing wavelet A^ 1 ' = 6(x — 1) — 6(x):
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In the wavelet jargon, this wavelet is generally referred to as the "poor man's wavelet" (Vergassola & Frisch 1991; Muzy et al. 1993a). Its extreme irregularity makes it a very unpleasant wavelet to work with. Indeed, T A (ij [/](&,«) =< /, A^ > is generally not defined when / is a distribution! In the particular case where / has some singular behavior corresponding to negative Holder exponents, (113) is theoretically not denned and the computation of the SF is numerically unstable. Thus using the SF method to analyze a signal which has discontinuities or stronger singularities may lead to unexpected results as the consequence of intrinsic drawbacks of this poor man's wavelet based approach. In Figure 27, we show the results of both the SF and the WTMM analysis of a signal that possesses some singularities with negative Holder exponents (Muzy et al. 1993a). The signal is a generalized devil staircase similar to the one illustrated in Figure 24a (the weights are pi = 0.69, p% = 0.46, ps = -0.46, ^4 = 0.31), but it has been fractionally differentiated with a coefficient /3 — 0.6 so that a part of the support of the D(h) singularity spectrum is below 0 (a fractional derivation of a signal with a coefficient /3 induces a translation of the singularity spectrum to the left by a factor /? (Schertzer & Lejevoy 1987)). In Figure 27a, one can notice the "jumps" in the signal which correspond to negative Holder exponents. The theoretical singularity spectrum is represented by the solid line in Figure 27d; it extends over the range [—0.1,1.36]. The analyzing wavelet is the Mexican hat ?/>( 2 '. The results obtained when using the WTMM method are reported in Figures 27b, 27c and 27d. As illustrated in Figure 27c, the numerical data for r(q) fall on a convex nonlinear curve which is particularly well-fitted by the theoretical T(q] spectrum. Its Legendre transform D(h) in Figure 27d is a single humped curve characteristic of a multifractal signal; it is also in remarkable agreement with the theoretical D(h) spectrum. The comparison between the SF and WTMM methods is shown in Figure 27e. As expected theoretically, the SF method gives a correct T(
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Figure 27: WTMM and SF analysis of a multifractal signal with negative Holder exponents, (a) Graph of the signal. This signal is a generalized devil staircase constructed from the recursive signed measure involving the weights p\ = 0.69, p% = —p3 = 0.46 and P4 — 0.31; it has been fractionally differentiated with a coefficient /? = 0.6. (b) Iog 2 (2(g,a)) versus Iog2 a for some values of q. (c) r(q) spectrum obtained with the WTMM method; the analyzing wavelet is ip^. (d) D(h) singularity spectrum obtained by Legendre transforming the numerical r ( q ) data in (c). (e) Comparison of the r(q] spectra obtained with the WTMM (•) and the SF (A) methods. In (c), (d) nd (e), the solid lines correspond to the theoretical spectra.
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method is clearly not well-adapted for analyzing signals with negative Holder exponents. The WTMM method uses very smooth wavelets and leads to robust and accurate results. b) Singularities with Holder exponents h > 1 As we have just pointed out, the increment Sfi(xo) of size /, at a point XQ, can be seen as the wavelet transform T^i)[f](xQ, I) of /, at the scale a = I and at the point 6 = .TO, using the analyzing wavelet A' 1 ) (see (113)). However, A^) is orthogonal to polynomials of order 0 (i.e., constant terms) only. Thus, as discussed in Section 3.2.2, the increments will be blind to any Holder exponent greater than 1, since only the first moment of A^ 1 ) vanishes (ft A (i) = 1). If we suppose that h(x0) > 1 at a given point XQ, the increment Sfi(xo) will be genetically dominated by a term of the form f'(xo)l (i.e., the first term in the Taylor series of / which is not constant). Each time the Holder exponent is greater than 1, the increments measure the exponent h — 1. Thus, for h > 1, the SF method leads to a degenerate singularity spectrum D(h = 1) = 1 corresponding to r(g) = q — 1. Actually, it is even worse: one can prove that, in most cases, there exists ft** verifying ft** = 1 - [1 - I>(ft**)]/£>'(ft**) < 1 (respectively q**) so that D(h) (respectively r(g)) is degenerate for any ft > ft** (respectively q < q**) (Muzy et al. 1993a). Let us note that these problems do not exist when using the WTMM method since the wavelet is always chosen so that the number of vanishing moments n^ is large enough (n^ > max/j{ft,D(ft) ^ —oo}). These considerations are illustrated on a specific example in Figure 28 (Muzy et al. 1993a). The analyzed function is a generalized devil staircase similar to the one shown in Figure 24a (the weights are p\ = 0.84, p2 = —ps = 0.36, P4 = 0.16), but it has been fractionally integrated with /3 = 0.7 so that the support of the D(h) singularity spectrum contains 1 (a fractional integration of a signal with a coefficient /? induces a translation of the singularity spectrum to the right by a factor f3). The function is represented in Figure 28a. The theoretical D(h) singularity spectrum of this function is represented by a solid line in Figure 28d; it lies on the interval [0.83,2.04]. The results obtained using the WTMM method are represented in Figures 28b, 28c and 28d. We used the fourth derivative V ( 4 ) of the Gaussian as the analyzing wavelet. The so-obtained D(h) and r(q) spectra are in excellent agreement with the theoretical spectra. The r(q} spectrum obtained for q > 0 using the SF method is shown in
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Figure 28: WTMM and SF analysis of a multifractal signal that possesses some singularities of Holder exponents h > 1. (a) Graph of the signal. This signal is a generalized devil staircase constructed from the recursive signed measure involving the weights p\ = 0.84, P2 = —ps — 0.36 and p4 — 0.16; it has been fractionally integrated with a coefficient /? = 0.7. (b) Iog2(-2(g, a)) versus Iog2 a for some values of q. (c) r(q) spectrum obtained with the WTMM method; the analyzing wavelet is t/A 4 ). (d) D(h) singularity spectrum obtained by Legendre transforming the numerical r(q) data in (c). (e) Comparison of the r(g) spectra obtained with the WTMM (•) and the SF (A) methods. The dashed line corresponds to the degenerate spectrum r(q} = q - 1. In (c), (d) and (e), the solid lines correspond to the theoretical spectra.
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Figure 28e for comparison. Along the line of the above theoretical discussion, below some critical value q < 7**, the data corresponding to the SF method systematically deviate from the analytical r(q) curve and follow the trivial behavior T(g) = q — 1. In the range q < g**, the scaling behavior of the structure functions is dominated by the weakest singularities which are misleadingly identified to h = 1. For q > q**, the WTMM and SF methods converge to a unique r(q) spectrum, in good agreement with the analytical curve. Remark: Would a higher order SF method work? We have just seen in the last paragraph that since the SF method uses explicitly the analyzing wavelet A^, it basically restricts the study of the singularity behavior to Holder exponents h G [0,1]. However, one could try to generalize this technique to "higher" or "lower" SF approaches which would allow us to capture other Holder exponents. For example, in Section 4.1, we took the "box function" A(°)(a;) = X[o,i](x) as analyzing wavelet in order to characterize multifractal measures. The corresponding Holder exponents were then found in the range [-1,0]; indeed we saw that h = a - 1 with a € [0,1] (see (90)). In the same way one could use the wavelet A^~ x ' = xX[o,i](x) to study the Holder exponents in the range [—2,—!] as well as the wavelet A( 2 )(z) = S(x + 1) - 2S(x + 1/2) + S(x) for Holder exponents in [1,2] and so on ... Thus one could imagine combining the singularity spectra obtained for each of the wavelets A^ 1 ), A' 0 ), A'1', A' 2 )... in order to get the entire D(h] singularity spectrum. Besides the fact that such a method would be numerically very "heavy", it clearly would not work because of what we could call "border effects". Indeed, we know from the former paragraphs that when using the wavelet A^^z), the Holder exponents are actually restricted to an interval [h*,h**] which is strictly included in [0,1]. Therefore one cannot really hope to combine all the "sub" spectra (obtained with the different AW analyzing wavelets) into a whole D(h) spectrum without expecting some drastic bias around all the integer values of h (Muzy et al. 1993a).
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A. Arneodo Not Everywhere Singular Fractal Functions
Smooth behavior induced phase transition in the singularity spectrum of multifractal functions.
In Section 5.1, we have pointed out that the WTMM method is very efficient as far as we use an analyzing wavelet with a number of vanishing moments n^ which is greater than hmax = max^j/i, D(h) / — oo}. Let us see what happens when this is not possible, i.e., if hmax = +00. It would mean that the analyzed function is C°° at some points. As we have just discussed, the structure function method would fail, in this situation, since the increments are unable to detect any Holder exponent greater than 1. Therefore let us concentrate on the influence of such C°° behavior on the singularity spectrum obtained with the WTMM method. For the sake of simplicity, we will consider that the fractal function /(x) = s ( x ) + r ( x ) is the sum of a multifractal singular part s ( x ) (whose maximum Holder exponent hmax is strictly smaller than +00) and a C'°° regular part r(x). Typically, s can be a generalized devil staircase, then hmax corresponds to the largest Holder exponent found on the support of the multifractal measure. The other points are not important since the wavelet transform at these points is 0 at any scale a small enough; in other words, these points do not contribute to the partition functions Z(q,a). Let T S (Q) and Ds(h] be the multifractal spectra which characterize the function s(x). We will use an analyzing wavelet tp with a number of vanishing moments n^ greater than the greatest Holder exponent hmax of s(x). Thus as r ( x } is C00, the wavelet transform of r is uniformly of the order of a™* when a is small enough (see (58)). On the other hand, as hmax is the maximum Holder exponent involved in s, wherever the wavelet transform of s is not 0, it is of an order which is uniformly larger than ahmax. As illustrated in Figure 29, the wavelet transform T^[f] is basically equal to either T^jY] (~ a"*), at the points where T^[s] - o(a n *), or T^[s\ (~ ah) anywhere else. Thus the set £/(a) of the maxima lines of / can be decomposed, at any scale a, into two subsets: the subset £ s (a) of the maxima lines of s (slightly perturbed by a term of the order of an^ due to the presence of r(x)~) and the subset Cr(a) of the maxima lines o f f around which T^[s] = o(an^} (e.g., the maxima lines which lie outside of the support of the multifractal
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Figure 29: Processing the singularities of a fractal signal perturbed by a C°° function, (a) Graph of the signal f ( x ) = s(x) + r(z), with r(x) = Rsm(8irx) and s(x) is the devil staircase constructed from the recursive measure distributed on the triadic Cantor set with the weights pi = 0.6 and p2 = 0.4. (b) WTMM skeleton computed with the Mexican hat t/>' 2 *; the small scales are at the top. The sets £ s (a) and £ r (a) of maxima lines are coded in medium gray and white respectively, (c) Along the maxima lines in £ r (a), T^.[f] ~ an* (ra,/, = 2). (d) Along the maxima lines in £ s (a), T^,[f] ~ ah where h fluctuates from line to line (h £ [hmin,hmax]).
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measure associated to the devil staircase). Thus the partition function of / splits into two parts (Bacry et al. 1993; Muzy et al. 1993a):
where Zs and Zr are the partition functions corresponding respectively to summing over the maxima lines in Cs and £r. Thus one deduces easily that there exists a critical value qcrn < 0 so that
One thus predicts the existence of a singularity in the r(q) spectrum. This nonanalyticity in the function r(q) expresses the breaking of the self- similarity of the singular signal s(x) by the C°° perturbation r(cc). In the context of the thermodynamical analogy, this phenomenon defines a phase transition (Badii 1987; Cvitanovic 1987; Grassberger et al. 1988). Below the critical value qcrn (which is the analog of the inverse of the transition temperature) one observes a regular phase, whereas for q > qcrn one switches to the multifractal phase. Let us note that this phenomenon is "meaningful" (as compared to the spurious phase transitions previously obtained with the modified SF method); indeed, it really detects the fact that there is a smooth behavior superimposed to the multifractal function. Moreover, (115) indicates that the r(q) spectrum in the "C100 phase" is governed by the number n$ of vanishing moments of the analyzing wavelet. Therefore, checking whether r(q) is sensitive to some change in the order n^ of the analyzing wavelet constitutes a very good test for the presence of a highly regular part in the signal. This phenomenon is illustrated in Figure 29. The analyzed function f ( x ) is the sum of r ( x ) = jRsin(87rx) and s ( x ) a generalized devil staircase which is the distribution function of a measure nonuniformly distributed on the triadic Cantor set with the weights p\ = 0.6 and pi = 0.4. The function f ( x ) is represented in Figure 29a. The r(q~) and D(h) spectra obtained with the WTMM method are displayed in Figures 30a and 30b respectively. Two different analyzing wavelets •iM1' (n^,(i) = 1) and ^(^ (n^/ 2 ) = 2), namely the first and the second derivative of the Gaussian function, were used to compute the wavelet transform. For q > 0, these two wavelets lead to the same numerical estimate for r(q) in perfect agreement with the theoretical curve. However,
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Figure 30: Phase transition phenomenon in the r(g) and D(h) spectra of the fractal signal perturbed by a C°° function shown in Fig. 29a. (a) r(q) spectrum obtained by the WTMM method using the analyzing wavelet ^W [(o) and (A)], V^2' [(°) and (•)] and i/>(^ [(o) and (a)]; the solid lines correspond to the theoretical curves (Eq. (115)); the dashed line corresponds to rs(q) for q < qcrit. (b) D(h) versus h from the Legendre transform of r(q). for q < 0, the results obtained with tp^ and ^M2) are different; they consist in two lines of respective slope 1 and 2. This corresponds to the phase transition predicted by the above theoretical speculations; the data for r(q) depend on n^\ r(q) = n^q (see (115)). The Legendre transform, D(h), of these two curves are represented in Figure 30c. One can check that they perfectly fit the predicted curve for q > qcrit(n^). For q < qcrit(n^), however, D(h] displays a linear fall off towards the limiting value h = 1 for ^^ and h = 2 for ^>( 2 ) (actually h = N for ij)(N)) where D(h} vanishes. This linear part is tangent to the theoretical D(h) spectrum (dashed line) and has a slope equal to qcrit(n^,). This is the signature of the phase transition phenomenon described above (Bacry et al. 1993; Muzy et al. 1993a). Remark. We have shown that a C°° component superimposed on a distribution that is not singular everywhere manifests itself in a phase transition phenomenon that masks the weakest singularities. However, since' the wavelet coefficients behave like a"* along the maxima lines created by the C00 function, by choosing n^ large enough
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and/or choosing a numerical threshold below which any local maximum is not considered, one can remove all the C°° maxima lines in Cr(a) and thus "numerically restore" the self-similarity of s ( x ) . The whole Ts(q] and Ds(h) spectra can then be estimated. To show that this procedure is actually operational, we have reproduced the WTMM analysis on the same signal but with the fourth derivative of the Gaussian function (^4)(a;)) as analyzing wavelet. The faster decrease of the wavelet coefficients along the maxima lines of Cr (7v,[/](.,a)| ( ~ a 4 ) makes more efficient the threshold discrimination of the maxima lines emanating from the singular part s ( x ) . The so-obtained T~(q) spectrum is shown in Figure 30a. Now the theoretical spectrum of the singular measure is recovered and no phase transition phenomenon is observed. Let us point out that the choice of such a threshold (or analyzing wavelet) is somewhat uncertain and strongly depends on various parameters like the number of sampling points, the relative amplitudes of r(x) and s(x) in the signal... Indeed, a more reliable way to proceed consists in choosing n^ large enough (as compared to the largest Holder exponent of s ( x ) ) so that the maxima lines induced by the regular part of the signal become easily distinguishable from the anomalously stiff decrease of the wavelet coefficients on the range of scale used to estimate the scaling exponents r(q). 5.3
Brownian Signals
As seen in Section 2.3.1, the fractional Brownian motions (fBm) BJJ(X} are Gaussian stochastic processes of zero mean; they are indexed by a, parameter H (0 < H < 1) (Mandelbrot & Van Ness 1968). Their correlation function is given by
Thus their increments are stationary Gaussian processes whose variance is
The classical Brownian motion is obtained for the value H = 1/2. The fractional Brownian motions are statistically self-similar in the sense that
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where ~ means that the two processes are equal in law (for fixed t and A). Thus, the exponent H is directly related to the Holder exponent which characterizes the realizations of the Brownian processes. One can prove that almost all the realizations are continuous, everywhere nondifferentiable and characterized by a single Holder exponent h = H (Levy 1965). In the multifractal formalism framework we will say that the fBm's are homogeneous, i.e., their singularity spectrum reduces to a single point
There have been previous attempts to analyze fBm with the wavelet transform (Argoul et al. 1989b; Bacry et al. 1991; Everson et al. 1990). When using a color coding similar to the one used in the previous sections, the wavelet representation in the space-scale half-plane of a fBm clearly displays some kind of fractal branching (Figure 3la) as observed in the analysis of generalized devil staircases (Figure 25). Recently, Vergassola & Frisch (1991) and Vergassola et al. (1991) have tried to pursue this analysis on a more quantitative basis, using the wavelet analysis to characterize the local Holder regularity of these random signals. But they have been facing the presence of random fluctuations, superimposed to the pertinent power-law behavior of the wavelet transform at small scales (see (57)), that makes quite uncertain the estimate of the local scaling exponents. In order to circumvent this difficulty, they have established an ergodic formula from which one can extract the Holder exponents from zoom-averagings over logarithmically varying scales. In fact, this ergodic formula indicates that in principle, Holder exponent measurement of fBm is possible. In practice, however, the available finite range of scales is not sufficient for the averaging process to converge and the above mentioned fluctuations result in an important scatter of the measured scaling exponents around the value h = H (Vergassola et al. 1991). In Figures 32 and 33, we report the results of a statistical analysis of the fBm using the WTMM method (Arneodo et al. 1991; Muzy et al. 1991). We focus on the fBm Bl/3 since it has a k~~s power spectrum similar to the spectrum- of the turbulent velocity signal investigated in the next section. This will allow us to clearly discriminate between these two fractal signals. The numerical signal
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Figure 31: Continuous wavelet transform of (a) top picture: a fractional Brownian signal 5 1 / 3 (i) (Fig. 32a) and (b) bottom picture: a turbulent velocity signal recorded in the Modane wind tunnel (Fig. 34a). The analyzing wavelet is the Mexican hat ^'2'. Same color coding as in Fig. 13. The small scales are at the top.
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Figure 32: WTMM skeleton of a Brownian signal, (a) A realization of the fractional Brownian motion BI/Z. (b) Wavelet transform maxima lines corresponding to the realization in (a). The analyzing wavelet is the Mexican hat ^>( 2 '. The large scales are at the top. was generated by filtering uniformly distributed pseudo-random noise in Fourier space to have the required k~s spectral density (Peitgen & Saupe 1987). A jfrt/3 fractional Brownian trail is shown in Figure 32a. The corresponding WTMM skeleton computed with the Mexican hat ^2\x), is illustrated in Figure 32b. When plotted versus g, the exponents r(q) extracted from the scaling behavior of the partition function Z(q,a) (see (92)), consistently fall on a line of slope h = 0.33 ± 0.01. Moreover, Figure 33b shows that the theoretical prediction
provides a remarkable fit of the data. According to Equation (120), [0,3 Z ( q , a ) } l / ( q ~ 1 } is expected to scale like a1/3, independently of q. The data reported in Figure 33a for different values of q are in good agreement with this general scaling law. The homogeneity of the fBm
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Figure 33: Determination of the r(g) and D(h) spectra for a fractional Brownian motion #1/3 using the WTMM method, (a) log 2 (a~ T ^'-2(g, a))/(g — 1) versus Iog2 a. (b) r(q) versus q. (c) Determination of the exponent h(q] from Eq. (97). (d) D(h) versus h, The analyzing wavelet is the Mexican hat t/^2'. In (b) the solid line represents the theoretical spectrum r(q) = g/3 — 1 (Eq. (120)). BI/Z signal is confirmed in Figure 33c where the direct estimate of the exponent h(q) from (97), for different values of q, does not reveal any significant g-dependence of this exponent: h(q] — 0.33 ± 0.01. Similarly, from (98) one gets D(h) - 1.001 ± 0.002 for all considered q values. These numerical results are in remarkable agreement with the theoretical singularity spectrum, h = // = 1/3; D(h — 1/3) = 1
Wavelet Analysis of Fractals
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(see (119)). As expected theoretically, we find that the fractional Brownian motion B l / ^ ( x ) is nowhere differentiate with a unique Holder exponent h = H = 1/3.
6
6.1
Wavelet Analysis of Fully-Developed Turbulence Data
Multifractal Approach of the Intermittency Problem
The central problem of three-dimensional fully developed turbulence is the energy cascading process. It has resisted all attempts at a full understanding or mathematical formulation. The main reasons for this failure are related to the large hierarchy of scales involved, the highly nonlinear character inherent in the Navier-Stokes equations and the spatial intermittency of the dynamical active regions (Monin & Yaglom 1971; Frisch & Orszag 1990). In this context, statistical and scaling properties have been the basic concepts used to characterize turbulent flows (Frisch 1985, 1991). One of the striking signatures of the so-called intermittency phenomenon, is the nongaussian statistics at small scales. The energy transfer towards small scales is related to the nonzero skewness of the probability distribution function (PDF) of the velocity increments and the large flatness of the PDF (kurtosis) corresponds to the presence of strong bursts in the energy dissipation. This fine-scale intermittency is responsible for some departure to the classical k~5/3 theory of Kolmogorov (1941) which neglects the presence of fluctuations in the energy transfer. Mandelbrot (1974) was the first one to advocate the use of fractals in turbulence. Some of his early multiplicative cascade models contained all the ingredients of the classical multifractal formalism described in Section 2.2.3. During the past few years, considerable effort has been devoted to the multifractal analysis of high Reynolds number turbulence (Frisch & Orszag 1990). But the problem of comparing the predictions of various multifractal cascade models (Novikov & Stewart 1964; Mandelbrot 1974, 1984; Frisch et al. 1978; Benzi et al. 1984; Nelkin 1989) with experimental data comes from the fact three-dimensional processing of turbulent flows is at the moment feasible only for numerical simulations which are unfortunately lim-
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ited in Reynolds numbers to regimes where the scaling just begins to manifest itself. Present experimental techniques have access to the two-dimensional structure of passive scalars (Prasad et al. 1989; Miller & Dimotakis 1991) and only to the one- dimensional structure of the velocity field (Anselmet et al. 1984; Gagne 1987; Gagne et al. 1988; Castaing et al. 1990; Meneveau & Sreenivasan 1991). Here, we will mainly elaborate on the statistical analysis of singlepoint data based on hot-wire techniques in the presence of a mean flow (wind tunnels, jets, etc..). There are mainly two approaches to multifractals in turbulence. The first one is based on the determination of the generalized fractal dimensions Dq and the /(a) singularity spectrum of the energy dissipation (Mandelbrot 1974; Paladin & Vulpiani 1987; Meneveau & Sreenivasan 1987, 1991; Chhabra et al. 1989). The second one consists in directly investigating the distribution of singularities of the velocity field itself via the scaling behavior of the structure functions (Van Atta & Chen 1970; Van Atta & Park 1972; Anselmet et al. 1984; Frisch & Parisi 1985; Gagne 1987; Gagne et al. 1988; Castaing et al. 1990). 6.1.1
Multifractal description of the energy dissipation
Following a tradition going back to Kolmogorov (1941), Mandelbrot (1974) originally proposed to investigate the fluctuations of the local energy dissipation. But the data obtained from point-probe measurements give only access to the stream (longitudinal) velocity component v(t). Using the Taylor's frozen flow hypothesis and further assuming that the gradient of only one velocity component in one direction is representative of the dissipation, the dissipation c is generally defined as (Meneveau & Sreenivasan 1987, 1991):
The scaling behavior of e (considered as a measure distributed over the real axis) has been studied over domains of size pertaining to the inertial range: /^ « / « / 0 , where /^ is the Kolinogorov dissipation scale and IQ the integral scale. Classical techniques (e.g., boxcounting algorithms) issued from the multifractal theory have been used to extract the generalized fractal dimensions Dq and the /(«)
Wavelet Analysis of Fractals
443
singularity spectrum (Section 2.2). The consistency of the measured spectra brings conspicuous evidence for the multifractal nature of the dissipation field (Meneveau & Sreenivasan 1987, 1991; Chhabra et al. 1989): Dq is a monotonic decreasing function of q while /(a) turns out to be a single humped function. The detailed comparison performed by Meneveau and Sreenivasan (1987, 1991) of their experimental results with the predictions of various cascade models clearly demonstrates the inadequacy of log-normal models (Obukhov 1962; Kolmogorov, 1962) and of cascade models with a single exponent (Novikov & Stewart 1964; Mandelbrot 1974, 1982) including the so-called /3-model (Frisch et al. 1978). Surprisingly, the simplest version of weighted curdling models proposed by Mandelbrot (1974), namely the binomial model, turns out to account reasonably well (at least at a certain level of description) for the observed multifractal spectra. Even though, owing to the degeneracy of the multifractal formalism (Feigenbaum et al. 1986; Chhabra et al. 1989), one cannot reasonably expect to extract the turbulent fragmentation process from the /(a)-spectrum information only, the binomial models give undoubtedly the simplest possible mechanism that reproduces most of the experimental observations recorded so far on the turbulent energy dissipation. 6.1.2
Multifractal description of the velocity field using the structure functions
An alternative description of the fine-scale intermittency phenomenon consists in investigating the singular aspect of the longitudinal velocity signal v ( x ) itself (Van Atta & Chen 1970; Van Atta & Park 1972; Anselmet et al. 1984). This approach pioneered by Frisch & Paris! (1985) relies on the determination of the spectrum D(h) of Holder exponents h of the velocity field from the inertial scaling properties of the structure functions (Monin & Yaglom 1971) (SF) of variable order (see (36)):
where 6vi(x) = v(x + /) - v(x] is the longitudinal velocity increment over a distance /. As later confirmed by Gagne and collaborators (Gagne 1987; Gagne et al. 1988; Castaing et al. 1990), the exponents (p were found to depend nonlinearly on p, deviating significantly
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from the prediction (p = p/3 of Kolmogorov (1941) based on the assumption that, at each point of the fluid, the velocity field has the same scaling behavior Svi(x) ~ I1/3, which yields the well-known E(k) ~ k~5'3 energy spectrum. This observation, however, does not rule out the notion of scaling invariance which might in fact be relevant at a local level. The idea of Frisch & Parisi (1985) was to interpret this nonlinearity as a direct consequence of the existence of spatial fluctuations in the scaling behavior of dvi(x) ~ lh(x) (see (34)). For each h there is a set in 1R of Hausdorff dimension D(h) for which Svi ~ lh. Thus, by suitably inserting this local scaling behavior into (122), one can bridge D(h) and (p by a Legendre transform (see (37)): which is, a priori, the counterpart of the Legendre transform which relates the /(a) and r(q) = (q—\)Dq spectra of multifractal measures (see (20)). It is clear from (123), that the nonlinear dependence of (p on p is equivalent to the assumption that there is more than a single scaling exponent h. However, as emphasized in Section 2.3.2, the structure function method has severe drawbacks and it cannot be considered as a reliable generalization of the multifractal formalism to fractal functions. Let us recall that the weaknesses of the structure function method were the main motivation that led us to introduce, in Section 4, the wavelet transform modulus maxima method as an alternative approach to the classical multifractal formalism originally proposed for singular measures (Arneodo et al. 1991; Muzy et al. 1991, 1993a, 1993b; Bacry et a/.1993). 6.1.3
The Kolmogorov's refined similarity hypothesis
Even though there is, a priori, no obvious relation between the D(h) singularity spectrum of the velocity signal and the /(a) singularity spectrum of the dissipation, the experimental results on the velocity structure functions (Anselmet et al. 1984; Gagne 1987; Gagne et al. 1988) and the energy dissipation (Meneveau & Sreenivasan 1987, 1991) converge to the same conclusion: a multifractal description of the intermittency of the fine structures is very appealing whereas "monofractal" homogeneous cascading models seem to be quite inadequate.
Wavelet Analysis of Fractals
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There has been, however, some attempts (Meneveau & Sreenivasan 1991; Frisch, 1991) to bridge the two above multifractal approaches by assuming that the local dissipation €f(x) averaged over a domain of size £ and the increment 8vt(x) are related locally by the following form (in the limit I —>• 0) (Kolmogorov 1962):
where the symbol ~ means that the two quantities have the same scaling laws, so that the scaling exponents are the same for corresponding moments of arbitrary order. Equation (124) is essentially the global scaling assumption of Kolmogorov (1941), reformulated in terms of fluctuating local quantities (Frisch 1991). The relevance of this local scaling relation turns out to be very much debated in recent numerical (Hosokawa & Yamamoto 1991; Thoroddsen & Van Atta 1992; Chen et al. 1992) and experimental (Stolovistky et al. 1992; O'Neil & Meneveau, 1992; Gagne et al 1994) studies. Taking this relation for granted one gets the following connection between the two multifractal approaches:
Previous attempts to check experimentally those relations have not revealed any inconsistency (Stolovistky et al. 1992; O'Neil & Meneveau 1992: Gagne et al. 1994). But these tentative comparisons are still a far cry from a quantitative experimental validation. Actually, beyond the technical difficulties related to point-probe measurements (Aurell et al. 1992) there are some limitations to the previous experimental analysis. In particular, it is clear that the squared derivative of the longitudinal velocity in (121) may not be fully representative of the 3-D local dissipation (Bershadskii & Tsinober 1993).
6.2
Wavelet Analysis of Single-Point Turbulence Data
In recent years, there has been increasing interest in applying the wavelet analysis to turbulence data (Farge 1992). In this section, we report on the first such analysis performed on single point velocity data from high Reynolds number 3-D turbulence (Argoul et al. 1989b;
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Figure 34: WTMM skeleton of a turbulent velocity signal recorded in the wind tunnel SI of the ONERA at Modane. (a) The velocity signal over about one integral scale, (b) WTMM skeleton denned by the maxima lines. The analyzing wavelet is the Mexican hat i j j ^ , The large scales are at the top. Bacry et al. 1991). The data were obtained by Gagne and collaborators (Gagne 1987; Gagne et al. 1988; Castaing et al. 1990) in the large wind tunnel SI of ONERA at Modane. A hot wire probe was located on the axis of the return section which was 24 in in diameter and 150 m in length. This probe was used to record the streamwise component of the velocity (Figure 34a). The turbulence level, expressed by the ratio of the r.m.s. value of the longitudinal velocity fluctuations to the mean velocity, was less than 7 %, which implies that time variations are equivalent to spatial variations (the Taylor hypothesis). The Reynolds number based on the Taylor microscale is R^ = 2720, i.e., more than one order of magnitude larger than the Reynolds numbers of the turbulent laboratory flows analyzed by Meneveau & Sreenivasan (1987, 1991). (Note that these authors have also studied atmospheric flows at higher Reynolds numbers: R\ ~ 1500.) The extent of the inertial range following approxi-
Wavelet Analysis of Fractals
447
mately the Kolmogorov &~ 5 / 3 law is almost three decades. The integral scale is /0 = 15 m, while the dissipation scale /<j = 0.3 mm is too small to permit accurate analysis in the dissipation range. The results reported in this section concern the analysis in the inertial range of about 100 integral length scales of the recorded velocity turbulent signal. 6.2.1
Local scaling exponents of a turbulent velocity signal
The application of the continuous wavelet transform to investigate the local scaling exponent fluctuations that characterize the multifractal nature of a turbulent velocity field at inertial range scales has been initiated by Bacry et al. (1991). Figure 31b illustrates the wavelet transform of a sample of the velocity signal of length of about one integral scale. The WTMM skeleton in Figure 34b is actually hardly distinguishable from the WTMM arrangement obtained in Figure 32b for a fractional Brownian signal BI/^(X) which has a similar A;~5/3 power spectrum. However, when using the additional information given by the WT amplitude (Figure 31b), this discrimination becomes easier since the coded wavelet transform displays a characteristic multifractal branching structure in the space-scale half-plane, that is different from the (homogeneous) fractal branching observed for the fractional Brownian motion. In particular, dynamically significant events having strong localized gradients appear as red tongues suggesting the existence of dominating singularities that are not seen in Gaussian processes. Actually, this is not surprising since it is now well-established experimentally that the small-scale statistics of the turbulent velocity signal is far from Gaussian (Anselmet et al. 1984; Gagne, 1987; Gagne et al 1988; Castaing et al. 1990). In a preliminary work, Bacry et al. (1991) have carried out a systematic investigation of the local scaling properties of the turbulent velocity signal, by analyzing the behavior of the wavelet transform as a function of the scale a at different positions XQ. According to (57), a log-log plot of |r^[w](a;o, a)| versus a should reveal the local Holder exponent h(xg). Examples of such log-log plots are given in Figure 35. In Figures 35a-c, the exponent h is estimated from the decay of the wavelet transform along vertical lines in the (a;, a) half-plane, i.e., for fixed XQ. We see on these figures that at scales smaller than the integral scale /Q (the scale we use for a is such that
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Figure 35: Local scaling exponents revealed in log-log plots of the amplitude |T^,[w](a;,a)| of the wavelet transform, of the Modane turbulent velocity signal versus the scale parameter a. In (a), (b) and (c), h is obtained by linear regression fit along a vertical cut x = x0 in the ( x , a) half-plane. In (d), h is estimated following a line of local maxima. The analyzing wavelet is the Mexican hat -^M2'. a = es corresponds approximately to /Q)> \T^,[v](x0,a) has a power law behavior with superimposed oscillations. Such oscillations are unavoidable in any situation where fractal branching occurs in the wavelet transform representation. (Note that for a larger than /o> this oscillatory scaling law breaks down simultaneously to the disappearance of the hierarchical branching process). Using a least squares fit to a power-law over inertial-range scales, one can nevertheless estimate the local scaling exponent (Vergassola fe Frisch 1991). The meaningful cases shown in Figure 35, have been selected from the set of investigated points (among the 4.106 points in our experimental sample) where a rather convincing scaling behavior is observed. As shown in Figure 35d, the determination of the local scaling exponent is much cleaner when following a line of local maxima in
Wavelet Analysis of Fractals
449
Figure 34b (see (61)). In contrast to the Brownian signal BI/S(X) (Section 5.3), the Holder exponent is found to fluctuate in a wide range between —0.3 and 0.7 (Arneodo et al. 1991; Bacry et al. 1991), thereby suggesting that the multifractal picture proposed by Frisch and Parisi (1985) is appropriate. Statistically, the most frequent exponents are close to the Kolmogorov value h = 1/3. We believe that the data are statistically significant for negative exponents down to —0.1 and beyond. Negative exponents do not seem to have been previously reported in the literature. One possible interpretation proposed by Bacry et al. (1991) is the occasional passage near the probe of slender vortex filaments of the sort observed in recent experiments (Douady et al. 1991) and 3-D numerical simulations (Siggia 1981; Brachet et al. 1983; Brachet 1990, 1992; She et al. 1990; Vincent & Meneguzzi 1991). These events with negative exponents are not encountered very often and surely not in every sample of integral scale length. In the framework of the probabilistic formalism recently developed by Mandelbrot (1988, 1989a, 1989c), these rare events are likely to contribute to the latent part of the D(h) singularity spectrum, i.e., to the part which corresponds to negative values of D(h). This might mean that the vortex filaments can approach the probe arbitrarily close so that there is a finite probability for a sphere of radius / (an inertial-range scale) to have nonempty intersection with both the probe and a filament; the decay of this probability with decreasing / gives negative dimension. But the examples reported in Figure 35 are only representative of the points in the experimental sample where a rather convincing scaling behavior is observed. Unfortunately, for most of the points, the scaling is not so well-defined, e.g., the superimposed oscillations are too large, and the determination of the Holder exponent is very uncertain. Even though we believe that the surprisingly large range of measured local scaling exponents is probably statistically meaningful, the quantitative determination of the D(h} singularity spectrum from local wavelet transform analysis turns out to be an elusive goal (Vergassola et al. 1991).
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A. Arizeodo
Figure 36: WTMM measurement of the r(q) and D(h) spectra of the Modane turbulent velocity signal, (a) log 2 (a~ T ' 1 'Z(g, a))/(q— 1) versus Iog2 a. (b) r(q) versus q. (c) Determination of the exponent h(q) from Eq. (97). (d) D(h) versus h. In (b) the symbols ( x ) correspond to i~(q) — C? — 1 obtained when computing the scaling exponent (g with the structure function method. In (d) the solid line corresponds to the average singularity spectrum obtained from dissipation field data via the Kolmogorov scaling relation (124). The results reported in this figure concern the analysis in the inertial range of about 100 integral length scales of the turbulent velocity signal.
Wavelet Analysis of Fractals 6.2.2
451
Singularity spectrum of a turbulent velocity signal using the wavelet transform modulus maxima method
Figure 36 shows the results of the multifractal analysis of the Modane turbulent velocity signal performed with the WTMM method (Arneodo et al. 1991; Muzy et al. 1991, 1993b). In contrast to the fractional Brownian motion (Figure 33), the r(q) spectrum obtained for the turbulent signal unambiguously deviates from a straight line. Let us note that the results obtained with the structure function method r(q) = (q — 1 (exclusively obtained for positive integer values of q] are in good agreement with the nonlinear behavior of r(q) found with the WTMM method. The values of h = dr(q)/dq when varying q from —30 to 30 range in the interval [0.10, 0.62]. The corresponding D(h) singularity spectrum obtained by Legendre transforming r(q) is shown in Figure 36d. Its characteristic single humped shape over a finite range of Holder exponents is a clear signature of the multifractal nature of the turbulent velocity signal. For q — 0, the largest dimension is attained for singularities of exponent h(q — 0) = 0.335±0.005, i.e., a value which is very close to the Kolmogorov (1941) prediction h = 1/3. Moreover, the corresponding maximum of the D(h) curve, D(h(q = 0)) = -r(0) = 1.000 ± 0.001 does not deviate substantially from Dp = 1. This strongly suggests that the turbulent signal is everywhere singular. This observation is corroborated by the robustness of the D(h) data with respect to changes in the shape of the analyzing wavelet: similar quantitative estimates of the r(q) and D(h) spectra are obtained when using the first (V->^), the second (t/J^) and the fourth (V>^) derivative of the Gaussian function and no wavelet dependent phase transition phenomenon of the type described in Section 5.2 is observed (Arneodo et al. 1991; Muzy et al, 1991, 1993b). In Figure 36d, the D(h) singularity spectrum of the wind tunnel velocity signal is compared to a solid curve which actually corresponds to a common fit of dissipation field data at lower Reynolds number (Meneveau & Sreenivasan 1991). This curve has been deduced from the experimental average /(a) spectrum of the energy dissipation e(ar) = (dv/dx)2 (see (121)) of laboratory and atmospheric turbulent flows by using the local Kolmogorov scaling relation (see (124)). The fact that, for similar statistical samples, one cannot discriminate between these two singularity spectra within the experimental uncertainty, can be interpreted a posteriori as an ex-
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perimental verification of the above Kolmogorov hypothesis. This observation can also be understood as an experimental confirmation of the universality of the multifractal singularity spectrum of fully developed turbulence with respect to Reynolds number. However, it is clear that considerable further work is needed before definitive conclusions can be drawn. In particular, long term statistical analysis must be carried out in order to capture more accurately the latent part (D(h) < 0) (Mandelbrot 1989c) of the singularity spectrum, including possible violent rare events that do not occur in every sample of inertial length scale. This analysis is currently in progress. It is likely to provide fundamental information about the true role played by the vortex filaments in the intermittency phenomenon of the fine structures in fully developed turbulent flows (Arneodo et al. 1994).
7
7.1
Wavelet Analysis of Fractal Growth Phenomena
Laplacian Growth Phenomena
Interfacial growth processes in systems far from equilibrium are a subject of considerable current interest (Pietronero & Tosatti 1986; Stanley & Ostrowsky 1986, 1988; Guttinger & Dangelmayr 1987; Feder 1988; Aharony & Feder 1989; Pietronero 1989; Vicsek 1989; Family & Vicsek 1991; Bunde & Havlin 1991; Ben Amar et al. 1991; Garcia-Ruiz et al. 1993). Growth phenomena that are controlled by diffusion processes are ubiquitous in many physical, chemical, biological and geophysical systems. Notable examples include viscous fingering, electrochemical deposition, dendritic solidification, colloidal aggregation, dielectric breakdown, the growth of bacterial colonies and neuronal outgrowth. In these systems, the growth results from the interaction between the growing interface and some surrounding field $ that satisfies the Laplace equation A$ = 0 (Evertsz 1989; Pietronero 1989; Vicsek 1989). In viscous fingering, $ is the pressure field, while in electrochemical deposition and dielectric breakdown, it is the electrochemical and electrostatic potential respectively. These systems have recently attracted considerable attention, since they exhibit a rich variety of geometrical patterns including fascinating frac-
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453
tal morphologies (Pietronero & Tosatti 1986; Stanley & Ostrowsky 1986, 1988; Guttiiiger fe Dangelmayr, 1987; Feder 1988; Pietronero 1989; Vicsek 1989; Ben Amar et al. 1991; Bunde & Havlin 1991; Garcia-Ruiz et al. 1993). In this context, the diffusion-limited aggregation (DLA) model, introduced by Witten and Sander (1981, 1983) in the early eighties, has played a major role since it has stimulated considerable experimental, numerical and theoretical effort (Meakin, 1988; Evertsz 1989; Pietronero 1989; Vicsek 1989; Bunde & Havlin 1991). In this prototype model, particles released one at a time from far away, execute a random walk until they touch a cluster site, stick to it immediately and permanently. This model accounts for physical situations where the characteristic time of the diffusion process is much larger than the characteristic time of the sticking reaction. The corresponding Laplace field is the probability P of visit of a site (AP = 0) and the normal velocity of growth of a local region of the interface depends linearly on the normal gradient of P: Vn oc (VP) • n. On-lattice and off-lattice computer simulations of the DLA model (Witten & Sander, 1981, 1983; Nittman & Stanley 1986; Meakin 1988; Vicsek 1989) have produced complex, apparently randomly branched aggregates that bear a striking resemblance to the tenuous tree-like structures observed in nature or experiments. But despite its appealing simplicity, the DLA model has resisted all attempt at a full physical understanding and no rigorous theory of diffusion-limited aggregation is available at the present time. One of the main obstacles to theoretical progress lies in the lack of structural characterization of the growing clusters. Most of the previous studies have mainly focused on the multifractal analysis of either the DLA geometry or the growth probability distribution along its boundary (Witten & Sander 1981, 1983; Meakin 1988; Vicsek 1989; Stanley et al. 1990). In particular, these aggregates were found to be homogeneous; within the numerical uncertainty, the generalized fractal dimensions are all equal and coincide with the fractal dimension Dq = D$LA = 1.60 ± 0.03, Vg (Meakin & Havlin, 1987; Argoul et al. 1988; Li et al. 1989; Argoul et al. 1990). This means that the mass locally behaves as a power-law of the length scale with a unique scaling exponent a ~ 1.60, which is independent of the point chosen on the cluster. Although these results definitely establish the statistical self-similarity of the DLA clusters, they are insufficient because they bring only limited information about the puzzling DLA architecture. In particular, it is still an open question whether or not
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some structural order is hidden in the apparently disordered DLA morphology. In this section, our purpose is to use the wavelet transform microscope explore the intricate fractal geometry of large mass off-lattice DLA clusters (Argoul et al. 1989a; Arneodo et al. f989b; Argoul et al. 1990; Arneodo et a/.1992b). When focusing this microscope on the internal "extinct" region of these clusters and zooming, one reveals the presence of Fibonacci sequences in the ramification process (Arneodo et al. 1992c; Arneodo et al. 1992d, 1992e; Arneodo et a/.1993b, 1993c). This analysis also indicates that this underlying hierarchy is likely to be intimately related to a predominant structural five-fold symmetry (Arneodo et al. 1992d, 1992e). These new structural ingredients strongly suggest an interpretation of the DLA morphology as a quasi-fractal counterpart of the well-ordered (crystalline) snowflake geometry (Arneodo et al. 1992d). 7.2
7.2.1
Wavelet Analysis of Fractal Measures over ]Rn
Definitions
When one intends to generalize to more than one dimension the whole continuous wavelet machinery developed in Section 3 for the ax + 6 affine group of translations and dilations, it is natural to consider theTC-dimensionalEuclidean group (translations and rotations) with dilations (Murenzi 1989, 1990). This group is a nonunimodular locally compact group. fZ(6, a, r) defined below is its most natural unitary representation in i2 (SLn,dnx), which turns out to be both irreducible and square integrable:
where V € i2 (lR",d"x), b e IR™ is the displacement vector, a the dilation parameter (a > 0) and r the n-dirnensional rotation operator (r 6 50(n)). The wavelet transform of a function /(£) G i2 (lR n ,d"a;) is then
Wavelet Analysis of Fractals
455
defined as follows (Murenzi 1989, 1990):
Throughout this section we mainly consider real analyzing wavelets if}. According to (127), the wavelet transform is basically the scalar product of the function with the mother function dilated by a, rotated with r and translated by b. In the Fourier domain, (127) can be rewritten in the following way:
where f ( k ) = / e~lkxf(x)dnx ss represents the Fourier transform of f ( x ) . Thus at a fixed scale a, the wavelet transform amounts to filtering the Fourier spectrum of the function / with the band-pass filter ^(ar~lk). This remark is at the heart of the principle of the optical device that has been designed and constructed at the Centre de Recherche Paul Pascal (Freysz et al. 1990; Pouligny et al. 1991; Arneodo et al. 1992b; Muzy et al. 1992) and which performs experimentally the 2D wavelet transform in real time. The Optical Wavelet Transform has been proved to resolve geometrical multifractality; in that respect it is a definite step beyond classical scattering techniques as emphasized by Arneodo et al. (1992b). Inversion formula
Similar to the 1-D case, for the wavelet transform to be invertible, the wavelet ^>(aT) must satisfy the admissibility condition
The reconstruction formula then becomes (Murenzi 1989, 1990):
where 6^(6, a, r) = a~(n+l>drdnbda is invariant under the action of the operators 0(6, a, r) denned in (126). The energy conservation
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formula becomes
Some examples of analyzing wavelets Among all the possible choices of 2D analyzing wavelets, we will mainly describe two wavelets which are currently used to analyze fractal aggregates grown in the plane. Example. The 2-dimensional Mexican hat. The 2D Mexican hat is denned in the following way (Murenzi 1989, 1990):
where A is a positive definite matrix and B = A"1. For the particular choice A = B = 11, one recovers the so-called radial Mexican hat:
The radial Mexican hat is illustrated in Figure 37. When the eigenvalues of the matrix A are different, the Mexican hat becomes an anisotropic wavelet (Antoine et al. 1991). The optical wavelet The simplest acceptable shape for the kind of filter to be used in the optical wavelet transform experimental set up is a binary approximation of the Fourier transform of the radial Mexican hat, namely the band-pass filter (Freysz et a/,1990; Pouligny et al. 1991; Arneodo et al. 1992b) (Figure 38a):
The corresponding analyzing function in the real space is sketched in Figure 38b; it is simply given by the expression:
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Figure 37: The radial Mexican hat defined in Eqs. (134) and (135).
jjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj where J\ is the first order Bessel function of the first kind.
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Figure 38: The optical analyzing wavelet, (a) Shape of the bandpass filters used in the optical wavelet transform set-up (Eq. (136)). (b) Shape of the corresponding analyzing wavelet in the real space (Eq. (137)). 7.2.2
Wavelets and local scaling properties of fractal aggregates
In this section we will mainly concentrate on the analysis of fractal aggregates embedded in a two-dimensional space. Local selfsimilarity of these aggregates means that the mass contained in an
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Figure 39: Wavelet transform of the deterministic (a-d) and random (a'-d 7 ) snowflake fractals shown in Figs 40a and 40b respectively. T!0(6, a) is coded using 32 shades from white (T,/, < 0) to black (maxT1,/, > 0) for each value of the scale parameter a = a* (a,a'), a*/3 (b.b7), a*/32 (c,c') and a*/33 (d ;; dd')- The analyzing wavelet is the Mexican hat (Eq. (134)).
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e-ball BgQ(e), centered at the point XQ, scales as
where the local scaling exponent a(xo) a priori depends upon the position XQ, i.e., the mass may scale differently from point to point. Now, let us generalize the definition of the 2D wavelet transform (127) to measures in the following way:
where we have implicitly supposed that the analyzing wavelet tp is radially symmetric. Then, in the same spirit we derived (63) from (62) in one dimension, the scaling behavior of the measure fj, (see (138)) is mirrored in the wavelet transform which scales, in the limit e -> 0+, like (Argoul et ai 1989a, 1990; Arneodo et al. 1992a)
provided the analyzing wavelet decays sufficiently fast at infinity. At this point, let us mention that on the basis of usual pointwise dimension calculations (138), we will disregard anisotropic effects in the local scaling behaviour of the measure under consideration, which explains our exclusive use of radially symmetric analyzing wavelets. In two-dimensions, according to the definition of (139) of the wavelet transform, the analogy with a mathematical microscope is patent: the translation parameter b and the dilation parameter a correspond respectively to the position and the inverse of the magnification of the WT microscope; the performance of the optics is determined by the choice of the analyzing wavelet 0. The optical wavelet transform (Freysz et al. 1990; Pouligny et al. 1991; Arneodo et al. 1992b; Muzy et al. 1992) can be seen as an experimental achievement of this analogy. From Equation (140), the WT microscope can thus be regarded as a very efficient singularity scanner (Argoul et al. 1989a, 1990; Arneodo et al. 1992a). It allows us (i) to locate the singularities of fj,: each singularity produces a cone-like structure in the wavelet transform representation pointing towards the point XQ when the magnification a~l is increased; (ii) to estimate the strength of the
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singularities of fj,: each singularity manifests itself through a powerlaw behavior of the wavelet transform in the limit a —>• 0+, with exponent a(£o)- Tne local scaling exponent a can thus be measured from a linear regression fit of the log-log plot of |Jy,[)u](xo,a)| versus the scale parameter a. Practically, the situation is somewhat more intricate due to the existence of a hierarchical distribution of singularities. Actually, the value of the exponent a(a?o) '1S governed by the singularities which accumulate at x0. This results in unavoidable oscillations around the expected power-law behavior of the WT amplitude. These oscillations contain fundamental information about the underlying multiplicative process (if any), but they can sometimes make the exact determination of a (from log-log plots on a finite range of scales) somewhat uncertain as previously pointed out when studying the one-dimensional cases in Section 4.1 (Argoul et al. 1990; Arneodo et al. 1992a). 7.2.3
Determination of the singularity spectrum of fractal aggregates from the wavelet analysis
Very much like in one dimension, the multifractal formalism can be revisited using wavelet analysis. The classical formalism relies on the computation of the partition function:
where the sum is over a partition of the support of the singular measure p, into boxes of size e. Since a wavelet can actually be seen as a smooth oscillatory variant of a characteristic function of a box, one can define a partition function in terms of the wavelet components (Arneodo et al. 1992b; Muzy et al. 1992):
from which one can again determine the Border generalized fractal dimensions in the thermodynamic limit a —> 0 + :
The f ( a ) singularity spectrum can thus be practically computed by Legendre transforming the partition function scaling exponents
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Let us note that the fact that the sum in (142) is restricted to the points that belong to the support of the measure is of fundamental importance to "stabilize" the computation of Z(q,a) for the values q < 1. In the spirit of the wavelet transform modulus maxima method introduced in Section 4.3 in the one-dimensional case, the sum should be restricted to the skeleton of the wavelet transform defined by its maxima lines. The implementation of a two-dimensional version of the WTMM method is currently in progress. From a practical viewpoint, this multifractal formalism revisited with wavelets has made the generalized fractal dimensions Dq and the /(a) singularity spectrum accessible experimentally to the optical arrangement that performs the wavelet transform using coherent optical spatial filtering (Arneodo et al. 1992b; Muzy -et al. 1992). In that respect, the optical wavelet transform is without any doubt, a fundamental step beyond classical scattering techniques (X-rays, neutrons or light), which are based on Fourier analysis and therefore are not able to resolve geometrical multifractality; these techniques actually give access to the two-point correlation dimension Dq = 2 only (Allain & Cloitre 1986; Teixeira 1986). 7.3
Wavelet Analysis of Snowflake
Fractals
The class of fractal aggregates possessing a recursive (mutiplicative) structure provides analytically tractable cases. It is thus welladapted to test the efficiency of the WT microscope to characterize the scaling properties of fractals. In this section, we will also illustrate its amazing ability to reveal the structural hierarchy of fractal and multifractal snowflakes that display a well-organized "crystalline" fractal architecture (Argoul et al, 1989a, 1990). 7.3.1
Deterministic and random snowflake fractals
Dendritic crystal growth enters the wide class of pattern formation in a diffusion field (Pietronero & Tosatti 1986; Stanley & Ostrowsky 1986, 1988; Guttinger & Dangelmayr 1987; Feder 1988; Vicsek 1989; Ben Amar et al. 1991). In general, dendrites have a well-organized
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Figure 40: Wavelet transform analysis of a deterministic (a), a ranr dom (b) and a multifractal (c) snowfiake. (d) illustrates the determination of local scaling exponents at different points of the deterministic and random snowflakes; the dashed lines correspond to the theoretical prediction a = In 5/In 3. In (e) and (f) are reported the experimental determinations of the generalized fractal dimensions Dq of the deterministic one-scale snowfiake (a) and the multifractal twoscale snowflake (c) respectively, when using the OWT set-up that performs the wavelet transform optically; the solid lines correspond to the theoretical spectra.
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branched structure, such as snowflake patterns. In Figure 40a, we show a one-scale snowflake fractal which is commonly thought of as a paradigm for self-similar fractal aggregates (Vicsek 1983). Its construction rule can be considered as a deterministic model for aggregation process. The configuration at the nitl stage is obtained by adding to the four corners of the (n — l)th stage configuration, the cluster corresponding to the (n — l)th stage. Therefore, from one stage of the construction to the next, the cluster size increases by a factor \L = 3, while the number of motifs is multiplied by a factor \B = 5. A straightforward calculation yields:
By Legendre transforming the Dq's according to (144), one quantifies the homogeneity of the deterministic snowflake into a single scaling index a = In 5/ In 3 with density f ( a = In 5/ In 3) = In 5/ In 3. In Figures 39a-d, we present an overview of the wavelet transform of the deterministic one-scale snowflake (Argoul et al. 1989a, 1990). Each picture corresponds to the wavelet transform at a given scale a; from Figure 39a to Figure 39d, one increases magnification. The wavelet transform "zooming" provides conspicuous information about the construction rule of the deterministic snowflake. At large scale (a = a*), one observes a single object of size P (Figure 39a). At a finer scale (a = a*/3), this object is divided into 9 identical pieces, each of which is a reduced version of the original object with length scale /*/3 (Figure 39b); 4 among the 9 pieces are removed, while the central piece with the 4 pieces at each corner are retained. Then, the same procedure is repeated at the next step for each of the 5 remaining pieces (Figure 39c). The snowflake is obtained by applying the same rule subsequently (Figure 39d), ad infinitum. Figures 39a'-d' report the results of a similar WT analysis of the random snowflake shown in Figure 40b. The multiplicative process used to construct this fractal aggregate is again clearly revealed. At each stage of the construction, an elementary motif is divided into 9 identical pieces, 5 pieces are retained as before, but their positions are now chosen at random among the possible configurations that preserve the connectivity of the aggregate. Thus, even though the deterministic and the random snowflakes have the same spectra of generalized fractal dimensions (see (145)), they are clearly distinguished and identified under the optics of the WT microscope.
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Figure 40d shows the results from the local WT analysis of the scaling properties of both the deterministic and the random snowflake fractals (Argoul'e* al. 1989a, 1990). According to (140), the local scaling exponent a(aT) is estimated at different points of the cluster by plotting Iog2 \T^[n](x, a)\ versus Iog2 a. Disregarding finite-size effects which occasionally affect the WT at large scales, T,/,[/i](a:,a) provides a rather accurate estimate of the exponent a = In 5/ In 3, in good agreement with the theoretical prediction. In fact, for both the deterministic and the random snowflakes, the scaling exponent is found to be unique, as expected for homogeneous fractals. However, as seen in the various log-log plots shown in Figure 40d, there exist oscillations which are superimposed on the expected WT power-law behavior. For the deterministic snowflake, these oscillations are periodic with period P — In A/, = In 3. This periodicity is a consequence of the exact recursive structure of this aggregate: the well-ordered morphology of the deterministic snowflake is invariant with respect to discrete dilations by a factor XL = 3. As far as the random snowflake is concerned, the nonperiodicity of the observed oscillations clues us in on the existence of some randomness in the construction process; the global self-similarity of the random snowflake is thus recovered at a statistical level when averaging over the fluctuations inherent to random morphologies. 7.3.2
Multifractal snowflakes
The main difference between the construction of multiscale and singlescale fractals is the fact that the starting object is now divided into XB parts which are not identical (Argoul et al. 1990). Indeed, each of these is a reduced version by different scale factors of the original object. In Figure 40c, we show a generalization of the deterministic self-similar snowflake fractal studied in the previous section. The configuration at the ntfl stage is obtained by adding the configuration at the (n — l)ih stage of the growth to the four corners of the twice enlarged version of the cluster corresponding to the (n — l)th stage configuration. Figures 40e and 40f present the data of a statistical analysis of the single-scale and two-scale deterministic snowflakes respectively; this study was actually performed with the OWT experimental set-up that computes the WT optically (Arneodo et al. 1992b; Muzy et al. 1992). The generalized fractal dimensions
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Dq extracted from the scaling behavior of the partition functions defined in (142) and (143) are in remarkable agreement with the theoretical predictions. The one-scale snowflake is homogeneous: up to the experimental uncertainty, the Dq are found to be all equal to Dq = In 5/ In 3. The data for the two-scale snowflake reveal, in contrast, a rather smooth decreasing dependence of the Dg curve, the hallmark of multifractality. These results clearly demonstrate that geometrical multifractality can be resolved, not only numerically, but also experimentally when using the optical wavelet transform.
7.4
7A.I
Wavelet Analysis of Diffusion-Limited Aggregates
Self-similarity of DLA clusters
As pointed out in the introduction of this section (Section 7.1), most activity in the context of irreversible growth processes has been focused on characterizing the geometry of growing aggregates (Pietronero & Tosatti 1986; Stanley & Ostrowsky 1986, 1988; Guttinger & Dangelmayr 1987; Feder 1988; Pietronero, 1989; Vicsek 1989; Bunde & Havlin 1991). In early numerical studies, on-lattice DLA clusters were found to have different scaling properties in the radial and the azimuthal directions, raising the question of selfaffinity (rather than self-similarity) for these fractal aggregates (Ball et al. 1985; Kolb 1985; Meakin fe Vicsek 1985; Meakin 1986; Meakin et al. 1987). Such an anisotropy can be induced by the presence of an underlying lattice. It may also result from finite size effects. Recent large scale simulations of off-lattice DLA clusters have demonstrated that the existence of two different scaling exponents is only a crossover effect that vanishes in the asymptotic limit of large mass (Li et al. 1989; Ossadnik 1992; Arneodo et al. 1992c, 1992d). Even though one cannot exclude some weak deviation from homogeneity (Vicsek et al. 1990), the statistical "monofractality" of DLA clusters is now well-accepted and rather accurately established by the measurement of the generalized fractal dimensions Dq using boxcounting and fixed-mass algorithms (Argoul et al. 1988, 1989a, 1990; Li et al. 1989). All the dimensions coincide: Dq = 1.60±0.03, \/q. Let us note that this estimate of the D q 's, is significantly smaller than the numerical estimate of the dynamical dimension Dd — 1.71 ±0.02,
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deduced from the evolution of the radius of gyration during cluster growth (Meakin 1988). Whether the fractal dimension D0 and the dynamical dimension Dd are strictly different or asymptotically converge to a unique value is still a very exciting numerical challenge far beyond the power of current computers (Argoul et al, 1988, 1989a; Li et al. 1989). As illustrated in Figure 41a (Arneodo et al. 1992e), a characteristic feature of diffusion-limited aggregation is the fact that most of the growth takes place in an "active" zone, near the outer radius of the cluster, which collects practically all the new particles. This active zone moves outward, leaving behind an "extinct" region that can be considered as asymptotic in the sense that it will no longer be modified by further growth (Plischke fe Racz 1984; Argoul et al. 1989c; Amitrano et al. 1989). This screening of the inner region by the tips is the basic reason for the fractal branching in DLA growth. In this section, we will focus the WT microscope on this internal frozen region that is likely to reflect the morphology of large mass DLA clusters (Arneodo et aU992c, 1992d, 1992e, 1993c). But to proceed to a quantitative fractal analysis of the DLA edifice, we need to investigate rather large DLA clusters so that their inner inactive regions contain several generations of branching. In Figure 41b, we show the central region of a 106 particles off-lattice aggregate generated using an efficient algorithm which combines the simplicity of the off-lattice algorithm designed by Derrida et al. (1991) with the speed of on-lattice hierarchical algorithms (Ball & Brady 1985). As illustrated in Figures 41c and 41d respectively, increasing the magnification of the WT microscope, reveals progressively the successive generations of branching. A first qualitative indication of the statistical self- similarity of the DLA clusters is that these branchings occur rather uniformly in space, without any preferential location and this at all scales. The results of box- counting and fixed mass dimension measurements in Figure 42 provide a quantitative confirmation of this qualitative finding: the generalized fractal dimensions are found to be equal to the fractal dimension (Arneodo et al. 1992c, 1992d, 1992e, 1993c):
The same quantitative estimate is obtained when investigating the scaling behavior of the partition functions 2(q, a) defined in (142) in terms of the wavelet components. Note that this numerical value
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Figure 41: Wavelet analysis of an off-lattice DLA cluster, (a) The early stages of a growing DLA cluster; the grey coding indicates when the particles arrived: medum gray(figure b, inner part of figure a, outer layer of figures c and d) represents the early arriving particles and red the late arriving ones (outer layer of figure a, innermost contours of figure b and c). (b) The inner frozen region of an offlattice DLA cluster of mass M — 106; about 8 104 particles are contained in a disk of radius R — 480 particle sizes. In (c) and (d), this frozen region is explored with the Mexican hat microscope (Eq. (134)) for magnification a~l and 2.2a~ 1 respectively. Intensity of the wavelet transform is grey-coded from black (T^ < 0) to medium gray (innermost contours, maxlw, > 0).
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Figure 42: (a) These graphs illustrate the determination of the generalized fractal dimensions Dq of the inner frozen region of the large mass off-lattice DLA cluster shown in Fig. 41b; the D^s are estimated from linear regression fits of Iog2 Z(q, e)/( 0) and fixed mass (q < 0) computation of the Dq's of the frozen region of the DLA cluster (A) and of the azimuthal Cantor set (•).
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is equal, to within numerical uncertainty, to the fractal dimension of the entire aggregate. This observation is consistent with the recent numerical demonstration that the subset of inaccessible sites of DLA clusters is a "fat fractal" that involves a finite proportion (~ 37 %) of the total number of perimeter sites (Argoul et al. 1989c; Amitrano et al. 1989). Figure 42 also shows the results of dimension measurement of the azimuthal Cantor set defined by the intersection of the DLA cluster with the circle that delimits its inner frozen region (Figure 41b) (Arneodo et al. 1992c). This azimuthal Cantor set is found to be statistically homogeneous with a fractal dimension D$ = 0.60 ± 0.03. According to the Mandelbrot (1984) rule for one-dimensional sections of homogeneous fractal sets, the fact that Dp = DpLA — 1 is an additional indication that DLA clusters are homogeneous fractal aggregates. 7.4.2
Structural five-fold symmetry of DLA clusters
In this section we want to push further our analysis of the DLA morphology by taking advantage of the remarkable ability of the WT to reveal the structural hierarchy of fractal aggregates, as previously tested (Section 7.3) on snowflake patterns that display a well-organized fractal architecture. As shown in Figures 41c and 41d, by choosing the appropriate magnification, the WT microscope provides an efficient way to define the concept of branches at each generation and, then, to measure the screening angles between bifurcating branches (Arneodo et al. 1992d, 1992e). Let us point out that these screening angles take into account the whole hierarchy of subbranches and thus differ from the angles between the stems of the branches (Feder et al. 1989; Ossadnik 1992). In Figure 43, we present the results of a systematic screening angle investigation of 4 off-lattice DLA clusters similar to the one shown in Figure 41b. Three colors are used to differentiate the distributions obtained for three different values of the magnification corresponding to successive branching generations. The three histograms are actually almost indistinguishable, which clearly suggests that the statistical self-similarity of DLA clusters is intimately related to the existence of a screening angle distribution which is scale invariant. This distribution displays a unique maximum at the value
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Let us mention that the presence of a five-fold symmetry at a macroscopic level in diffusion-limited aggregation was originally noticed by Ball (1986) and Halsey et al. (1986). The demonstration in Figure 43, that this pentagonal symmetry is actually statistically dominant at all scales in the structure of DLA patterns is likely to be a clue to a hierarchical fractal ordering. We refer the reader to some of our previous works (Arneodo et al. 1992d) where further evidence for the existence of a structural five-fold symmetry in the internal frozen region, as well as in the active outer region of the interface of DLA clusters, was reported.
Figure 43: Histogram of screening angle values at the branching bifurcations in the wavelet transform representation (Figs 41c and 41d) of 4 off-lattice DLA clusters, for magnifications a~l (leftmost bar), 2.2 a"1 (center bar) and 2.2 2 a~ x (rightmost bar) corresponding respectively to three successive generations of branching. A single maximum is observed for 9* ~ 36° = ir/5.
7.4.3
Fibonacci hierarchical ordering in the DLA azimuthal Cantor sets
The relationship between five-fold symmetry, Fibonacci numbers and the golden mean cj> = 2cos(?r/5) = 1.618 • • •, has been well-known for
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a long time (Garland 1987). The angle defined by the sides of the star and the regular pentagons is 6* = 36°, while the ratio of their lengths is the golden mean. The recent discovery of "quasi-crystals" (Michel fe Gratias 1986) in solid state physics, is a spectacular manifestation of this relationship. This new quasi-periodic organization of atoms in solids, intermediate between perfect order and disorder, generalizes to the crystalline "forbidden" symmetries, the properties of incommensurate structures. Similarly, there is room for "quasifractals" between the well-ordered fractal hierarchy of snownakes and the disordered structure of random aggregates (Vicsek 1989). This section is devoted to the demonstration that DLA clusters are possible "quasi-fractal" candidates that display quasi-periodic scale invariance (Arneodo et al. 1992c, 1992d, 1992e, 1993b, 1993c). Fibonacci sequences are naturally generated by the recursive process (Garland 1987):
If one starts with the species B at the generation n = 0, one gets A at the generation n = 1, and successively AB, ABA, ABAAB, ABAABABA, • • • The population Fn at the generation n can be deduced from the two preceding populations, Fn-\ and F n _2, according to the iterative law:
Note that F n _i and Fn-i are also the respective populations of A and B at step n. A remarkable property of the Fibonacci series {Fn} = {1,1,2,3,5,8,13,21,34,---} is that the ratio of two consecutive Fibonacci numbers converges to the golden mean:
Evidence of Fibonacci sequences are reported in various contexts as diverse as mathematics, art, architecture, sciences and technology (Garland 1987). In particular, the golden mean arithmetic has been shown to play a fundamental role in the growth of phyllotactic patterns in the botanical world (Jean 1984; Sleeves & Sussex 1989; Donady fe Couder 1992). But there are more intriguing geometries in nature that involve Fibonacci numbers. Some trees, root systems, algae, blood vessels and the bronchial architecture do appear to exhibit morphologies that are strikingly similar to DLA fractal patterns
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(Pietronero & Tosatti 1986; Stanley & Ostrowsky 1986, 1988; Garland 1987; Feder 1988; Vicsek 1989; Bunde & Havlin 1991). It is thus tempting to speculate how far the search for Fibonacci ordering can be pushed in the context of fractal growth phenomena. At a first step of our demonstration (Arneodo et al. 1992c, 1993c), we will focus our wavelet analysis on the azimuthal Cantor set defined by the intersection of the DLA cluster with the circle of radius R that delimits its extinct region (Figure 41a). As previously emphasized in Section 4.1.1, the wavelet analysis of singular measures actually does not require the analyzing function if} to be of zero mean. In this section, we will mainly use the Gaussian function il'(x) = e~x / 2 as analyzing Wavelet.
Figure 44: Wavelet transform skeleton denned from the local maxima of \T^(x,a)\ considered as a function of x. (a) The uniform triadic Cantor set. (b) The azimuthal DLA Cantor set. The analyzing wavelet is the Gaussian function ^(x) = e~~x / 2 . For pedagogical purposes, we first compare in Figure 44 the wavelet transform of the azimuthal Cantor set of a 106 particle offlattice DLA cluster (Figure 44b) to the wavelet transform of the uniform triadic Cantor set (Figure 13a). Indeed, only the skeletons defined by the positions of the local maxima of |T^,(a;,a)| are represented in this figure. As discussed in Section 4, although we have considerably reduced the amount of data for the representation, the so-obtained tree like pattern is likely to reveal the hierarchical structure (if any) of the considered fractal object. This is clearly what is observed in Figure 44a for the uniform triadic Cantor set: at the
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scale a = ao3~ n , each one of the k02n modulus maxima simultaneously bifurcates into two new maxima, giving rise to a cascade of symmetric pitchfork branchings in the limit a —> 0+ (OQ and k0 are constants that depend on the specific shape of ip). The fractal dimension Dp = In 2/ In 3 of the uniform triadic Cantor set can be directly obtained from the branching ratio A# = 2 and the scale ratio A£ = 3, according to the formula
At first sight, when looking at the WTMM representation of the DLA azimuthal Cantor set in Figure 44b, one does not see any conspicuous recursive structure in the wavelet transform skeleton. One can, however, proceed with a systematic investigation of the value of the scale ratio A/, between two successive bifurcations (black dots in Figure 44b). The results of a statistical analysis of 50 azimuthal Cantor sets, similar to the one in Figure 44b, are reported in Figure 45. The distribution of scale ratios displays a main maximum at the value \*L — 2.2 ± 0.2. With the additional numerical information obtained by box-counting or wavelet computation of the generalized fractal dimensions of the azimuthal Cantor set (Figure 42), one can insert the numerical values Dp = 0.60 ±0.03 (see (146)) and X*L = 2.2 ± 0.2 into (151); doing so, one gets a branching ratio A£ ~ (2.2)0'6 ~ 1.61. This numerical value is significantly different from 2, which unambiguously discards the possibility of an exact binary branching process. The most striking feature is that the value Ag is remarkably close to the golden mean
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Figure 45: Histogram of values of the scale ratio A^ separating two successive bifurcations ((•) in Fig. 44b) in the WTMM skeletons of 50 DLA azimuthal Cantor sets. A single maximum is observed for \l = 2.2 ±0.2. indicated in Figure 46b, some deviations from the Fibonacci ordering are observed at small scales, but this is not surprising since, at scales a of the order of a few particle sizes, the azimuthal Cantor set is very sensitive to small changes in the radius R of the circle which delimits the frozen region of the DLA clusters. The overall Fibonacci ordering is, however, rather robust with respect to the arbitrariness of the choice of this circle. As illustrated in Figures 46a and 46b, by assigning a symbol A or B to each maxima line issued from a bifurcation, one obtains a coding of the WT skeleton that complies with the recursive law (148). However, a systematic investigation of this coding for our statistical sample reveals some randomness in the relative position of the symbols A and B at each bifurcation A —»• AB. Out of 1586 bifurcation points for which the coding has been achieved, 747 (47%) correspond to the A branch being on the left and 839 (53%) to the A branch being on the right. Moreover, the analysis of the correlations between two successive bifurcation points does not indicate any memory effect. Actually, within the statistical uncertainty, one cannot distinguish the random occurrence of the symbols A and B at each bifurcation point from a fair tossing coin. The spreading of the histogram in Figure 45 around A* ~ 2.2
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Figure 46: Enlargements of the WT skeleton of the azimuthal Cantor set in Fig. 44b, corresponding to three distinct main branches of a large mass off-lattice DLA cluster. The horizontal lines mark the scale an = ao\*L~n with A£ — 2.2. In (a) and (b), the number of WTMM at each generation follows the Fibonacci series (Eq. (149)); moreover, a symbol A or B can be assigned to each of these maxima according to the Fibonacci recursive process (Eq. (148)). (c) illustrates some local defect to the Fibonacci structural ordering.
indicates the existence of important fluctuations in the scale ratio value. These fluctuations result, for example, from the existence of some arbitrariness in the spatial location of A when B proceeds to B —» A- as discussed in the next section, this arbitrariness is likely to result from local fluctuations in the value of the screening angle (about 6 = 36°) in the DLA branching morphology. These fluctuations can produce some local departures from the Fibonacci ordering as shown in Figure 46c. A close examination of the WT skeleton in Figure 44b reveals the presence of many of these defects. The presence of these structural defects raise the question of the statistical relevance of this Fibonaccian architecture. Figure 47 reports the results of a systematic analysis of the WTMM skeletons associated with the 240 main branches identified in our statistical sample of 50 offlattice DLA clusters. The angular width of each of these branches has been normalized to 1 before computing the WTMM skeleton. The different histograms represented in Figure 47 correspond to the
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Figure 47: Statistical distributions of the number of WTMM lines that exist in the WT skeletons of the 50 DLA azimuthal Cantor sets at scales an = (2.2)~n for successive generations from n = 1 to 7. statistical distribution of the number of WTMM lines that exist at scales an — \*^n = (2.2)~™ for successive generations. Each of these histograms displays a well-defined maximum at the value Fn given by the Fibonacci series. This is a quantitative confirmation that the Fibonacci structural ordering is a generic statistical characteristic of the azimuthal DLA Cantor sets and not some feature recognized on particular realizations (Arneodo et al. 1993d). This observation corroborates our previous estimate of the branching ratio \*B ~ <^, the asymptotic limit of the ratio of two successive Fibonacci numbers. In that respect, our wavelet analysis provides the first numerical evidence for the existence of a "Fibonaccian" quasi-fractal structural ordering in the DLA clusters (Arneodo et al. 1992c, 1993b, 1993c, 1993d). 7.4.4
Golden mean arithmetic in the fractal branching of DLA clusters
A fundamental step in our demonstration is now to return to the DLA cluster itself and to point out Fibonacci sequences in its disordered
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Figure 48: Three main extinct DLA branches are seen through the Mexican hat microscope. The color coding is as in Figs 41c and 41d. The magnification is such that three successive generations of branching are identified. In (a) and (b), at each branching, a symbol A and B has been assigned to the outcoming branches in order to reveal the Fibonacci structural ordering, (c) illustrates a structural defect induced by some local fluctuation of the screening angle 0 away from 6>* = 36°.
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branched morphology (Arneodo et al. 1992d, 1992e). In Figure 48, we use the two-dimensional WT microscope to explore the internal structure of three main branches of a 106 particle off-lattice DLA cluster. The analyzing wavelet is the so-called Mexican hat and the magnification is such that three successive significant branchings are resolved. One can see that these branchings proceed according to the Fibonacci recursive law (148), as identified by assigning a symbol A or B to the branches of successive generations. In Figure 48a, the original branch A gives two branches A and 5; both these branches bifurcate into new branches, but one of them, issued from B, dies before reaching the reference circle that delimits the frozen region of the DLA cluster. This peculiar electrostatic screening actually governs the growth process and originates in a statistical Fibonacci structural hierarchy with a branching ratio that converges asymptotically to the golden mean. As previously noticed in the WT skeleton of the azimuthal Cantor set, this Fibonacci architecture contains, however, some randomness in the relative position of the branches A and B at the bifurcations A -+ AB. Note that the observation of a perfect Fibonacci ordering in Figures 48a-b, coincides with a succession of screening angles that do not significantly deviate from 0 = 36°. But the histogram in Figure 43 is rather widely spread around 0 = 36°, which clearly indicates the existence of important fluctuations in the screening angle value. As illustrated in Figure 48c, these fluctuations can produce some local departure from the Fibonacci structural ordering. The screening angle B ~ 90° at the primary branching is so large that there is enough space between the two outward branches for a new branch to grow and split almost immediately into two new branches with screening angle 9 ~ 36°. Despite the presence of many of these local structural defects, the DLA branching organization does exhibit a fascinating prevalent tendency to be Fibonaccian. A systematic investigation of the actual relationship between the Fibonacci branching ordering and the structural five-fold symmetry is currently in progress. A very important issue is to elucidate whether these structural defects to the Fibonacci hierarchical ordering are the consequence of the randomness of the DLA algorithm used to mimic Laplacian growth processes. In the spirit of Penrose's tiling (quasi-periodic tiling of the plane), one could reasonably imagine a "quasi-fractal" snowflake model of DLA, with a screening angle between branches strictly equal to 36° and a Fibonaccian branching structure satisfying (148) (quasi-periodicity
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with respect to dilations). By removing the observed randomness in the branching process A —» AB, one could even conceive an exact deterministic "quasi-fractal" version of the DLA clusters (Arneodo et al 1992c, 1992d,1992e, 1993b,1993c, 1993d).
8
Prospects: Solution to the Inverse Fractal Problem from Wavelet Analysis
To summarize, we have presented in this tutorial, a first theoretical step towards a unified theory of singular distributions including multifractal measures and multifractal functions. Rigorous results have been mainly derived under specific hypothesis concerning the signal under study as well as the shape of the analyzing wavelet (Bacry et al. 1993). But the WTMM method is likely to remain valid under less stringent conditions. The results of some numerical applications strongly suggest that our theoretical results may extend to more general multifractal functions such as the realizations of some stochastic processes. Preliminary investigations in this context indicate that fractional Brownian motions (Flandrin 1991a, 1991b) could well be amenable to such a rigorous treatment relying on the wavelet decomposition. Moreover, this wavelet-based multifractal formalism provides algorithms for determining the D(h) spectrum of Holder exponents directly from the considered fractal distribution. From the comparative applications of the WTMM and SF methods on specific examples, we have pointed out the severe intrinsic limitations of the later that fails to fully characterize the scaling properties of a multifractal signal. In contrast, we have demonstrated that the former does not introduce any bias in the estimate of the scaling exponents of some partition functions which are at the heart of this "generalized" multifractal formalism. The efficiencies of the WTMM method originate from two main ingredients; on the one hand, the partition functions are based on discrete scale-dependent summations: the skeleton of the wavelet transform defined by the local maxima of its modulus provides a practical guide to achieve a scale adaptive partition; on the other hand, one uses sufficiently smooth and localized analyzing wavelets with an arbitrary large number of vanishing moments, which makes the entire range of singularities accessible to this method, even in the presence of regular behavior in the signal. The reported results of a prelimi-
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nary analysis of a fully developed turbulent velocity signal show that this method is readily applicable to experimental situations. Indeed we believe that the WTMM method for determining the singularity spectrum of fractal signals is likely to become as useful as the wellknown phase-portrait reconstruction, the Poincarre section and the first return map techniques for the analysis of chaotic time series. Applications of this method to turbulent dynamics generated by fractal growth phenomena, critical fluctuations in colloidal systems, surface roughening in noise driven growth processes and DNA "walks" nucleotide sequences are currently in progress. Moreover, as illustrated in Section 7, its generalization to two-dimensions looks very promising in various experimental contexts, e.g., percolation, fractal growth processes, fracture patterns, nucleation and coalescence phenomena, two-dimensional melting and turbulent flows. But beyond the statistical "thermodynamic" description of scale invariant objects that provides the multifractal formalism in either its (box counting) classical version or its wavelet-based generalized version, there is a need to get deeper insight into the complexity of such objects and eventually to extract some "microscopic" information about their underlying hierarchical structure. The discovery of a Fibonacci structural ordering in the fractal branching of DLA clusters reported in Section 7, is a first encouraging step made towards this goal using the wavelet transform machinery. On general grounds, in many cases the self-similarity properties of fractal objects can be expressed in terms of a dynamical system which leaves the object invariant. The inverse problem consists in recovering the dynamical system (or its main characteristics) from the data representing the fractal object. This problem has been previously approached within the theory of Iterated Function Systems (Barnsley & Demko 1985; Barnsley 1988; Handy & Mantica 1990). But the methods developed in this context are based on the search of a "best-fit" within a prescribed class of IFS attractors (mainly linear homogeneous attractors). In that sense, they approximate the self-similarity properties more than they reveal them. But as emphasized in Section 4.1 when analyzing the multiplicative structure of Bernoulli invariant measures of expanding Markov maps (Collet et fil. 1987; Rand 1989), the space-scale unfolding that provides the wavelet transform generally enlightens the hierarchical structure of a fractal object and thus can possibly be used to reveal the renormalization operation which accounts for its construction process. To
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conclude this survey, we will describe a wavelet-based tree matching algorithm that we have recently implemented and which provides some openings towards solving the so-called inverse fractal problem (Arneodo et al 1993a, 1993b). 8.1
A Wavelet-Based Tree Matching Algorithm to Solve the Inverse Fractal Problem
The class of fractal objects we will use to carry out our demonstration are the invariant measures of "cookie-cutters". A cookie-cutter (Rand 1989) is a map on A = [0,1] which is hyperbolic (\T'\ > 1) and so that T~ 1 (A) is a finite union of 5 disjoint intervals (Ak)i
where Y^Pk = 1- Let us note that these self-similar (Bernouilli invariant) measures have been widely used for modeling a large variety of highly irregular physical distributions; notable examples include strange repellers which characterize the transient behavior of nonlinear dynamic systems (Rand 1989) and the spatial distribution of the dissipation field in fully developed turbulent flows (Paladin & Vulpiani 1987; Meneveau & Sreenivasan 1991). Using a simple "smoothing wavelet" T/J(X) = exp( — x 2 ) , one can reproduce the straightforward calculation carried out in Section 4.1 to derive (70) and (78); to first order in a (a « 1) one gets:
where T^1 is the first derivative of T^"1. In the case where the T^1 are linear, i.e., T^l(x) = r^x + tk (f'/t < 1), we thus obtain:
These relations can be interpreted as self-similarity properties of the wavelet transform itself as previously illustrated in Figures 13a, 14a and 16. Indeed, in the linear case, the wavelet transform on the rectangle [0.1]x]0.a 0 ] of the space-scale half-plane (6, a) is "similar" to
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the wavelet transform on each of the rectangles [s/,, r^ + tk] x]0, rk(io] (O,Q is ai1 appropriate coarsest scale which actually depends on the analyzing wavelet). Our goal is to study the self-similarity properties of /J, through those of its wavelet transform T^[fj,]. For that purpose, we are not going to deal with the whole wavelet transform but only with its skeleton. One can easily prove that the self-similarity relation (153) still holds when restricted to the set of modulus maxima of the wavelet transform (Figs 13b and 14b). For more details, we refer the reader to our previous work (Arneodo et al. 1993a, 1993b; Bacry et al. 1993) and to a recent preprint by Hwang & Mallat (1993) where an alternative approach to recover the affine self-similarity parameters through a voting procedure based on (153) is reported. Let us illustrate our purpose on a particular example. For the sake of simplicity, we choose s = 2, p\ = 0.7, pi ~ 0.3 and the TVs to be linear: T\(x) = 5x/3 and TZ(X) - 5x - 4. The corresponding invariant measure is shown in Figure 49a and the position of its wavelet transform modulus maxima in Figure 49b. As previously noticed, one can see that the part of the space-scale half-plane displayed in Figure 49b (the entire rectangle [0,1] x]0, OQ]) '1S "similar" to the two rectangles delimited by the dashed lines ([0,3/5]x]0,3a 0 /5]) and [4/5,1] x]0, «o/5]) (up to a global rescaling of the modulus of the wavelet transform). Let us describe on this particular example our technique for recovering from the wavelet transform modulus maxima, the discrete (cookie-cutter) dynamical system T. We call bifurcation point any point in the space-scale half-plane located at a scale where a maxima line appears and which is equidistant to this line and to the closest longer line. The bifurcation points at coarse scales are displayed in Figure 49b using the symbol (•). They lie on a binary tree whose root is the bifurcation point at the coarsest scale. Each bifurcation point defines naturally a subtree which can be associated with a rectangle in the space-scale half-plane. This root corresponds to the original rectangle [0, l]x]0, CQ] whereas its two sons correspond to reduced copies delimited by the dashed lines. As illustrated in Figure 49b, the self-similarity relation (154) amounts to matching the "root rectangle" with one of the "son rectangles", i.e., the whole tree with one of the subtrees. More generally, this relation associates any bifurcation point (xn,an) of an order n subtree to its hierarchical homologous (a; n _i,a n _i) of an order n — 1 subtree. It follows from (154) that xn = r^Xn-i + t^ and an = r^o-n-i • Thus by plotting .Tn_i versus xn, one can expect to recover the initial cookie-cutter
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Figure 49: (&) Invariant measure of the two branch cookie-cutter TI(X) = 5x/3, T2(x) = 5.T — 4, distributed with the weights p\ — 0.7, p2 = 0.3 on the interval [0,1], (b) Position in the (a;, a) half-plane of the local maxima of the modulus of the wavelet transform of the measure shown in (a), using a gaussian analyzing wavelet; the large scales are at the top. According to the self-similarity relation (Eq. (154)), the maxima line arrangement in the two dashed rectangles is the same as in the original rectangle. The bifurcation points associated to each rectangle are represented by the symbols (•). Arrows indicate the matching of these bifurcation points according to the self-similarity relation (Eq. (154)). (c) ID map that represents the position xn--\ of an order n— 1 bifurcation point versus the position xn of the associated order n bifurcation point following the tree matching denned in (b). The graph of this map corresponds exactly to the original cookie-cutter, (d) Histogram of scale ratios r = an/an-\ between t h e scales of two associated bifurcations points, (e) Histogram of amplitude ratios p = T^i\(xn, a n )|/|T,/,[^](x n _ 1 , a n _ i ) | computed from two associated bifurcation points.
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T. This reconstructed 1-D map is displayed in Figure 49c. As one can see, the two branches T\ and TI of the cookie-cutter T provide a remarkable fit of the numerical data. Let us point out that the nonuniform repartition of the data points on the theoretical curve results from the lacunarity of the measure induced by the "hole" between the two branches T\ and T%. In Figure 49d, we show the histogram of the (contracting) scale ratio values between the scales of two bifurcation points of successive generations r = an/an^i as computed when investigating systematically the WTMM skeleton. As expected, it displays two peaks corresponding to the two slopes n = 3/5 and r2 = 1/5 of Tf1 and T^1 respectively. Note that the peak corresponding to the smallest value of r is lower than the other one; this is a direct consequence of the finite cut-off we use in our wavelet transform calculation at small scales. On a finite range of scales, the construction process involves less steps with the smallest scale ratio r-2 than steps with the largest one r\. (The so-computed histogram can be artificially corrected in order to account for these finite size effects; actually it suffices to plot JV(r)ln(l/r-) instead of N(r}}. Figure 49e displays the histogram of amplitude ratio values
One clearly distinguishes two peaks in good agreement with the weights pi = 0.7 and p^ = 0.3. Let us mention at this point, that the distribution N(r) of scale ratios is in a way redundant with respect to the 1-D map since it is basically made of two Dirac functions located at the inverse of the slopes of the two branches of this piecewise linear map. On the contrary, the distribution N(p) of amplitude ratios brings a very important piece of information which is not present in the 1-D map: the repartition of the weights at each construction step. When this repartition is uniform, we get an histogram N(p) which reduces to a single point p — 1/2. When the repartition is not uniform, as in Figure 49, one can furthermore study the joint law of p with r in order to find out the specific "rules" for associating a p with a r. In the case where s is no longer equal to 2, one can easily adapt this technique by trying to match not only the root bifurcation point on its sons but also on its grandsons and so on... For instance, in the case s — 3, we will match the root with one of its sons and with each of the two sons of its other son. The general algorithm, that
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we have developed, uses a "best matching" procedure that automatically chooses the matching which is the most consistent (e.g., such that the different derivatives of T^[/J,] follow the same self-similarity rules as T^[/J]). Thus the algorithm is not looking for a given number s of branches that the user would have guessed a priori, it automatically comes up with the "best" value of s. Figure 50 shows the 1-D map, the histogram of scale ratios and the histogram of amplitude ratios obtained in the linear case where s = 3, pi = p2 ~ ps = 1/3 and r\ = 0.2,f2 = 0.3,rs = 0.5. All these values are very accurately recovered by our algorithm. Let us notice that, so far, we have only considered measures which do not involve any "memory" effect, in their hierarchical structure, i.e., the successive (backward) iterations always consist in applying the same dynamical system T, independently of the previous iterations. However, in a certain way, a construction rule involving a finite memory can be accounted by increasing the number of branches of a "no memory" map T. As illustrated in Figure 50, this class of dynamical systems is directly amenable to our WT algorithm procedure. Nevertheless, it is meaningless to look for some dynamical systems with a rather high number of branches; generally, there would not be enough scales in the data to ensure the theoretical validity of the resulting discrete map.
8.2
Wavelet Transform and Renormalization of the Transition to Chaos
In the former examples, we have described our technique to solve the inverse fractal problem for cookie-cutters made of linear branches. Since (153) locally (in the space-scale half-plane) looks like (154), we can apply exactly the same technique for nonlinear expanding maps. (Let us point out that the hyperbolicity condition is a priori required for fmiteness of the first derivative of Tj^1 involved in the right-hand side of (153).) Figure 51a displays the 1-D map extracted from the wavelet transform modulus maxima skeleton of the uniform Bernouilli measure associated with a nonlinear cookie-cutter made of two inverse hyperbolic tangent branches. Once again, the numerical results match perfectly the theoretical curve. In this case, the histogram of amplitude ratios is concentrated at a single point p = 1/2. The histogram of scale ratios, however, involves more than simply two scale ratios as before, since the nonllnearity of the map implies
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Figure 50: (a) Inverse problem for the invariant measure of the three branch cookie-cutter TI(,T) = 5x, T^(x} = 10z/3 — 2/3,13(0;) = 2x — 1, distributed with equal weights p\ — p% = ps — 1/3 on the interval [0,1]. (b) Histogram of scale ratios r = an/an-i. (c) Histogram of amplitude ratios p = |T^,[^](a; n ,a n )|/|r^[//](a; n _i,a n _ 1 )|. that new scale ratios are actually operating at each construction step. As a first application of our wavelet-based tree matching algorithm to a physical problem, we show in Figure 51b, the results obtained when analyzing the natural measure associated with the iteration of quadratic unimodal maps at the accumulation of perioddoublings (Section 4.1.2). A well-defined 1-D map with two distinct hyperbolic branches is numerically reconstructed. A computation at finer resolution would reveal that the left-hand branch is linear with a slope 1/r = !/$*(!) ~ —2.5, whereas the right-hand one is nonlinear. The computation of the scale-ratio histogram confirms this observation. One can also compute the amplitude ratio histogram and find that the weights associated with these two branches are equal: P! = p2 = 1/2. The period-doubling natural measure can thus be seen as the invariant measure of the (hyperbolic) cookie-cutter displayed in Figure 51b. The solid lines shown in this figure corresponds to the dynamical system defined in (76) and which was proved by Ledrappier & Misuirewicz (1985) to have the same invariant measure as the fixed point mapping $*(x) (see (75)) of the renormalization operation (74). Our numerical data are in remarkable agreement
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Figure 51: (a) Inverse problem for a nonlinear cookie-cutter made of two inverse hyperbolic tangent branches. The data are obtained from the same wavelet transform tree matching analysis as in Fig. 49. The original dynamical system (solid lines) is recovered accurately, (b) Inverse problem for the invariant measure associated to the period doubling dynamical system $* (see text). The solid lines represent the theoretical prediction (Eq. (76)). Finer resolution computations would reveal that the right-hand branch is nonlinear. The left-hand branch is linear with a slope 1/$*(1) ~ —2.5. with the theoretical prediction (Arneodo et al. 1993a, 1993b).
8.3
Uncovering a Multiplicative Process in One-Dimensional Cuts of Diffusion-Limited Aggregates
The wavelet-based tree matching algorithm introduced in Section 8.1 provides a very attractive method to push further the analysis of the azimuthal DLA Cantor sets carried out in Section 7.4.3 and to extract some "average 1-D map" which could explain and quantify the presence of a predominant statistical Fibonaccian structural hierarchy in the DLA morphology. In order to adapt the tree matching WTMM technique to the presence of the statistical left-right symmetry observed at each bifurcation A —> AB in the WTMM skeleton (Figure 46), we flip, as illustrated in Figure 52, the relative position of A and B whenever the A branch is found on the right of B, so that the skeleton actually processed is made only of A branches emerging on
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Figure 52: WTMM skeleton of a part of the azimuthal Cantor set corresponding to one main branch of an off-lattice DLA cluster (Fig. 46). The analyzing wavelet is the Gaussian function; the large scales are at the top. (a) Symbolic coding of the WTMM skeleton according to the Fibonacci recursive process; the horizontal lines mark the scales an = r*n with r* — 0.44. (b) Illustration of the tree-matching algorithm after transforming the bifurcation points ( x ) in such a way that the symbols A emerge systematically on the left. According to the self-similarity relation (153), the 2 dashed rectangles are mapped into the original rectangle. Arrows indicate the matching of the ( ) and TB ( :) respectively. the left (Arneodo et al. 1993b). Then our tree-matching algorithm consists in extracting the map T which is made of two branches TA and TB and which leaves /j, invariant (i.e., ^ o T^1 = p^/J and /i o Tgl = PB^} from the "self-similarity" relation (153). The 1-D map T(x) reconstructed from scanning the 50 azimuthal Cantor set WTMM skeletons is shown in Figure 53a. The data points obviously do not fall on a well-defined 1-D map. However, the set of data points clearly separates into two distinct "noisy" branches. The solid lines
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in this figure correspond to the piecewise linear 1-D map:
where X*A ~ r*A"1 ~ 2.2 and X*B ~ r*g~l ~ X*A2 ~ 4.8. This ].-D map is made of two linear branches whose slopes correspond to the inverse of the preferential scale ratios found when splitting the histogram N(r) in Figure 54a into two histograms N(rA) and N ( r g J . These two histograms account for the scale ratio fluctuations observed in the WTMM skeletons when one computes the scale ratio for the A branches and the B- branches separately. They both display a maximum for r\ = 0.44 ± 0.03 and r*B = 0.21 ± 0.03 respectively. The fact that r*B ~ r*A has a remarkable consequence. A straightforward computation shows that if this equality holds, then the number of cylinders (subintervals) of a given size rkA generated by iterating T"1 is exactly the Fibonacci number Fk (Arneodo et al. 1993b). A 1-D map model as simple as the piecewise linear map (see (155)) therefore provides a rather natural understanding of the origin of the Fibonacci structural hierarchy discovered on individual realizations (Figures 46a and 46b). The accumulation of data points around the solid lines in Figure 53a can thus be regarded as a quantitative indication of the existence of a statistically predominant multiplicative process, whereas the "noise" around these lines is the signature of the statistical importance of the structural defects to the Fibonacci ordering. There is moreover some randomness in this multiplicative process since at each bifurcation point in the WTMM skeletons, there are as many chances for T(x) (A on the left) as for its "flipped" version T(x) = 1 — T(l - x] (A on the right) to be iterated backward (Arneodo et al 1993b). The WTMM tree matching algorithm gives also access to the histogram of amplitude ratio values
The histograms N(pA) and N ( P B ) , corresponding respectively to the histograms of scale ratios N(rA) and N(rg~), are shown in Figure 54b. Let us note that for the Bernouilli invariant measures of the piecewise linear cookie-cutter model (see (155)) to be homogeneous, the respective weights pA and pg, distributed multiplicatively at each
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Figure 53: (a) ID map extracted from the WTMM skeletons of 50 DLA azimuthal Cantor sets using the tree matching algorithm described in Fig. 52b. The solid lines correspond to the two branches of the linear cookie-cutter (155) with the respective slopes A^ = 2.2 and A£ = X*A = 4.8. (b) Inp versus In r where r = an/an-\(p = \ T ^ [ p ] ( x n , a n ) \ / \ T ^ [ ( j ] ( x n - i , a n - . i ) \ ) is the ratio between the scales (amplitudes) of two bifurcation points that are associated by our tree matching algorithm. A linear regression fit of the data provides a slope a = 0.61 ± 0.03. iteration, must satisfy the requirement p*A = p*B, Since p*A + p^ = 1, one gets exactly p\ = <j>-1 ~ 0.618 and p*B = (j>~2 ~ 0.382. Both histograms in Figure 54b display a maximum in very good agreement with those expected values for p*A and p*B. This is an indication that the DLA azimuthal Cantor sets are likely to be homogeneous fractals. Furthermore we show in Figure 53b that the random variables In r and In p are strongly correlated according to the law p = Crom This result is in remarkable agreement with previous WT measurements of the local scaling exponent a = Dp = 0.61 ±0.03 of the DLA azimuthal Cantor sets (Figure 42b). The scatter of points around the solid line in Figure 53b might explain some weak multifractal departure from statistical homogeneity as noticed in previous box-counting calculations (Vicsek et al. 1990). To summarize, we believe that the wavelet tree matching algorithm described in Section 8.1 is without any doubt a very promising
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Figure 54: (a) Histogram N(r) kfldgujiofugiofuuifdogu ifdfodifdgfffa ratio between two successive bifurcation points in the WTMM skeletons of 50 DLA azimuthal Cantor sets. A maximum is observed for r* = I/A* ~ 0.44 (A* ~ 2.2). N(rA)(dgjlfidjgoifdguiouiiiiiiiifdguffgd the histograms obtained when considering bifurcation points mapped by TA and TB respectively, (b) The corresponding histograms of amplitude ratio values N(PA) (lkdfhglkfgfklgflkf;dghfd;lghfdg step towards solving the inverse fractal problem. To our knowledge this is the first time that some statistical evidence for the existence of a multiplicative construction process hidden in the DLA geometry is reported in the literature. The cookie-cutter T defined in (155) accounts for the presence of a statistically predominant Fibonacci structural ordering. This is a spectacular manifestation of the statistical relevance of the golden mean to fractal growth phenomena (Arneodo et al. 1993b). We are convinced that further applications of this wavelet-based technique will lead to similar major breakthroughs in various fields where multiscale phenomena are ubiquitous. The implementation of a tree matching algorithm that generalizes our wavelet-based method from 1-D to 2-D is currently in progress.
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Acknowledgements
The material reported in this review paper relies on collaborative works with F. Argoul, E. Bacry, J. Elezgaray, E. Freysz, G. Grasseau, J.F. Muzy and B. Pouligny. Some parts of the text have been extracted from previous review papers: Arneodo, A., Argoul, F., Bacry, E., Elezgaray, J., Muzy, J.F. and Tabard, M., 1993, in Progress in Wavelet Analysis and Applications, editors, Meyer, Y. and Roques, S. (Editions Frontieres, Gif sur Yvette), 21. Muzy, J.F., Bacry, E. and Arneodo, A., 1994, Intl. J. of Bifurcation & Chaos 4, 245.
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Index bifurcation point, 475, 483, 485 Boltzman probability measure, 416 boundary, 8 boundary conditions, 14, 16, 19 homogeneous, 299 nonhomogeneous, 299 periodic, 20 boundary value problem, 231, 234, 237 boundary-layer, 332, 333 box counting, 364, 442, 466, 474 box function, 48, 55, 56, 58, 62, 7678, 409 branching, 447 binary, 474 fractal, 467, 477, 481 morphology, 476 symmetric, 474 Brownian motion, 373, 375, 436, 449 fractional, 375, 376, 425, 426, 436, 437, 447, 451 homogeneous, 437 Burgers equation, 17, 252, 292, 305 non-periodic, 17 periodic, 17
A-averaging, 332 accuracy, 5, 10, 14, 291 accuracy threshold, 205, 240, 251 adapted approximation space, 291 adaptive algorithms, 24, 26, 252 admissibility condition, 319, 383, 455 aeroacoustics, 3 affine transformation, 371 algebra, 139 algorithms adaptive, 291 convolution, 299 multipole, 185 segmentation, 331 statistical multiscale, 322 zero-crossing, 322 analysis tool, 8 approximation semi-discrete, 288 approximation error, 11 atomic decomposition, 143 attractor, 405 B-spline, 50 basis, 46 B-spline, 30 biorthogonal, 133 Fourier, 184, 266, 317 Haar, 184, 188, 189, 196, 254 two-dimensional, 190 HUbert, 124 Meyer, 133 multi-wavelet, 195-198 orthogonality of, 15, 50 orthonormal, 5, 46, 50, 125, 126, 167, 192, 240, 266 tensor product, 206 unconditional, 126, 135, 146, 154, 167-169, 173 Bedrossian theorem, 113, 114 Bernouilli numbers, 86 best approximation, 280 best basis, 346 bifurcation period-doubling, 404 subharmomc, 405
Calderon factorization, 146 Cantor set azimuthal, 470, 473, 474, 477, 479, 488, 491 generalized, 375, 407, 417 period doubling, 404 triadic, 359, 362-364, 369, 374, 401,411, 418, 420,473, 474 capacity, 358 Carleson condition, 149, 150, 169, 170, 174 cascade algorithm, 59, 60 CFD, 3 coherent state, 184 collocation methods, 22, 30 collocation points, 24 combustion, 4 commutator, 157 completeness, 41, 43, 45 compression, 3, 10, 30, 55
503
504
lossless, 11 cone of influence, 109, 320, 384, 460 conservation laws, 19 continuous spectrum, 376 continuous wavelet transform, 318, 381, 447 convergence, 12, 56, 135, 280, 281, 285, 286 cookie-cutter, 482, 483, 490 Daubechies, 6 David and Journe theorem, 213 decomposition average scale, 323, 332, 338, 341, 345 local energy, 321, 324 normalized scale, 328 normalized standard deviation, 329 dendritic crystal growth, 462 derivative, 25 of dilation equation, 70, 71 devil staircase, 378, 418, 427 generalized, 375, 429, 437 differentiation, 14 diffusion processes, 452 diffusion-limited aggregate, 453, 466 homogeneity, 453 diffusion-limited aggregation, 467 dilation, 2 anisotropic, 373 discrete, 395, 465 dilation equation frequency domain, 73, 75, 79 time domain, 62 dimension box, 358, 370 capacity, 367 dynamical, 466 fractal, 368, 474 generalized fractal, 352, 366, 369, 442, 453, 461, 466, 467 Hausdorff, 352, 353, 355, 357360, 363, 364, 370, 378, 414, 420, 444 information, 367 negative, 425 topological, 357-360 discrete wavelet transform, 24
Index discretization, 10 time, 288 displacement vector, 454 dissipation field multifractal nature of, 443 dissipation scale, 447 distribution Dirac, 379 fractal, 380, 392, 413 Lipschitz, 392 wavelet transform of, 394 DLA clusters, 466, 473 active region, 467 extinct region, 454, 467, 479 statistical self-similarity, 453, 467, 474 DLA model, 453 Dq spectrum, 368 dyadic dilation, 87 dyadic interpolation, 132 dyadic interval, 48, 144, 320 dyadic point, 62, 66 dyadic refinement, 26 dynamical systems, 355, 361, 481 inverse problem, 481 period doubling, 407 edge detection, 3 energy, 269, 294 energy cascade, 441 energy dissipation, 442 energy plane, 321 energy spectrum, 323 equation dilation, 11, 50, 51, 55, 57, 58, 66, 68, 69, 79, 126, 129 elliptic, 187, 280, 288, 306 constant coefficient, 273 fixed-point, 62, 63, 67, 80 heat, 22 Laplace, 452 nonlinear, 252, 303 parabolic, 287, 288, 294 refinement, 51 Sturm-Liouville, 273 two-scale, 55, 189 wavelet, 52, 59 Euclidean distance, 156 Euclidean group, 454
Index fast wavelet transform, 245 Fibonacci numbers, 471, 472, 477 Fibonacci ordering, 473, 476, 481, 492 Fibonacci sequence, 454, 472, 474 filter autoregressive, 81 averaging, 56, 57 coefficients, 39, 59, 307 discrete, 39, 271 filter bank two-channel, 39 filter banks, 42 orthqnormal, 52, 78 filters, 39, 54, 284 band-pass, 323, 381, 455, 456 biorthogonality, 39, 44, 287 halfband, 58 high-pass, 5, 20, 42, 59, 74 low-pass, 5, 54, 74, 79 multirate, 73 orthogonality, 39 finite-difference, 14, 32, 288, 312 finite-element, 32 forkings, 402, 412 Fourier analysis, 50 Fourier basis, 3 Fourier coefficients, 77 Fourier transform, 9, 24, 73-75, 85, 184 fractal snowflake, 462 fractal aggregate, 461, 464, 466 self-similar, 458, 464 fractal growth phenomena, 452, 473 fractal morphology , 453 fractal set, 360 fractals, 352, 353 fat , 360, 470 homogeneous, 465, 491 inverse problem, 482, 492 self-similar, 481 fractional derivatives, 230, 231 frames, 380 frequency low, 5 function accretive, 165, 171 band-limited, 123
505 biorthogonal, 278, 286 Chebyshev, 299 C°°, 325 Fourier, 299 fractal, 371, 376 homogeneous, 379, 426 Gaussian, 117, 384, 395, 401, 412 generalized intermittency, 343 Green's, 233, 275 Haar, 171, 190 Heaviside, 388 HSlder, 108, 325, 326 intermittency, 338 Lipschitz, 325, 386 multifractal, 379 para-accretive, 171 partition, 354, 355, 366, 378, 413, 415, 419, 434, 439, 461, 466, 467, 480 periodic, 9 probability density, 353 self-affine, 373, 376, 378, 417 singular, 353 square, 86, 143 structure, 353, 376, 378, 424, 442, 443 function localization, 283 functional analysis, 140 functions multiplication of, 251, 252 Galerkin method, 164, 279-281 Galerkin projection, 30 Gaussian process, 375, 436 golden mean, 471, 472, 474, 477 Gram-Schmidt algorithm, 80, 196 grid, 8, 9, 17 adaptation, 307 collocation, 304 dyadic, 269 refinement, 24 staggered, 231 Haar wavelet, 53 hat function, 49, 50, 57, 65, 72 hierarchical structure, 397 Hilbert transform, 109, 114, 141 Daubechies wavelets, 229
Index
506
non-standard form, 228 histogram method, 366, 367 Holder exponent, 355, 376, 386, 388, 389, 395, 414,415, 420, 426, 429, 432, 437, 447, 449 negative, 426, 427, 449 non-uniqueness, 424 Hurst exponent, 374-376, 394, 395 local, 376, 378 inertial range, 446 integral scale, 442, 447 interfacial growth, 452 intermittency spatial, 441 intermittency factor, 338 invariance by scaling, 271, 296, 444 by translation, 271, 296 inverse generalized, 244 inverse fractal problem, 355 invertibility, 134 iterated function systems, 356, 481 Jaffard and Journe theorem, 153 Jaffard and Meyer theorem, 153 Karhunen-Loeve decomposition, 317 Kato's conjecture, 164, 171, 174 Khinchtine inequality, 140 Kolmogorov dissipation scale, 442 Kolmogorov scaling, 451 Kolmogorov spectrum, 441, 447 departure from, 443 Lagrange interpolant, 303 Laplacian growth processes, 479 Legendre transform, 352, 353, 355, 367, 368, 370, 378, 415,420, 424, 425, 444, 451, 461, 464 Lemarie algebra, 152 Lemarie-Meyer theorem, 139 Lipschitz curve, 172 Lipschitz graph, 164 Littlewood-Paley decomposition, 44 theorem, 86, 87 local asymptotic scaling, 326 local energy, 324
local mean scale, 328 local scaling, 325, 458 local scaling exponent, 465 localization physical space, 2 ip-atom, 142 Lyapunov exponent, 406 Mallat and Meyer theorem, 124 matrix condition number, 187, 232-234, 236, 237 differentiation, 12, 15, 30 inversion of, 160, 164 near diagonal, 156 non-standard, 151 second derivative, 234 generalized inverse, 246 standard, 151 matrix functions, 250 maxima line, 390, 392, 402, 411, 419, 432, 434, 435, 448 maximum local, 109, 322, 355, 473 measure, 361, 366, 398 Bernouilli, 407, 411, 413, 486, 490 binomial, 370 density component of, 395 equilibrium, 361 fractal, 352, 361, 370, 371, 401, 411, 417, 454 Hausdorff, 357, 358 homogeneous, 363, 369, 374, 411 invariant, 361, 482 Lebesgue, 358, 361, 363 multifractal, 354, 371, 394, 395, 404, 413, 434, 444 nonhomogeneous, 363, 370, 375, 402 nonuniform, 401 probability, 140 self-similar, 417, 482, 483 singular, 352, 353, 361, 362, 409, 410, 413, 414, 444, 473 uniform, 397 moments, 353 negative order, 355
Index vanishing, 6, 9, 190, 192, 198, 210,213,299, 326,388,389, 395, 432 multifilters, 49 multifractal analysis, 453 multifractals, 352, 353, 361, 366, 368, 407, 447 turbulence and, 449, 451 wavelets, 409 multigrid, 26, 33, 187 multiplicative structure, 481, 490 multiresolution biorthogonal, 271, 275, 278, 287 r-regular, 146, 147, 165, 166, 267 multiresolution analysis, 23, 28, 41, 42, 44, 45, 47, 48, 53, 121, 146, 184, 191, 192, 278,380 translation, 46 non-standard form, 151, 164, 169, 186, 187, 200, 206, 207, 209, 212,213,215,221,225,230, 244 multiplication of, 241 nonlinearity quadratic, 255, 294 norm L 2 , 320 Sobolev, 330 numerical analysis, 14, 265 numerical methods collocation, 303, 305, 307 oblique projection, 286 operator Calderon Zygmund, 134, 137, 146, 185,207, 209, 238,240, 242 inversion of, 152, 164, 175 compression of, 209, 249 condition number, 281 continuity, 149, 158, 159 convolution, 216, 225, 228 multidimensional, 258 differential, 215 symbol, 210, 228, 273-275, 278 elliptic, 233, 278, 279, 312
507
constant coefficient, 279, 280 finite-difference, 16, 233 homogeneous, 227, 278 integral singular, 134 inversion of, 159-161, 281 kernel, 147, 189 antisymmetric, 148 Calderon-Zygmund, 151, 169, 190 linear, 289 maximal-accretive, 172, 173 non-standard, 147 nonlinear, 303 parabolic, 279 pseudo-differential, 157, 185, 186, 207, 209, 210, 238, 240, 242, 245 rotation, 454 square root, 249 standard, 147 unbounded, 160 optical wavelet transform, 455 orthogonal complement, 41, 43 orthogonality double-shift, 51 frequency domain, 77 orthonormal basis, 2 oscillation, 2 paramultiplication, 161 periodic differential operator, 233 periodic domain, 13 periodicity, 9, 10, 13, 193 phase multifractal, 434 regular, 434 phase transition, 432, 434, 435, 451 point wise analysis, 108 power law, 345, 453 preconditioning, 164, 187, 265, 281, 282, 286 diagonal, 224, 233 Proper Orthogonal Decomposition, 317 pseudo-wavelet, 274, 276 pyramid algorithm, 195, 257, 278, 282 quadratic map, 408
508 quadrature mirror filters, 6, 184, 194, 199, 206, 219 quasi-fractal, 454, 472 random morphologies, 465 reconstruction formula, 383 two-dimensional, 455 regime global, 341 intermittent, 317, 334 laminar, 334 oscillatory, 332, 334, 337 turbulent, 334, 337 regularity local, 341, 343, 344, 437 renormalization group theory, 26 renormalization groups, 406 reproducing kernel, 384 Reynolds number, 337, 338, 341, 344 critical, 337, 338 ridge, 116, 322 ridge extraction, 117 Riesz theorem, 86 rotating disk, 332, 333, 344 scalar product, 271, 283, 307 scale intermittency, 323 scale intermittency factor, 323 scaling function Daubechies, 65 scaling functions, 23, 25, 51, 54, 55, 126 boundary, 22 orthogonality, 23 Schwartz class, 87 screening angle, 470, 479 self-affininity, 371, 466 self-similarity, 360, 371, 397, 398, 409, 434 global, 465 Sherman-Morrison formula, 237 shift invariance, 46 shock waves, 3 signal asymptotic, 113, 115 Brownian, 353 signal processing, 6 singular distributions, 355
Index singularities, 269, 293, 307, 308, 354, 355, 401 singularity complex, 325-327, 345 isolated, 110, 345, 392 location, 205, 255, 266 nonoscillatory, 112 oscillatory, 112, 386 strength of, 352, 362 singularity detection, 401 singularity exponent, 362, 368, 409 singularity spectrum, 353, 354, 361364, 367, 368, 411, 413415, 417, 424, 442,461, 481 dissipation, 444 multifractal, 452 support, 362, 364, 424 velocity, 444, 451 skeleton, 116, 401, 411, 419, 447, 473 skewness, 441 snowftake quasi-fractal, 479 snowflake fractal, 470 deterministic, 464, 465 one-scale, 464 random, 464, 465 snowflake geometry, 454 space adapted, 291, 308 approximation, 121, 265, 266, 294 Besov, 151 BMO, 141, 142, 331 boundary, 293, 294 complex interpolation, 161 cone-structured, 272, 293, 305 functional, 135, 266 Hardy, 145 HHbert, 152, 267 Holder, 325 low-pass, 72 L", 86, 331 I 2 , 266 metric, 357 multiresolution, 265 biorthogonal, 272 Sobolev, 145, 171, 267 wavelet, 268 spectral line extraction, 119
Index spectral lines, 114, 322 spectral method, 12, 25, 308 Chebyshev, 31 Fourier, 31, 273, 305 spectral radius, 290 spline functions, 22, 122 cubic, 49, 50 stability, 12, 281, 285, 286 standard estimate, 136, 138, 149, 168, 170 standard form, 186, 187, 206, 207, 221, 225, 240, 244 multiplication of, 240 stationary point, 116 statistical mono-fractality, 466 statistics nongaussian, 441 stochastic process, 374 structure function method, 376, 431, 451 drawbacks, 379, 380, 427, 431, 432 superconvergence, 14, 15, 30 symmetry five-fold, 454, 471 forbidden, 472 T(b) theorem, 170 T-l theorem, 150, 186, 211 T spectrum, 366, 369, 416, 424, 434 Tauberian theorem, 108 Taylor hypothesis, 442, 446 telescopic series, 241 time continous, 40, 55 discrete, 40 transition, 317, 332, 333 translation, 2 tree algorithm, 270, 271, 284-287, 304 tree matching algorithm, 482, 487, 489, 491 trigonometric polynomials, 41, 194, 217, 224 turbulence, 4, 317, 332, 353, 361 turbulent bursts, 345 turbulent spot, 317 two-scale identity, 79 uncertainty principle, 2, 273, 321
509 vaguelette, 274 viscous fingering, 452 von-Neuman algebra, 152 vortex filaments, 449, 452 wavelet, 2, 3 adapted, 174, 305 analytic, 116 analyzing, 383, 389, 455, 466 anisotropic, 456 basis, 121, 135 biorthogonal, 44 cancellation, 129, 165 coefficient, 16, 90, 255, 267, 286, 318, 319, 354, 401 modulus, 135 compact support, 271, 384, 398 continuous, 454 Daubechies, 4, 16, 32, 54, 133, 196, 220-222, 384 finite interval, 295, 299 Mexican hat, 386, 401, 427, 439 radial, 456 two-dimensional, 456, 479 Meyer, 384 Morlet, 118, 119, 384 mother, 23, 380, 455 optical, 456 orthogonal, 299 orthonormal, 8, 174 partition function, 412 periodized, 295, 296 r-regular, 273 radially symmetric, 460 spline, 24, 29, 33, 132, 269, 275, 304, 384 tensor product, 302 two-dimensional, 302 wavelet collocation, 25 wavelet curve, 117 wavelet packet, 185, 346 wavelet plane, 318, 319 wavelet transform, 85, 318, 354, 380, 388, 394, 395, 397, 409, 424, 447, 454, 483 power-law, 461 semi-discrete, 109 two-dimensional, 455
510 wavelet transform microscope, 454, 460, 479 wavelet transform modulus maxima, 355, 390, 392, 413, 416, 417, 424, 426, 432, 434, 437, 444, 447, 451, 462, 480, 483 wavelet transform skeleton, 462, 474, 475, 480 Wavelet-Optimized Finite-Difference, 16
Index wavelets, 12, 54, 90 turbulence analysis with, 445 wavepacket, 3 weak boundedness property, 148, 151 weak cancellation condition, 211 WOFD, 16, 17 two-dimensional, 21 WTMM skeleton of, 476, 485, 486, 489 two-dimensional, 462
