Pseudosolution of Linear Functional Equations: Parameters Estimation of Linear Functional Relationships

Figure 2.2-2. Comparison of the LSM for simple regression model with a background (a) and the MLM for model of simple active-regression experiment (b).

2.2.3 Degenerated ~ o d eof l Active-Regression Experiment

The degenerated model is obviously "nonsense" at the representation of the predictor matrix and the theoretical H, that is when experiment beforehand is planned and it is possible to provide all troubles. But for the applications concern-

115

Pseudosolution of Linear Functional Equations

ing linear integral equations of the first kind, such supposition is normal, therefore it deserves to study. Assumption 2.2.3. We use the linear influent-regression stochastic model (2.2.1) of active experiment of an incomplete rank, which uses Eq. (2.2.0). Definition 2.2.3. The vector

(B$,s$) T is called a normal solution (normal

vector) of the SLAE (2.0.2) with the incomplete-rank matrix [@,HI if T

(Bg,6$) = Arg

min

1012 +16r

0,6:@0+HS=y,

where the matrix [@,HI and the vector cp are known exactiy. Definition 2.2.4. Let 0,6:(@0+Hs - i)T(EuuT)-l(@O + H S- i ) + lndet EUUT 2 m is the set of admissible estimates (pseudosolutions),where

m2 = E(@G+ HS -

-1 (EUUT)

(@0+ ~g -

a) + lndet

EUU'

Problem 2.2.3. Knowing the approached vector = @0 + H6 + i 0 + Z as realization q, the exact matrices @, Z, P, and that fact, that the matrix [@,HI of the SLAE (2.0.2) has incomplete rank, estimate unknown values of an approximation vector to the normal vector so that the sum of squares of this approximation was minimum on the set of admissible estimates ^T (t^T ,d

)m = Arg rnin l0r +16(2. 0,Sn

Theorem 2.2.3. The solution of the Problem 2.2.3 exists, is unique, satisjies to the relation

(i';dT)

T m

=~ r g

min

p12 + 1612 .

0,6:(@0+~S-$~Y-'(@0+HS-ij)+ln(det Y)+n In2n=m2

The proof of this theorem is similar to item 2.1.4.

116

Alexander S. Mechenov

2.2.4 Regularization in Case of the Homoscedastisity of Errors

We consider the most popular case when errors of a matrix and of a response have the same variance. Assumption 2.2.4. We use the linear injuent-regression stochastic model (2.2.2) of incomplete rank. Thus, perturbations have a scalar covariance matrix as well in errors of a righthand side e as in the matrix J. Then the Problem 2.2.3 has the form. Problem 2.2.4. Knowing the vector { = @0 + H6 + k3 + E as realization q, the exact incomplete-rank matrix A=[@,H] of the SLAE (2.2.0) and values a, p, estimate the approximate vector to the normal vector so that the sum of squares of this approximation was minimum on the set of admissible pseudosolutions

(i' ;QT)

T zu

= Arg

min

~ , s : ( @ ~ + H s (- ~ u ) ~u ~ ~ ~ ( @ 0 + ~ 6 - ~ )Euu')+n1n2&mz +ln(det

ler +IS? .

Corollary 2.2.4. The stable approximation to the normal vector exists, is unique, satisfies to the relation

and the SLAE

where a=lli220 is a numerical Lagrange multiplier. Proof. For the proof it is enough to use 2.2.2 and 2.2.3. Remark 2.2.4. The given problem can be simplified not only as in item 2.1.5, simplifying residue principle, but also knowing that fact, that only one of matrices
Pseudosolution of Linear Functional Equations

2.3. Analysis of Passive-Active Experiment Assumption 2.3. We are given a linearfinctional relation

I

where 8 = [E,l,...,E,rn]is an unknown measured matrix, 9= +l,...,+p] is a known precisely prescribed fill-rank

p = ( p I ,---, /3rn)T and

matrix,

8 = (O1,...,1 3 ~ ) ~are

rankA = rank [E@]= m+ p, unknown

parameters,

r = ( P I ~ - . P ~ ) is an unknown response vector. Also we construct for this linearfinctional relation a structural relation

where i = (il ..,in)T is an unknown measured response vector, F = f 1 , ,a

a random matrix and J is an error matrix. We consider briefly this yet universal experiment description.

I

..fp ] is

2.3.1 Maximum-Likelihood Method of the Parameter Estimation of Linear Model of Passive-Active Experiment

Assumption 2.3.1. We shall assume the linear stochastic model of passiveactive experiment offill rank, which uses the finctional relation (2.0.3) and the structural relation (2.3.0):

and we shall assume that the errors are normally distributed. We assume that the experiment is both passive and active, that is, the researchers prescribe the exact predictor matrix Q, measure the matrix E, and estimate the parameters 8 and p. The researchers have at their disposal a random response vector

118

Alexander S. Mechenov

that is further corrupted by measurement errors e, the random regressor matrix X and the exact predictor matrix @. That is, the model contains structural relations of the Eq. (2.3.0) on which the measurement response errors e are imposed. Models of such type in view of complexity of their parameter estimation were considered by nobody. Though a variant of experiment where the researcher prescribes something, and measures something passively, it seems in many respects natural. The scheme of such an experiment is shown in Figure 2.3-1.

Figure 2.3-1. Scheme of passive-active experiment.

We shall study the estimation of the unknown parameters by the MLM. Problem 2.3.1. Knowing one realization q = EP + a 0 +& + Z of the ran-

=

dom variable q, one realization % = + (rank X=m) of the random matrix X , the nonrandom matrix @ (rank @=p), and covariance matrices M, T , Z , P, estimate the unknown parameters p, 0 of the linear confluent-influent stochastic model (2.3.1) by the MLM. Theorem 2.3.1. The estimates 6,i of the parameters P, 0 of the linear confluent-influent model (2.3.1) of passive-active experiment from Problem 2.3.1 can be computed by minimizing the functional

where Y = {oik +pThilikp-(TJ + T ~ ) +eTpike] P , Z~ = {oik + e T pike). Proof. We change model and statement of the problem. Instead of the matrix X and the vector q we consider the vector z = ( x , - ~ ~of) dimension ~ mn+n. To do this we arrange the matrix X by rows into a row and adjoin to it the row -qT , and we do the same for the vectors l, = (%-cpT)T and w = ( ~ , - u ~ Then ) ~ . the

Pseudosolution of Linear Functional Equations

119

original linear confluent-influent model reduces to a linear regression model with linear constraints T<=-@8

where the constraint matrix r is the same, as in item 1.4.1 and the covariance matrix has the form: zp = {oik +B'P~~o) (its computation is similar to item 2.1.1). The Problem 2.3.1 thus reduces to the following two stage minimization problem: estimate the true values of parameters so that the negative double log-likelihood function

<

is minimized subject to the constraints I?< 4 8 , and then find the minimum of this form over all possible values of the parameters P, 8:

Consider the first stage. We use the method of undetermined Lagrange multipliers, i.e., multiply the constraints r<+W=O by the vector 2h=2(A1,&..., add the product to the negative double log-likelihood and come to the Lagrangian minimization:

$=

min

(z-<) Tn-1 (z-<)+2f ( r ( + ~ ) + h & t ( ~ ) + ( n m + n ) l n 2 n .

~BR~,&R~":T<=-@O

A necessary condition of minimum is equality to zero of derivatives with respect to and h

<

120

Alexander S. Mechenov

To construct the estimator for the vector of undetermined Lagrange multipliers, we left-multiply the first equation of the augmented normal equation by the matrix rC2. Since

<

the estimator of the vector is calculated from the relationship

and its covariance matrix from the relationship

At any gangs of parameters P, the matrix r is a full rank because of presence of the submatrix I. Therefore the minimization problem of the negative double loglikelihood in linear constraints always has the unique solution. Then the original variational problem is rewritten in the form independent of 3 and 9:

R2 = ( 2 - i)T C ~ - ' C ~ Q( -5~- i)+ lndet(C2)+ (nm+ n)ln2n = iTl?C21'Ti+

lndef(C2)+ ( n m + n ) l n 2 ~

= (q - %p - @€I) T Y -1

(i - %p- ale) + lndef(C2)+ (nm+ n)ln 2x

where Y = {oik + pT~,,$ - (T; + T;)P + B'P~~B), Mik are elements of the cell with number ik and dimension mxm of covariance matrix M(mnxmn), Tik is the segment k of dimension m of the row i of the matrix T(nxnm),Pikare the elements of the cell with number ik and dimensionpxp of the covariance matrix P(pnxpn). Differentiating this expression with respect to the parameters P, 8, we obtain the SNAE

Pseudosolution of Linear Functional Equations

121

-T Y -1-X),p-u T -T Y -1-Y-u+2(X aJ 1- -T Y -1Q,),0=2(X T -T Y-@j,j=l,m; 1 2(X

@i

aY -=--

dSZ - {(Pik + ~ l ) T e. )Since the original matrix is nonsingular, the esti-

a, a,

mator 6,i, is the unique solution of Problem 2.3.1. Now, using (*), we easily obtain the estimators of the regressor matrix E and the response vector cp respectively

Theorem 2.3.1.1. The mean of RSS is given by

Eii TY -1-u = n . The obtained result satisfies completely to the correspondence principle. Really, when the matrix E is absent in Eq. (2.3. l), this estimation method is transformed to the MLM [Fedorov 1968, Mechenov 19881, and when the matrix Q, is absent, it is transformed to the LDM [Mechenov 19881.

2.3.2 Homoscedastisity in Experiment Assumption 2.3.2. We assume the following well-posed linear stochastic confluent-influent model of passive-active experiments (the Jirnctional relation (2.0.3), the structural relation (2.3.0))

122

Alexander S. Mechenov

We assume that experiment is both passive and active. Also each random matrix has the own error variance. We consider an estimation of required parameters by the MLM. Problem 2.3.2. Assume that we have one realization ii of the random variable q and one realization 2 of random matrix X and the predictor matrix a. We also have their variances 2, j.?, $. It is required to estimate the parameters p, 8 of the linear stochastic model (2.3.2). Theorem 2.3.2. The estimates of the parameters P, 8 of the linear model (2.3.2)from Problem 2.3.2 minimize

and are calculatedfrom the SNAE

We call this SNAE the normal equation of the passive-active experiment. Proof. We apply the same proof scheme, as in item 2.3.1. Following it, we consider that place where the vector estimate is calculated from the equation (*). Having taken advantage of an estimate for (*), we rewrite the original variational problem in the form independent of 9 and of cp:

<

<

Differentiating this relation with respect to the parameters P, 8, we obtain the SNAE (**), whence we can calculate the required estimate supplying the solution to the Problem 2.3.2.

Pseudosolution of Linear Functional Equations

2.3.3 Singular Model

The degenerated model is obviously nonsense at the representation of the predictor matrix (9 but for the matrix E such supposition can take place in practice therefore it deserves study. Assumption 2.3.3. We use the following ill-posed linear stochastic model of passive-active experiments (theJitnctional relation (2.0.3), the structural relation (2.3.0))

Definition 2.3.3. The vector

T

(pT;eT) is called the normal solution of the 0

SLAE (2.0.3) with the degenerated matrix [E, (91, if

(P T $3T )T = Arg 0

min

lpr +

$,0:E$+@Q=(p

at the known exact matrices E, @ and the vector 9. Before to put a problem, we alter model. As in item 2.3.1, we reduce the original linear confluent-influent model to linear regression model with the linear constraints ry =-@€I

where the matrix r(nxmn+n) has the same appearance, as in item 1.4.1 and the matrix xp = {aik+ eTp*e) (its computation is similar 2.1.1.). Definition 2.3.4. Let r i s the set of admissible pseudosolutions (estimates) b,t: where

min

(;:T<+@t=O

2

1%-&l,+lndetR+(nm+n)lnlais

Alexander S. Mechenov

m2 =

min-

<:T;t;+oe=o

IE

-&la2 + lndet +(nm +n)ln2n

Problem 2.3.3. Knowing the realization vector E containing the fill-rank matrix %, the exact matrices CD, C, T , M,P, the valui d and thatfact, that the matrix [E, CD] of the SLAE (2.0.3) has incomplete rank, calculate the approximation vector to the normal vector so that the square of this approximation was minimum on the set of admissible pseudosolutions

Theorem 2.3.3. The solution of the Problem 2.3.3 exists, is unique, satisjes to the relation

gT;iT

T

(

=Arg A

A

-

2

min

1612 +lif

b, t:,xb+@t-i&, +nlndet C2+(mn+n)1nk=m2

Proof. We take advantage of the Theorem 2.3.1 and rewrite the original variational problem in the form

Since the minimum is attained on the set boundary [Tikhonov & Arsenin 19791, the inequality can be replaced with an equality: T

iT;iT

(

= Arg

b ,t : ~ b + @ t - $ A,.

min

1612 +lif .

+nlndet~+(mn+n)ln2z=m~

We use the method of undetermined Lagrange multipliers. For this purpose we multiply the constraint by the multiplier A, add the product to the minimized quadratic form and come to Lagrangian minimization

The estimates, calculated from this relation, supply the solution of Problem 2.3.3.

Pseudosolution of Linear Functional Equations 2.3.4 Degeneracy and Homoscedastisity

Assumption 2.3.4. We use the following incomplete-rank linear stochastic model of passive-active experiments (the functional relation (2.0.3), the structural relation (2.3.0))

We reduce the original linear confluent-influent model (2.3.4) to the linear relatively 5 regression model with the linear constraints I'c =-me of the form:

Problem 2.3.4. Given the realization vector E, the exact matrix
(6'; iT)T= Arg tD

min

b,tcr

+ /tr.

Theorem 2.3.4. The solution of the Problem 2.3.4 exists, is unique and satis$es to the relation

(6',ir); = ~r~

min

6,i:

P*-f

+nm*$ a2+p21b12+p21t12

+nl n ( d

+dlf )t(nrn+n)1n2n=3

Proof. We take advantage of the Theorem 2.3.3 and we rewrite an original variational problem in the form:

126

Alexander S. Mechenov

(I,^ T ,t^T)

=A&

min

m

6,i:

1612 + 1i12.

+nmlnr2+n fn[02+p21tr) +(nrn+n)fn2n=m2 a2+p21b(2+p21t)2

We apply for a minimum computing the method' of undetermined Lagrange multipliers. For this purpose we multiply the constraint

on the multiplier A,we add the product to the quadratic form and come to minimization of the quadratic form

Differentiating this expression with respect to b, t and A,we obtain the SNAE

where a=lli2>0 is the numerical Lagrange multiplier. The obtained estimate will supply the solution of Problem 2.3.4. In the given paragraph the problem of a parameter estimation of models of well-posed and ill-posed passive-active experiments is solved. For completeness of research it is necessary to supplement only the model investigated above in free (regression) parameters, as it will be done in the following paragraph.

127

Pseudosolution of Linear Functional Equations

2.4. Analysis of Passive-Active-Regression Experiment Having connected all models it is easier to see a real difference between them and to understand their interaction. Assumption 2.4. We consider the following linearjimctional relation

where 8 = [E,l,...,E,m]is an unknown measured matrix, 6 = 41,...,4p] is a

[

known prescribed matrix, H = [ q ,-..,q k ] is a known theoretical& matrix with k, ~=(Q~,-..,B~)~ common r a n k ~ = r a n k [ E , 6 , ~ ] = m + ~ +p=(pl,..-,pm)T, T

and 6 = (Ljl ,...,Ljk) are unknown parameters, cp = ,...,fpn)T is an ~nknown response vector. Wewe enter also the structural relation

corresponding Eq. (2.0.4), where i =(il,...,in)T is an unknown measured re-

[ ...,f p ] is a random matrix and J is its error.

sponse vector, F = f,,

That is the structural relation is a functional relation in which one of three matrices contains an additive random error. 2.4.1 Maximum Likelihood Method of the Parameter estimation of Linear Model of Passive-Active-Regression Experiment

Assumption 2.4.1. We shall assume the fill-rank linear confluent-influentregression stochastic model of passive-active-regression experiment

based on thefinctional relation (2.0.4) and the structural relation (2.4.0). We assume that the experiment is both passive and active, that is, the events occur when E, 0,H, p, 0, 6 have certain values. The researchers have at their disposal the predictor matrix 0,the matrix of theoretical values H, the random response vector

Alexander S. Mechenov

with error vector u=J8+e, and the random confluent matrix X with error matrix C. This model is generated by structural relations of the form i=EP+FB+HG, on which the measurement errors e of the response i are imposed. In such model all nuances of carrying out of experiment are taken into account completely, it contains a confluent, influent and regression part. The scheme of such an experiment is shown in Figure 2.4-1.

In what follows we shall assume that the errors are normally distributed and study the estimation of the unknown parameters by the MLM. Problem 2.4.1. Suppose we know one realization = EP + (DO + H6 + i 8 + Z and = E + (rank X=m) respectively of the random variables q and X , the values of the matrices
A

Pseudosolution of Linear Functional Equations

tional

where Y = {oik + P ~ M -~(T; ~ + ~ T;)P J + B ~ P ~ OZP } , = toik+ e T Pike}.

Proof. We alter model and statement of a problem. Instead of the matrix X and the vector q we consider the vector z = of dimension mn+n. To do this we arrange the matrix X by rows into a row and adjoin to it the row -qT, and we do the same for the vectors = (3,-cpTlT and w = ( ~ , - u ~ Then ) ~ . Eq. (2.4.1) reduces to the linear relatively regression model with linear constraints T<=-@8H6

<

<

where the matrix r is the same, as in item 1.4.1 and the matrix Z = {oik+ 0 T~ikO)(its evaluation is similar 2.1.1). The Problem 2.4.1 thus reduces to the following two-stage minimization problem: estimate the unknown values of parameters so that the negative double logarithm of likelihood function is minimized subject to the constraints T<=-08H6, and then minimize this negative double logarithm of likelihood function over all possible values of the parameters P, 8, 6

<

Let consider the first stage. We use for its solution the method of undetermined Lagrange multipliers. For this purpose we multiply the constraints TC;+
i2= min

min ( i - < ) T R - 1 ( - i - < ) + 2 h T ( r g + O + ~ 6 ) + l n d e t ( ~ . n e nt;rnwn

130

Alexander S. Mechenov

A necessary condition of minimum is equality to zero of the derivatives with respect to the vector C; and the vector h

To construct the estimator for the vector of undetermined Lagrange multipliers, we left-multiply the first equation of the expanded normal equation by the matrix rC2. Since

the estimator Z of the vector C; is calculated from the relationship

and its covariance matrix is equal to

Since the matrix r is nonsingular by virtue of presence of unit submatrix I, the estimator Z is the unique solution of the first stage of Problem 2.4.1. Having taken advantage of the estimator Z (*), we rewrite the initial variational problem in form independent of the unknowns E and
where Y = {qk+ pTlulikj3 - (T: + T;)P + e T p i k 8 ) Mik are the elements of a cell with number ik and dimension m*m of covariance matrix M(mn *mn), Tikis a ?

Pseudosolution of Linear Functional Equations

131

k segment of dimension m of i row of a matrix T (n*nm), Pikare the elements of a cell with number ik and dimension p*p of covariance matrix P,, ., Differentiating this expression with respect to the parameters P, 0 and 6, we obtain the SNAE

where

m

---= -

aY

5= q - @ - @0- HG.

m

aj aj

-= {(Mik + M $ ) T ~- ( T ~ -) (~T ~ ~ ) ~ ) ,

43j

{(Pik + P;)T~) . Since the original matrices

-

A

R , ~ are H non singu-

A

lar, the estimators b,t,d, are the unique solution of the Problem 2.4.1. Having taken advantage of a relation (*), it will be easy to calculate an estimation of the regressor matrix E and of the response vector cp accordingly

Theorem 2.4.1.1. The mean of the RSS is equal to

Eii T Y -1- u = n . 2.4.2 Homoscedastic Experiment

Assumption 2.4.2. Given the functional relationship (2.0.4) and the finctional-structural relationship (2.4.0), we consider the following homoscedastic well-posed linear conjluent-injluent-regression stochastic model of the passiveactive experiment

132

Alexander S. Mechenov

The experiment is homoscedastic, i.e., events occur when e, C, J have respectively constant variances. We consider an estimation of required parameters by the MLM. Problem 2.4.2. Assume given one realization of the random variable q and one realization of random matrix X :

the matrixes @, H. We also know the corresponding variances 2, j?, 8.It is required to estimate the unknowns cp, Z and the parameters P, 0, 6 of linear stochastic model (2.4.2) by the MLM. Theorem 2.4.2. The estimators of the parameters P, 0, 6 of linear model (2.4.2) for Problem 2.4.2 minimize

They are calculatedfrom the SNAE

Proof. We apply the same proof scheme, as in the item 2.4.1. Following it, we consider that place where the estimator of the vector C, is calculated from the relationship

where

133

Pseudosolution of Linear Functional Equations

Then the initial variational problem is rewritten in form independent of E and 0:

Differentiating this expression with respect to the parameters P, 8, 6, we obtain the above-stated SNAE. Remark 2.4.2. It is easy to construct an iterative process that generates the sought estimators. The mean weghted RSS is used as the zeroth approximation. Substituting this value in the SNAE, we solve the SLAE, recalculate the weighted RSS, and so on. 2.4.3 Singular Model

The singular model obviously nonsense at the representation of a predictor matrix Q, or theoretical H, but for the matrix E such supposition is normal, therefore it deserves study. Assumption 2.4.3. We use the following linear stochastic model of an incomplete rank for passive-active-regression experiment (the linear jimctional equation (2.0.4), the structural relation (2.4.0))

Definition 2.4.3. The vector (PT;8T;6T)T,-,refers to as a normal solution of the SLAE (2.0.4) with the degenerated matrix [E,@,H], if T

( ~ ~ ; 8 ~o = ; Arg 6 ~ )

min

P,0,6:BP+@B+H6=(p

+ l8f +16r

134

Alexander S. Mechenov

at known exactly matrixes E, a,H and the vector cp. Before to put a problem, we alter model to the form

(its evaluation is similar 2.4.1). Definition 2.4.4. Let the set of admissible estimators T b,fd:

min

1?-~~+lndet~+(nm+n)ln2x~d

<:rcmt+Hd=O

where

Problem 2.4.3. Given the approached vector z, and also exact matrixes
(6';i';d~)~ = Arg w

min lb12 +lt12 +1d2,

b,t,da-

Theorem 2.43. The solution of the Problem 2.4.3 exists, is unique, satisfies to the relation:

(b^ T , t^T , d^ T )T = Arg

2

min

lbp +lt12 +Id12 .

b,t,d:IXb+@t+~d-~~~+nlndet~+(mn+n)ln2r=w~

Proof. We take advantage of the Theorem 2.4.1 and we copy an initial variational problem as

GT;zT;dT

(

T

)"

= Arg

min lbf +1tl2 +1d12 . b,t,d:I~b+@t+Hd-q(, +nlndet~+(mn+n)ln2r~a?

135 It is known [Tikhonov & Arsenin 19791, that the minimum is attained on the set's boundary, that is

Pseudosolution of Linear Functional Equations

We apply for its evaluation a method of undetermined Lagrange multipliers. For this purpose we multiply the constraint on the multiplier A, we add the product to the quadratic form and in a result we minimize the Lagrangian

Differentiating this expression with respect to parameters and A, we obtain the SNAE and the calculated from this SNAE estimate is a unique solution of the Problem 2.4.3. Remark 2.4.3. But in this case the value a? hardly gives in to an estimation as S2 depends on unknown parameters. Therefore the variant containing only a residue, seems more preferably, especially in view of that fact, that asymptotically they lead to the same outcome. As the RSS estimate is known from the Theorem 2.4.1.1. and is equal to n then Ew T S2 -1 w In Then last equation of this SNAE is possible to replace on

.

2.4.4 Singularity and Homoscedasticity Assumption 2.4.4. We use the following incomplete-rank linear stochastic model of passive-active-regression experiment (the linear Jitnctional equation (2.0.4), the structural relation (2.4.0))

We reduces Eq. (2.4.4) to the linear relatively 6 regression model with linear constraints T&=-@e-H6

136

Alexander S. Mechenov

where the matrix I? is the same, as in item 1.4.1. Problem 2.4.4. Given the approached vector z, and also exact matrixes a, H, corresponding variances d,p2,p2, the value a? and that fact, that the matrix [=,@,HI of the SLAE (2.0.4) has incomplete rank, calculate the unknown values of estimator vector to the normal vector (pT;8T;6T)T so that square of this estimator was minimum on the set of admissible estimators:

(b" T ;t"T ;d"T) L = ~ r gmin \b(Z+I$ +1d2. b,t,d€T

Theorem 2.4.4. The solution of the Problem 2.4.4 exists, is unique, satisjes to the relation:

min

bI2+ltlZ +ld2

1pm+EkI-q2 b,W Srmthp~(2+~~t~~+(lon+n)h2IC;3 2+IUzlbI2+p21tl2

and the SNA E

where e l l b 0 is a numerical Lagrange multiplier. Proof. We take advantage of the Theorem 2.4.3 and we rewrite an initial variational problem in the following form:

Pseudosolution of Linear Functional Equations

137

We apply for an evaluation of a minimum the method of undetermined Lagrange multipliers. For this purpose we multiply the constraint

on the multiplier A, we add the product to the quadratic form and in a result we minimize the Lagrangian

Differentiating this expression with respect to the parameters and the numerical Lagrange multiplier (having replaced a=1/DO),we obtain the required SNAE and the estimator is the unique solution of the Problem 2.4.4. Remark 2.4.4. As the mean of the weighted RSS is equal to n:

that, having replaced compulsorily last line the SNAE and corresponding values, we receive the SNAE

138 Alexander S. Mechenov which solutions asymptotically are identical previous, but it is a little bit easier for computations. We remark that for precise matrices (influent and regession parts) the negative displacement of diagonal elements is not present. For a matrix of a passive part there is a negative displacement. So formulas of regularized solution computation do not coincide. 2.4.5 Summary Table of Experiments

All results are shown in small tables that give a total characteristic of an experimental material and methods of their parameter estimation. By completely simplifjring the description of all the errors occurring in the model (or in the system of linear algebraic equations), that is, assuming they are homoscedastic and have variance 1, one can construct a summary table of the models, quadratic forms, and equations for estimating the parameters (for computing the pseudosolutions) of all possible combinations of measurements and prescriptions of the initial data. Table 2.4.1 contains the different models in presence of l~omoscedastic errors. In Application the tables of models in the presence of the homoscedastic errors (Table 1) and in the presence of the heteroscedastic errors (Table 2) are shown. Table 2.4.1. Models of errors for systems of the linear algebraic equations and methods of a computation of their pseudosolutions. There is the input data in column D and the unknown values in column V.

dodels cp=& A

e equations and quadratic forms, the authors

1

=[$@,HI

[Gauss 1809, Legendre 18061

YP Rep

-u

I I

(RTR- g[%-711)~ =R ~ Y

s2 = m i n 1 % ~ -112 = -A'["-Y] [Pearson 1901, Mechenov 19881

139

Pseudosolution of Linear Functional Equations

[Mechenov 19911

k'@+n1-s21

[Fedorov 19681

)

mTa+nl-s21 e + m T ~ 6 = m T q

H ~ C D € I + H ~=H S~ ~ r j

[Mechenov 19961

p+ZToe=%l'"4

+kTm +nI-

oTWp

I

S ~ I

[Mechenov 19961

( g T t- s 2 1 ) p + gT@9+ g T H 6 = g T i j d t p + ( ~ D T Q+ n1- S ~)eI+ Q I ~ H =S @'a H ' X ~ + ~~m+ H ~ = H ~

[Mechenov 19961

Alexander S. Mechenov

2.4.6 Inference of Chapter 2

The basic result of the second Chapter is construction of model of passiveactive-regression experiment. Thus, the picture of exposition of experimental researches in frameworks of confluent-influent-regression models is completed. Such gradation of experiments allows the contributor to understand better to itself a picture of researches and correctly to carry out a parameter estimation. Really appreciated of usual Gauss transformation, the parameters turn out "underestimated" in case of passive experiment and "overstated in case of active. The method of effective correction of rounding errors is constructed also at the SLAE solution and at the parameter estimation on the computer. They are developed regularized methods of an estimation for a case of the singular matrices. The given approach has ample opportunities of development in a multivariate confluence analysis, in nonlinear models, in model with linear constraints, with the not Gauss error, etc. Tikhonov has applied the regularization term for computation of a solution of integral equations of first kind and the singular SLAE, that is, for problems with strict and beforehand known singularity. In them, for deriving a solution (in view of their infinite number), the additional (a priori) condition of selection of unique solution is necessary. It does not concern to the badly stipulated (nevertheless, stipulated) SLAE, to the account of the a priori information on a solution though in a result these methods can lead to the equations for an evaluation of the solution, similar among themselves. Problems of a solution of integral equations are considered in Chapter 3.

Chapter 3 LINEAR INTEGRAL EQUATIONS

Abstract In this chapter we allow for deterministic and random errors in variational methods that construct the pseudosolution of linear integral equations of the second kind and the regularized pseudosolution and quasisolution of the linear integral equations of the first kind. We consider both passive errors (i.e., errors during observation or measurement) in the right-hand side and passive or active errors (i.e., errors during specification) in the core. We consider the representation methods of the a priori information on a sought pseudosolution using the mixed models and the statistical regularization methods. We construct the numerical realization of these methods.

3. ANALYSIS OF PASSIVE AND OF ACTIVE EXPERIMENTS In Chapter 1 and 2 [Mechenov 1991, 1997, 19981 we have considered the estimation of the pseudosolution (the parameters) of linear functional equations of the first kind such as systems of linear algebraic equations in the presence of various deterministic and random errors in the input data. In this chapter we consider a more complicated form of linear functional equations [Mathematical encyclopedia 19851, namely linear integral equations. Solving a linear integral equation of the first kind involves an ill-posed problem [Tikhonov 19631, whereas a linear integral equation of the second kind has solution from the well-posed problem [Zabreyko, etc. 19681. 3.1 Variational Problems for the Construction of Pseudosolutions of Linear Integral equations

142 Alexander S. Mechenov In this paragraph we compute the pseudosolutions of linear integral equations of the second kind in presence of nonrandom passive or active errors in the core and nonrandom passive errors in the right-hand side. Similarly to [Mechenov 19971, we start with the well-posed problem. 3.1.1 Pseudosolutions of Linear Integral equations of tlie Second Kind Assumption 3.1.1. Take the Fredholm linear integral equation of the second krnd [Zabreyko, etc. 19681

with a suflciently smooth nondegenerate core without singularities. We assume that this relationship is maintained exactly. WeJitrther assume that the constant A is exactly known and is not contained in the spectrum of the integral operator with the core K ( X , S ) . 3.1.1.1 Pseudosolution in the Presence of Deterministic Passive Errors Consider the equation (3.1.1) in the presence of nonrandom passive measurement errors in the right-hand side and in the core. Assumption 3.1.2. We use the following linear model ofpassive observations for the Fredholm linear integral equation the second krnd (3.1.1) in the presence of an LZnonrandompassive measurement error in the core and in the right-hand side

If we have a single observation with an error y" only for the right-hand side of Eq. (3.1.2), the pseudosolution is computed from the standard variational problem (an analog of the least-squares method)

We first consider the construction of the pseudosolution for the case when both the core and the right-hand side are measured with an error.

Pseudosolution of Linear Functional Equations

143

Problem 3.1.1. Given one realization of the approximate right-hand side y" and the approximate core of model (3.1.2), compute the pseudosolution that minimizes the sum of squares of the deviation norms of the approximate input data from the sought data for the equation (3.1.1)

Theorem 3.1.1. Assume that the conditions of the Problem 3.1.1 are satisfied. Then the variational problem (3.1.3) to construct the pseudosolution takes the form

and the pseudosolution tion

(3.1.3) of equation (3.1.2) is computed from the equa-

Proof. Passing from the integral to Darboux sums, that is, making an approximation (without loss of generality, on uniform grids), we replace the equation (3.1.1) with the SLAE

where Kth ,It are matrices such that [Mechenov 19771

and the x- and s-partition of the interval [a,b] are identical, rp, is a vector such that

144

Alexander S. Mechenov

L,, = Ch are the solution vectors on the grids t and h, respectively. Assuming that the approximation errors are much smaller than measurement errors [Mechenov 19771 and therefore can be incorporated in the of input data errors, we consider the variational problem of computing the pseudosolution with a known approximate right-hand side and the matrix yt,kthfor the SLAE (approximated as in Eq. (3.1.1'))

This system has been solved in [Mechenov 19911 (the proof is similar to the Corollary 1.4.3), and its solution minimizes the functional

Differentiating, we find that to estimate the solution we need to solve the equation

Passing to the limit as t+O, h+O, that is, passing from Darboux sums back to integrals, we write the expression in functional form, that is, as the solution of the variational problem (3.1.4). This method is applicable to construct numerically the pseudosolution of linear integral equations of the second kind in the presence of measurement errors in the core and in the right-hand side. 3.1.1.2 Pseudosolution in the Presence of Deterministic Active Errors Let us consider the methods for computing the pseudosolution in the presence of nonrandom active errors associating with the realization of an exactly specified core and the passive errors associate with measurement of the right-hand side. Assumption 3.1.3. We use the following linear model of active observations for the linear integral equation of the second kind (3.1.1):

Pseudosolution of Linear Functional Equations

145

where j(x, s) E LQ ([a,b]x [a.b]] are the realization errors of the exactly specifed core K ( X , SE) & ([a,b] x [a,b]). Exactly specified cores that are realized with an error (an active experiment [Mechenov 19971) are considered here for the first time in the literature. We provide a formal statement of the construction of the pseudosolution. Problem 3.1.2. Given the approximate right-hand side q", the error norm = CT, the exactly specified core K ( X , S ) and the norm of its realization

11~11~

error

llJll

= p for model (3.1.5), compute the pseudosolution.

The total error of the right-hand side u" = q" - C + AKC = -fir + Z depends on the sought solution, and therefore the minimization problem cannot be solved by standard methods, i.e., as

Its solution requires an equivalent of the statistical approach [Mechenov 1997, 20011. Using the result of [Mechenov 19971, we construct a minimizing functional by a method equivalent to the maximum likelihood method. We construct a functional similar to the functional of maximum likelihood method [Kendall & Stuart 19681. For simplicity, we consider the analog of the negative double logarithm of likelihood function, without insignificant constants [Mechenov 19971.

Theorem 3.1.2. Assume that the conditions of the Problem 3.1.2 are satisJied The solution of the variational problem

146

Alexander S. Mechenov

to construct the pseudosolution exists, the pseudosolution is unique and it is computed from the equation

The pseudosolution computed in this way takes into account the active errors in the core. Let us now consider linear integral equations of the first kind, which must be solved using the regularization method. 3.1.2 Regularized Pseudosolution of the Linear Integral equation of the First Kind in the Presence of Measurement Errors in the Right-Hand side Assumption 3.1.4. We use the following Fredholm linear integral equation of the first ktnd

with a suflciently smooth nondegenerate core without singularities, so that the solution is unique. We assume that this relationship is maintained exactly. The Volterra integral equation of the first kind

can be regarded as a Fredholm equation if the core is augmented with zero in the region s> x. The problem of computing the solution of this equation is Hadamard ill posed. The main feature of the various approaches to the solution of this problem with a known error in the right-hand side is that the approximate solution in principle cannot be computed without information about the norm of the right-hand side error [Tikhonov 19631 or the norm of the solution [Ivanov 19631. We will describe these approaches for the case when the right-hand side is measured with an error and then compare them with the previous results for a core with observation or specification errors. We thus assume that only the right-hand side of the equation (3.1.6) contains nonrandom observation errors.

147

Pseudosolution of Linear Functional Equations

Assumption 3.1.5. We use the following linear model of passive observations for the equation (3.1.6) in the presence of a nonrandom passive measurement error in the right-hand side:

where y(x) is an r2[c,d]known function measured with an error. We start by reviewing the essentials of a regularization method [Tikhonov 19631 and statement of the corresponding variational problems [Morozov 1967, 1987, Ivanov 19631. 3.1.2.1 Regularized Pseudosolution. Let us consider a method that utilizes the known norm of the right-hand side errors to compute the pseudosolution (the residue principle [Morozov 19671). Assumption 3.1.6. The set of admissible pseudosolutions u of the equation (3.1.7) is deJned by the inequality

where the error norm o is known in advance. The problem of constructing the normal pseudosolution as the argument minimizing the functional

on the set of admissible pseudosolutions kv has the form

However, this pseudosolution has only weak convergence to exact solution as 04 The .distance of the approximation z(s) from the unknown function ~ ( s is)

1 1 ~ 1 (,1, ~ =

b

therefore measured in the space w2(1) [a.b]:

W2

z2 (s)+ zt2(s)ds [Tikhonov

a

19631. The selection of regularized pseudosolutions (that is, pseudosolutions that strongly converge to the exact solution as 0 4 ) relies on the choice of set of suf-

148

Alexander S. Mechenov

ficiently smooth pseudosolutions [Tikhonov 19631 (a set in the space w2(1) [a,b] [Sobolev 19501). Problem 3.1.3. Given the approximate right-hand side 7,lly"- = o , an exactly specwed nondegenerate core K(x,s)of the equation (3.1.7), and the error norm a,Jind the regularized pseudosolution as the argument that minimizes the stabilizing functional [Tikhonov 19631

qll

on the set of admissible pseudosolutions

~x (3.1.8) has

the form

This approach has been systematically developed in [Tikhonov & Arsenin 19791, [Morozov 19871. It is proved in [Tikhonov & Arsenin 19791, that the regularized pseudosolution obtained in this way stably converges to the exact solution as 04. We write out the solution method used in what follows and the required Euler equation. Since the infinum is attained on the set boundary [Tikhonov & Arsenin 19791, the inequality in Eq. (3.1.9) can be replaced with an equality. Applying the method of Lagrange indeterminate multipliers, we pass to problem

A necessary condition for a minimum of functional Ma[4 is that its first variation with respect to 6 and A, vanish: A M ' [ ~ = OThis . leads to the Euler equation

where L is the Sturm-Liuville operator with boundary conditions that specify equality to zero of a sought solution and (or) its derivatives:

Pseudosolution of Linear Functional Equations

149

If inf r(il)=r(co)
where yis a priori known value of the norm of the exact solution. Problem 3.1.4. Given one realization of the approximate right-hand side 9,lP - = c , an exactly specijied nondegenerate core K of the equation (3.1.7), and the value y, compute the regularized quasisolution as the argument

rll

minimizing the residuefunctional

I$ - K

~ P on the set v =2

The solution of this problem is similar to above-described procedure and only involves a modification of the equation for the Lagrange multiplier. In principle, however, this quasisolution is not associated with the norm of the right-hand side error a. In [Mechenov 1988, 19911 we constructed the normal pseudosolution of the SLAE that allows also for errors in the matrix. We can accordingly extend this result to the problem of constructing the regularized pseudosolution and the regularized quasisolution of linear integral equations of the first kind with the observation errors in the core and in the right-hand side. 3.1.3 Regularized Pseudosolution and Quasisolution in the Presence of Nonrandom Passive Errors in the Core and in the Right-Hand Side

In practice, in addition to measurement errors in the right-hand side we also have to deal with passive measurement errors in the core or errors associated with the realization of the core. We have to distinguish clearly between these two problems. We start with a model of purely passive nonrandom observations (that is, we

150

Alexander S. Mechenov

assume that the core and the right-hand side are observed with errors). A lot of attention was given to this problem [Tikhonov, etc. 19711, [Tikhonov & Arsenin 19791, [Morozov 19871. Assumption 3.1.8. We use the following linear model ofpassive observations [Petrov 19741 for the linear integral equation of the jrst kind (3.1.7) with L2 nonrandom measurement errors in the core and in the right-hand side

3.1.3.1 Regularized Pseudosolution We develop a method that utilizes the known norm of nonrandom passive errors in the right-hand side and in the core to compute the pseudosolution (the distance method [Mechenov 19911). To this end, we define again the set of admissible pseudosolutions. of the equation Assumption 3.1.9. The set of admissible pseudosolutions (3.1.11) is dejned by the inequality

t~ =

{(,

inf

[lly- ~ J (+LK -K11:2]] 6 o2 + p 2 } ,(31.12)

q~,K:Kr=y,

where the error norms o, p are known in advance. Problem 3.1.5. Given one realization of the approximate right-hand side - q~ll= o and of the approximate nondegenerate core - =p ,

K-, I K - K I

y,-l y -

where a, p are known values, jnd the regularized pseudosolution of model (3.1.11) inf

Theorem 3.1.3. Assume that the conditions of the Problem 3.1.5 are satisjed. Then the variational problem to construct the regularized pseudosolution takes the form

Pseudosolution of Linear Functional Equations

zP = arg

inf

Ilw:2

6:

l+llse2

=a2

II~[,~ 2

and the regularized pseudosolution of the equation (3.1.1 1) satisjes the Euler equation

Proof. Passing from the integral to Darboux sums, as in item 3.1.1 (that is, making an approximation), we rewrite the equation (3.1.6) in the form

Ignoring the approximation errors, which by assumption are much smaller than the measurement errors [Mechenov 19781, we consider problem

Its solution has obtained in [Mechenov 19911, [Mechenov 19881. Noting that the sought value of the pseudosolution is attained on set boundary, we can write in form

Passing to the limit as t+O, h+O, that is, passing from the Darboux sums to integrals, we can write the expression in integral form. Solving this variational problem by the method of Lagrange undetermined multiplier, we obtain

Alexander S. Mechenov

Differentiating, we find that for the computation of the regularized pseudosolution requires solving the Euler equation (3.1.14) simultaneously with the equation for the Lagrange undetermined multiplier A. If the second equation in Eq. (3.1.14) is not satisfied at any A> 0, then the pseudosolution is computed for A+@ [Tanana 19811. Theorem 3.1.3a. Assume that the conditions of the Problem 3.1.5 are satisJied. Then the regularized pseudosolution is a stable approximation to the exact solution of the equation (3.1.7). Proof. Define the sequence 6k = { o k , p k ) such that "

llyk - -,Il

l$ -KI

= ok,

= pk

[Morozov & Grebennikov 19921. Let

=2

exact solution. Using the definition of the element z6, ,we obtain

Using the relationship

we have

5 be the

153

Pseudosolution of Linear Functional Equations

Hence we obtain that the sequence zak ,k = 1,2,.... is bounded from above

It follows that we can isolate a weakly convergent subsequence from the sequence zak . Without loss of generality, we assume that zgk A?. Passing to the limit k+

in the inequality

we obtain that lim

ak+o

- (011

%

=

1

-

1

%

= 0 and B is the solution of the

equation (3.1.7). Then it follows from Eq. (3.1.15) that lim llzgk k-m

wil

=~

~.

In the Hilbert space wf) , weak convergence and convergence of the norms imply strong convergence (Arcela theorem- [Kolmogorov & Fomin 1972]), so that lim zg., = i and by uniqueness i = 6 . k+oo

Remark 3.1.2. 1. The Euler equation for. this problem differs from the Euler equation (3.1.10) primarily in the following sense. With respect to the SturrnLiuville operator LZ= z - Z" in the stabilizing operator of the equation (3.1.14)

~

Alexander S. Mechenov

the smoothing part becomes dominant, and the bias is suppressed. This is consistent with common sense: since the core contains an error, that is, is less smooth than the exact core, a better (smooth) solution is obtained by increasing the influence of the smoothing part. 2. The common approach to construction problems of the regularized pseudosolution bases on sets of the admissible pseudosolutions of the form

that is, the functional Q from a pseudosolution less or equal to the functional Q from the exact solution. In principle, we can compute other regularized pseudosolution on other sets of admissible pseudosolutions. For example, the set

which, after carrying out of the same computations, as in Theorem 3.1.3, has the form

allows to calculate the best pseudosolution, but demands knowledge

11k?z- jf

12

llqL. If there are these values we can take advantage of the formula (3.1.12"). If there is only (lqf we can take the formula L2

and

Pseudosolution of Linear Functional Equations

155

At last, taking into account that value of the exact solution norm beforehand very seldom is known, we take advantage of a Cauchy-Bunyakovskii inequality

that in result we receive the formula (3.1.12) of the set of admissible pseudosolutions. The application of the following iterative process for computing the pseudosolution: use the pseudosolution from Eq. (3.1.13) for computing the pseudosolution from Eq. (3.1.12') and then the pseudosolution from Eq. (3.1.12') for computing the pseudosolution from Eq. (3.1.12'7, can improve quality of the last. 3. The Euler equations (3.1.14), (3.1.12') and (3.1.12") for regularization parameter (the Lagrange multiplier) also differ from previously proposed equations [Tikhonov, etc. 19711, [Tikhonov & Arsenin 19791, [Morozov 19871, [Motozov & Grebennikov 19921. The previous procedures [Tikhonov, etc. 19711, [Tikhonov & Arsenin 19791, [Morozov 19871 and many other methods were applied to the problem with passive errors in the core and were strictly heuristic. They were not the result of the variational problem solution on minimization of quadratic functional~on compact sets. In particular, when discussing such a problem with A.G. Jagola back in 1969, the author proposed a clearly similar to the inequality [Morozov 19671

formula for computing the regularization parameter of the Euler equation (3.1.10) from the condition

Of course, this condition does not follow from any variational problem, but it has been tested in practice and as has been shown subsequently [Tikhonov, etc.

156

Alexander S. Mechenov

19711, some modifications of this condition lead to the necessary asymptotic expression, which has been useful for solving a number of physical problems. At that point however, the variational problem (producing the regularized pseudosolution) in the presence of observation errors in the core and in the right-hand side had not been developed. Its algebraic solution for SLAE was only obtained in [Mechenov 19911. Let us compare the result with that proposed in the formula of the generalized residue principle [Tikhonov & Arsenin 19791, [Jagola 1979, 1979a, 1980, l98Oa], which requires solving the following Euler equation

where c is the measure of inconsistency of the equation (3.1.11). Here the same goal (increasing the smoothness of the solution) is achieved by increasing the value of the regularization parameter a in addition to plliall we also introduce the

r,

inconsistency measure in the equation (3.1.10) for the regularization parameter. The inconsistency measure is not needed for computing the regularized pseudosolution, because the following Cauchy-Bunyakovskii inequality holds:

and so

where equality is attained only for p = oll&

I(

L2

.

4. The error norms are the weakest link in the regularization theory, and summation of two squared error norms reduces the probability of an error in estimating the norm. We know [Tikhonov, etc. 19731that the residue principle produces a parameter value that is somewhat greater than optimal.

157 5. In principle, we can easily consider the case when the equation (3.1.7) additionally contains a pure regression part (for instance, in the form of an unknown constant), as has been done in [Mechenov 19911 and in the Table 3 and 4 of Application. 6. This is applicable to nonlinear integral equations of the first kind and to other forms of the operator equations.

Pseudosolution of Linear Functional Equations

3.1.3.2 Regularized Quasisolution Let us construct the solution method that utilizes the known solution norm in the presence of nonrandom passive measurement errors in the right-hand side and in the core (a generalization of the regularized quasisolution method). Problem 3.1.6. Given one realization of the approximate right-hand side y"

and of the approximate core of the model (3.1.1 1) and given the value y bounding the set of admissible quasisolutions, compute the regularized quasisolutions

Theorem 3.1.4. Assume that the conditions of the Problem 3.1.6 are satisJied. Then the variational problem to construct the regularized quasisolution take the form

and the regularized quasisolutions is stable approximation of the exact solution ofEq. (3.1.7). Proof. Again passing from the integral to Darboux sums, we repeat the calculations of Theorem 3.1.2. Passing from the discrete problem to continuous problem, we obtain the equation (3.1.18). Applying the Lagrange method with the multiplier a , we obtain the functional

158 Alexander S. Mechenov Differentiating this functional, we obtain the corresponding Euler equation. The proof of stability of the regularized quasisolution is similar to Theorem 3.1.4. 3.1.4 Allowing of Nonrandom Active Errors in the Core and Passive Errors of the Right-Hand Side

Let us construct methods that compute the solution in the presence of nonrandom active errors arising during the realization of the given core, and passive errors associated with the measurement of the right-hand side. Assumption 3.1.10. We use the following linear model of active observations for a linear integral equation of thefrst kind (3.1.6)

where j(x,s) is the realization error of an exactly specijied core in the experiment. To the best of our knowledge, exactly specified cores that are realized with an error (an active experiment [Mechenov 1997, 1988]), have not been considered in literature. Let us formulate the construction of the residue-based regularized pseudosolution and the regularized quasisolution. 3.1.4.1 Regularized Pseudosolution We construct a method to find the regularized pseudosolution given the norm of nonrandom active errors that arise during the realization of the specified core and the norm of passive measurement errors in the right-hand side (further to us it can be demanded the exact solution norm). Problem 3.1.7. Given the approximate right-hand side 8,118- KG - k,ll= G, the exactly specfled nondegenerate core, the core realization error norm llSll=p for model (3.1.19), and the value q,p, fnd the regularized pseudosolution. Since the total right-hand side error

depends on the sought solution, the set of admissible pseudosolutions

159

Pseudosolution of Linear Functional Equations

will lead to the minimization problem with the nonquadratic functional, that is, to zl = arg

1

inf

IMI'

J,

E-"Y: w2 (~llsll, +.)

Even if we take advantage of the set of admissible pseudosolutions of the form 22

= arg

then, in both cases, we construct a method similar to the method for passive errors, that contradicts sense of the problem. Besides the given methods do not coincide with the statistical approach in which that fact is essentially used, that the full error turns out depending from a sought solution. To solve this problem we apply an equivalent of the statistical approach [Mechenov 19971. Using the result of [Mechenov 19971, we construct a functional equivalent to the MLM [Kendall & Stuart 19691 and we realize it for finding the set of admissible pseudosolutions:

where

This functional distinguishes from a functional implying from the MLM on some constants. In particular, in a finite-dimensional case, after the computing the logarithm of the determinant of expectation of a correlation error matrix, before the logarithm there is a value n, which is more or equal to expectation of the RSS. For the set of admissible pseudosolutions T this value is equal to 1. With the pur-

160

Alexander S. Mechenov

pose to compare the results with the previous results, we write the variational problem for the regularized pseudosolution in the following form. Problem 3.1.7'. Find the regularized pseudosolution as argument of functional injnum

where the value rn is equal to

Theorem 3.1.6. Assume that the conditions of the Problem 3.1.7 are satisjed. Then the variational problem for construction of a regularized pseudosolution has the form: zp = arg

inf

(3.1.22)

Il~~-q"ll:~+(u2+p2)ln (l+\~dl:~) =m2

6:

l+II$

and the regularizedpseudosolution of the equation (3.1.19) satisjes to the Euler equation

and it is unique in area

Pseudosolution of Linear Functional Equations

161

Proof. The proof is similar to Theorem 3.1.2. Theorem 3.1.7. Assume that the conditions of the Problem 3.1.6 are satisJied. Then the regularized pseudosolution is a stable approximation to the exact solution of equation (3.1.6). Proof. Define the sequence 6 k = {ok ,pk such that [Morozov & Grebennikov 19921

Let

be the exact solution. Using the definition of the element isk,we obtain

Hence we have

that is, the sequence z6, ,k = 1,2,.... is bounded from above

It follows that we can isolate a weakly convergent subsequence from the sequence z6, . Without loss of generality, we assume that

Passing to the limit k +co in the inequality

162

Alexander S. Mechenov

we obtain that Nm

6, +o

1 1 - vll~'k ~1 1 -~ 1 =

"L

=0

and i is the solution of the

equation (3.1.7). Then it follows from Eq. (3.1.24) that /'Jlz6,

iw4,

=I lil,!~)

.

In the Hilbert space wz(l), weak convergence and convergence of the norms imply strong convergence (Arcela theorem- [Kolmogorov & Fomin 1972]), so that lim z6, = i and by uniqueness 5 = 4. k-m

Remark 3.1.3. 1. The Euler equation (3.1.23) for this problem differs from the Euler equation (3.1.10) in the following sense. With respect to the Sturm-Liuville operator LZ= z - Z" in the stabilizing operator of the equation (3.1.23)

the smoothing and biasing parts become multiplying by a constant, and biasing part becomes increasing.

Pseudosolution of Linear Functional Equations

163

2. Since the value m is complicated for computing (it depends from the norm of the exact solution), we propose to compute the regularized pseudosolution from the Euler equation (see Eq. (3.1.19))

The changed mode of the parameter choice can be preferable. In this case, the parameter choice does not depend from the exact solution norm. 3.1.4.2 Regularized Quasisolution Let us construct a method that utilizes the known solution norm in the presence of nonrandom passive measurement errors in the right-hand side and of nonrandom active errors in the prescribed core (a generalization of the regularized quasisolution method). Problem 3.1.8. Given the approximate right-hand side q,llg"- KC, - k,l= o,

the exactly specijed nondegenerate core, the core realization error norm p in model (3.1.19) and values a,y;find the regularized quasisolution. Since the total right-hand side error G = - KC = 74+ Z depends on a sought solution, the L2 residue norm inadequately represents the situation and the minimization problem cannot be solved by standard method, that is, in the form

Therefore, correct description of the error norm required applying the equivalent of the statistical approach [Mechenov 19971. Using Eq. (3.1.20), we construct the problem of computing the regularized quasisolution

164 Alexander S. Mechenov Theorem 3.1.8. Assume that the conditions of the Problem 3.1.8 are satisJied. Then the variational problem to construct the regularized quasisolution takes the form

<,=argf

inf

and the regularized quasisolution of the equation (3.1.6) satisJies to the Euler equation

and it is unique in interval

Proof. The proof is similar to Theorem 3.1.2. 3.1.5 Sturm-Liuville Differential Operator

In practice, the use of the Sobolev spaces of higher order than the first is desirable. We study connection between functions from the Sobolev space and the solutions of the Sturm-Liuville differential equation. 3.1.5.1 Sturm-Liuville Differential Operator of the First Order We consider an ordinary linear differential operator

165

Pseudosolution of Linear Functional Equations

Let function K(S)2O, but is not equal to zero everywhere, &)>O and we consider variety twice continuously differentiable functions a s ) such, that r(a)=r(b)=O. It is obvious, that unique function which satisfies to equation Lc=O, at ~(s)=const,S(s)=const, and to these boundary conditions, it will be identically equal to zero. It is known, that the Sturm-Liuville operator is unlimited. We consider the following functional [Tikhonov 1963, Tikhonov & Arsenin 19791: the norm in Sobolev space

5(1) [a,b][Sobolev 19501

Integrating by parts at ga)=c(b)=O, we have

3.1.5.2 Sturm-Liuville Differential Operator of the Order n Similarly above-stated, we consider the function norm in the Sobolev space

P$)[a,b]

c(s)

with the conditions q(s)>O, i 0 , l and function such, that &(a)=&b)=0, i=O, 1,...,n- 1. Fulfilling an integration by parts, we have

where L(") is the Sturm-Liuville differential operator of the order n.

166

Alexander S. Mechenov

3.2 About the Appurtenance of Gauss-Markov Processes to Probability Sobolev Spaces 3.2.1 Random Processes

The given item is necessary for an account of some facts from the theory of random processes that will be used further. Let { S 2 , 7 , ~ )is a probability space, where n={o} is a space of simple events, 7 is the a-algebra of its subsets, which elements refer to as random events, P is a probability (the measure on {S2,7) such that P{n}=l). Also is present the measurable space (W,q in which all singletons are measurable also which elements refer to as conditions [Ventsel 1975, Malenvaud 19701. The random variable w is measurable map of space {Q7,P) in (W, 9 . Under the a-algebra generated by system of random variables w, with values in (Wa,?a), we understand the a-algebra generated by events of an aspect {w,EG}, GE?,, CF {w.}=cF {{w,EG}, GE?,}. Distribution of a random variable w is the measure F, on (W, a), assigned to the relation

A joint distribution of random variables x, y..., z, accepting values in spaces (X, 4, (Y, a),..., (2, $Z) is a distribution of the random vector (x, y..., z)

The random defined on the set T function is function wt(u) which at everyone f ET is measurable on w. Random process with discrete time is a random function, for which T is or set of all whole: f c Z or set of the whole positive: f d . Taking into account a finite number of values f: M = {tl,f2,...,t, ) c T , the distribution F of random variables wt, (a),wt2(o),...,wtm(o) (that is, the vector (wl,w2,...,w,)~) is by definition the probability P {w €A} for set A E F

These distributions at every possible f 1, f2,...,fm€ T carry a title of finitedimensional distributions of a random function. Random process wt(o) refers to stationary, if

Pseudosolution of Linear Functional Equations

identically relatively u, t and at anyone T. Only random process by definition is formed of sequence of independent random variables with the same distribution. Distributions of this process are the following

It is obvious, that this process is stationary.

The moments of the first and second sort of random process will be

The expectation is calculated with the help of a cumulative distribution function O f ,a covariance is calculated with the help of OtIt2. If process is station-

"

ary, the condition Ot,+, = Otl leads to that Ew(t)= is equal to a constant, not dependent from t, and K(t1,t2)=K(tl-t2)=K(u)depends only on a difference of arguments and refers to as an autocovariance. We put rco = a2 ,/ci = cr2f i ,i = 1,. ..,GO. Thus, performances of the first and second order of stationary random process are: expectation ~ i r ,a variance a2, and a correlogram p i ,i = 0,.. oo . All autocorrelation coefficients of correlogram of only random process are equal to zero except for the first. Let random process a,

be stationary random process without correlations. Process of the form wt = yget + ylet-1 +...+Yket-k refers to as sliding average process, where yo,^,...,yk are the fixed numbers. It is easy to calculate its correlogram

168

Alexander S. Mechenov

Hence, the correlogram is equal to zero since valuej=k+ 1. 3.2.2 Norms of Autoregression Processes

We consider the equation

where ,$,b,...,Pk are the fixed numbers, and wt is a stationary random process. We study solutions of this classical linear finite difference equation with constant factors. We replace w, on zt, where z is a complex number

where z,, j=1,. ..,k, are the radicals of this equation. Three types of radicals differ: 1. Iz,l>l, j=l, ...,k, the solutions miss, when t will increase. 2. Iz,l
Stationary random process w, being a solution of the equation

where et is a stationary random process with zero expectation and without correlations, refers to autoregression process. In the case 2, when all radical modules of the equation (3.2.1) are less than unit, it is possible to calculate the following expansion in a series

Pseudosolution of Linear Functional Equations

and then

That is process w,notes as process of sliding average process e, , teZ. From this formula immediately follows, that

Above-stated result allows consider the problem evaluations of association between norms of processes. 3.2.2.1 Norm of Autoregression Random Process of the First Order We consider more in detail autoregression random process of the first order

where the factor of an autoregression p should be less unit: O
that process w, = pw,-l

+ e, can be written in the form

whence any term of a covariance matrix is written in the form

Thus for autoregression processes of the first order we have model

Alexander S. Mechenov

where

Lemma 3.2.2. The sum of squares of elements ("Euclidean norm") of the stationary random process e, with zero expectation and without correlations of the model (3.2.2) is equal to

that is to "Sobolev norm" [Sobolev 19501 of autoregression stationary random process w , where'the norm in 6 f ) is written in a dfirence form (norms are taken so-called as operation of expectation is not written out). Proof. We note, for convenience of the calculations, above-stated model (3.2.2) in the finite number of points t= 1 ,..., k+ 1

or in the following matrix form

Pseudosolution of Linear Functional Equations T We calculate eTe= w T ApApw. AS

that, applying the notation of finite differences, we have [Tikhonov 19631

For stationary random processes factors I-$ and p are positive. 3.2.3 Construction of Gauss-Markov Process 3.2.3.1 Gauss Process We consider construction of random process on a known set of the finitedimensional distributions coordinated naturally. Let {R,,d,&T} is a metric space. As it is known, the direct product of a finite number of metric spaces {R,,d,tcME T,M
Alexander S. Mechenov

PdA *Rm)=PN(A), NcM, A c%.

The measure PN refers to as a projection of the measure PMon (RN,?~). We define still measurable space RT = Rt . For final set M c T and

n t€T

Ac?, the s e t A * R T d T refers to as the cylinder with foundation A, and ?T is the least a-algebra containing all cylinders. Theorem 3.2.3a. ([Kolmogorov 19561, [Ventsel 19751, [Klimov & Kuz'min 19831) Let R, for everyone t ET be fill separable metric space and the on the Jjnal sets M f l satisfi to coset of probability measures PM on (RM7gM) herence condition. Then there is the unique probability measure P on (RT,?T), which projection on (RM7gM)coincides with PM for any set M d , that is P(A*RTw)=PIM(A)for everyoneAc?M.. Corollary 3.2.3. We assume, that conditions of the Theorem 3.2.3a are carried out. Then there is a random process w={w, t e n such that: 1. The random variable w, for everyone t ET accepts values in R, 2. For any final gang M= (tl,...,t,,) c T

for all 13-gl x ? . ..X

C

?

~

where gk = gtk , that is finite-dimensional distribu-

tions of process w coincide with corresponding probability measures of the set coordinated set of measures. I Proof. Really, it is enough to put w,(w)=o(t) for teT, weRt. As w={o(t): t€T) E RT ,then

3.2.3.2 Gauss-Markov Process We consider random process with discrete time. We assume, that w k ac+ ~ on the value wk and does not cepts values from set R,. Let distribution w ~ depends depend on the preceding values we,, Wk.2,. ... That is wk+l=hk(wk,ek), where the Bore1 functions h k are given, and values ek,k1,2,..., m ,... are those, that Ee,ek=O, Ewkek=O. Then this sequence refers to as Markov process.

173

Pseudosolution of Linear Functional Equations

(el

.., Let some vector ( = and a symmetric positive definite matrix k , j ~ M are ) given. We show, that there is a probability space {Ll,7,P) and random process zf, such that ,a

1) Finite-dimensional distributions of process are normal. k= 1,2,...,m; j=1,2,...,m. 2) Ezk= rk, COV(Z~,Z,)=W~, Such process is called normal (Gaussian). We put R ~ = R for ' k= 1,2..., m; Rk = Rm. For any set M={t,,t2,...,tm) we put Q,,={wb; k,jjeM), RM = tk &f

&(G ,&,. ..,em) also we define probability measure PMon (RM, ?M)

(Rm, F), supposing PpN(&&). The coherence condition of the set of measures PM, ME[0,4 at IM 1
It is enough to consider the case ILI=m-1, that is L={tl ,..., tm.l) and if the random vector z = (zl ;.., random

vector

T

2,)

has the normal distributionPpN(&,aM), then the

z =( z , , z

T

has

the

normal

distribution

N , - ~ ( ~ ~ ~ , ~ , ~ - " ) = N , , ~ ( ~ ~where , R ~ )the= P matrix ~ ,

T={s,,

i= 1,. ..,m- 1;j= 1,. ..,m) . Therefore PIM('BMR')=P~ZE 'BMR')=PM(~zE Q=PL(~. NOW according to the Corollary 3.2.3, exist random process z(t), t~ [O,a with values in IR' such that for any final set M={tl,...,tm)je [O,TJ, the distribution of the random vector z=(z(tl),...,z(tm))is the normal distribution PpN(&,QM). In particular, as COVZtl ( ( ).Zt2 ( )) Theorem 3.2.3b. Gaussian process for which the covariance matrix is equal to

is as well Markov process. Proof. [Klimov & Kuz'min 19851. Because of previous it is enough to check up, that for anyone m numbers tl
Alexander S. Mechenov

is positive definite. This matrix can be written as

The matrix A is positively defined by Sylvester criterion [Shikin 19871, because all matrices

have d e t ( ~ k=)al (a2 - al),...,(ak - ak-1) > 0 . For an evaluation of a determinant, we subtract from the last row the penultimate row, etc. Precisely also all matrices

have det(Bk)= (bl - b2)(b2- b3),...,bk > 0 (for an evaluation of determinant it is subtracted the second row from the first row and so on). We show, that matrix W=C?A@B, where A> 0, B> 0 are also positives, is defined at given A and B. Really, we present the symmetric matrix A in the form

Pseudosolution of Linear Functional Equations

175

A=QAQ~,where Q is an orthogonal matrix and A={h&ki} is a scalar matrix. Then

as all b & - 0 . 3.2.3.3 Canonical Expansion Definition 3.2.3. Any representation of process w(t) in the sum of elementary randomfunctions, that is in the form:

where ,u(t), (t),i = 0,m; are nonrandom functions (called coordinate functions), and ei are the random variables (called random factors and satisjed the conditions Eei=O, Eeiei=Di>O,Eeie,=O at iZj), is called canonical expansion of random process w, This implies canonical expansion of correlation function

and variance ~ ( tt ),=

CDiy? ( t ).

The convergence of this positive functional series in each point, for example, an interval [0, is necessary and sufficient for existence on this interval of canonical expansion. That is, for convergence, in sense of average quadratic, of some elementary random functions to the centered cut wp ,t E [0, T] the given process.

176

Alexander S. Mechenov

For construction of the approached canonical expansion of process wt on an interval [0, T ] with known correlation function K(t,s), the interval [0, T ] is divided by points tl,...,t,,. .. in the parts. For random factors of expansion ei are taken the linear combinations of the centered cuts of process w, in the moments tl,...,tm..with constant factors z, . For which evaluation, the following SLAE is solved

applying operation of term-by-term multiplication of expectation and of variance. Theorem 3.2.3~.Gauss-Markov process is the autoregression random process of thejrst order. Proof. We consider Gauss-Markov process with zero expectation and correlation function cov(w,,ws)=~e~4~s~. We discover the approached canonical expansion of this process, having taken for random factors linear combinations of cuts of this process in the moments t=(k-l)h, k=L. .,m; and solving system with unknowns pf Obviously, Ewl=Eel=O, Del=Dl. Further, multiplying termwise the first and the second equations and applying expectation operation, we discover

As el and e2 should be uncorrelated, we receive Eele2=0, whence we discover y 2 1 = e-44 .

(

De2 = D 1-e - 2 al h l )

Further

we

have

2 ~ 2= Y ~ ~2 D + whence E W E ~ ~ ,

. Multiplying now the third equation in turn on the first and

the second, and applying expectation operation, we obtain

etc. Substituting all these values yif in a system of equations and solving they relatively ei, we obtain

Pseudosolution of Linear Functional Equations

177

That is it is shown that Gauss-Markov process will be at the same time autoregression process of the first order. 3.2.3.4 Ergodicity Definition 3.2.3. The random process w(t) refers to ergodic concerning the expectation, if 1) Its expectation is constant: Ew(t) =paonst; 2) The limiting relation takes place: lim f T w(t)dt = p . T+m 0

The following condition lim

T

T+m 0

(I- $ ) ~ ( t ) d =t 0 is necessary and sufi-

cient condition for an ergodicity concerning expectation of the stationary process. The condition k(t)4 at t - m . is sufficient. Theorem 3.2.3d. Markov random process is ergodic. The proof is obvious, as correlation function of this process has the form: k(t) = e-alhl also aspires to 0 at t +m.. From ergodic Birhoff-Hinchin theorem follows, that for a determination of expectation, of variance and of autocorrelation of an ergodic process with probability unit it is possible to be limited only to one realization. 3.2.4 Random Process of Errors of the Right-Hand side

We designate PN a probability density of normal distribution with zero expectation and a variance d.Let on probability space (n,~, p N } the random function e, ,t E T is given. We enter the o-algebra %generated by this randonl function. With o-algebra 3 it is possible to connect the various spaces of random variables generated by a random function e, ,t E T . Let L ~ , Tbe a Hilbert space of the square integrable random variables, generated by e, ,t E T

1

such that e: pNdt < m converges. T

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Alexander S. Mechenov

In this Hilbert space of the square integrable random variables ~2 {Q, 7,PN ) ,the scalar product is defined through (e,c)L2= EeF .

If the random function e, ,t E T is square integrable, then it is defined the scalar product

that expectation and correlation function define a random function, that has one to one correspondence with respect to isometric linear transformation of the space ~2 (Q.7, P N ) that are leaving invariant the vector 1. If T is a final set M then L2,T = R~ also the Hilbert norm becomes Euclidean. 3.2.5 A Priori Distribution of the Required Pseudosolution

The regularization was constructed initially in w2(1) [a,b] [Tikhonov 19631. Various authors on miscellaneous tried to solve a problem of the probability representation of the a priori information about appurtenance to this space of the integral equation solution. In [Turchin 1967, 1968, 19691, [Turchin & Nozik 19691, [Turchin, etc. 19701 " the statistical ensemble of smooth functions " (we underline, that statistic is a function of sample) as "statistical" exposition of the Sobolev space w2(1) [a,b] without engaging probability analog of this space was introduced. In work [Korobochkin & Pergament 19831 the parameter exposition of a solution is offered. We designate PN M a probability density of Gauss-Markov distribution with zero expectation, with variance c? and covariance p = e-"It-'I . Let on probability space ( $ 2 ,p~N.P M ]the random function w,,teT is given. We enter aalgebra YT generated by this random function. With a-algebra YT it is possible to connect the various spaces of random variables generated by a random function

Pseudosolution of Linear Functional Equations

179

(9 be a Hilbert space of the square integrable random variables and of w,. Let w2,T the square integrable their derivatives, generated by w,, t ET

such that

j

W:

pNeMdt< m converges.

T

In this Hilbert space of the square integrable random variables and of the square integrable their derivatives w2(1) {S2,7,pNSM),the scalar product is determined through

If the random function w,, at anyone t€T, is square integrable together with the derivative, then the scalar product which is defined, according to the Lemma 3.2.2, is equal to

As the functions also belong to Lebesgue space, the expectation and correlation function of the random function w, and of its derivative give a random function, that has one to one correspondence with respect to isometric linear transformation of Lebesgue space leaving invariant the vector 1. If T is a final set M, then L2,T = R ,also the Hilbert norm becomes Euclidean power norm. For construction of distribution of these processes we consider the theorem. Theorem 3.2.5. The functions w(rJ with the limited norm in space w$$ with constant factors K 1 and K O

such that wf(a)=w'(b)=Oor w(a)=w(b)=O at a difference approximation of their norm by conservative difference schema of thejrst and second types on uniform

180

Alexander S. Mechenov

grids generate system ofjnite-dimensional distributions of Gauss-Markov random processes. Proof. Under boundary conditions w'(a)=wt(b)=O or w(a)=w(b)=O this norm can be written in the form [Tikhonov 19631

We consider approximation of this norm on a uniform grid, not limiting a generality, for example, the difference scheme of the first type. Then for preservation of symmetry accordirig to the Theorem 3.5.1 it is necessary to approximate an integral on a trapezoid rule and we receive the following difference analogue

where the matrix SZ-'has the form

with boundary conditions w'(a)=w'(b)=Oor it is represented in the form

for the boundary condition w(a)=w(b)=O. From here

where

Pseudosolution of Linear Functional Equations

1

We present p in the form p = e-ah , where a = - -En p. Then, applying h the Theorem 3.2.3b, and supposing the expectation vector is equal to w=(0,0...,o ) and its covariance matrix is equal to

we receive a set of required Gauss-Markov distributions. For the proof termination it is enough taking advantage of the Lemma 3.2.2. 3.2.5.1 Gauss-Markov Distributions of the Order n Let the random variable zs is given on probability space @,7, P M } . We enter a-algebra 7,generated by this random function. With a-algebra 7, we connect space of random variables with integrable in square derivatives up to the order n, generated z,, s ES,

For the representation of a scalar product we consider the theorem. Let w(s)=z(s)-a s ) .

(4

Theorem 3.2.5a. Functions w ( s ) ~% [a,b] with the limited norm in Sobolev space with constantfactors K,, i=O..., n 2

~:i~~(%] ds<w

i=O

ds'

such that w(')(a)=w(')(b)=0,i=1,2,...,n-I, at a dflerence approximation of their norm by conservative drfference scheme of thejrst and second types on uniform grids generate system ofjnite-dimensional distributions of Gauss-Markov random processes of the order n (or otherwise normal autoregression processes of the order n) The proof is similar to the Theorem 3.2.5 with use of the mathematical induction method. So, the scalar product is defined.

~

182

Alexander S. Mechenov

3.3 Fredholm Linear Integral Equations with the Random RightHand Side Errors We consider the pseudosolution estimation of the linear integral equations of the second kind also the regularized pseudosolution and quasisolution estimation of the linear integral equations of the first kind in the presence of random passive measured errors in the right-hand side. Let the random processes of the measured right-hand side belong to Hilbert space of square integrable ergodic random functions. This supposition allows to take advantage of ergodic Birhoff-Hinchin theorem. The theorem states: for estimating the mathematical expectation, the variance and the autocorrelation of an ergodic process with probability unit is possible to be limited only to one realization. 3.3.1 Solution Estimator of Fredholm Linear Integral Equations of the Second Kind Assumption 3.3.1. According to item 3.2, we use the following linear model of passive observation for the Fredholm linear integral equation of the second kind (3.1.1) in the presence of the Lz random errors in the right-hand side: p = 6 - AKc

-

11

q(t) = ~ ( t-) ~ ( s)c(s)ds, t, t E [a,61, a

(3.3.1)

It agrees the item 3.5.1, the approximation of the linear model (3.3.1) leads to the linear regression model

The procedure of a statistical estimation of the pseudosolution, at corresponding extrapolation in the case s , # ti ,is reduced to the LSM. 3.3.2 Regularized Estimators of Fredholm Linear Integral Equations of the First Kind

Pseudosolution of Linear Functional Equations

183

Assumption 3.3.2. According to item 3.2, we use the following linear model of passive observation for the Fredholm linear integral equation of the Jirst kind (3.1.6) in the presence of the L2 random errors in the right-hand side:

It agrees the item 3.5.1, the approximation of this linear integral equation leads to linear regression model

We construct the statistical analog of the variational problems considered in item 3.1 for computing the estimator of the regularized pseudosolution and of the regularized quasisolution of equation (3.3.2). For computing the error norm of the right-hand side we take advantage of the item 3.2.4 where the scalar product is defined. According to this definition, the weighted norm of errors is equal to

3.3.2.1 Estimator of the Regularized Pseudosolution from the Residue Principle Assumption 3.3.2a. The set of admissible pseudosolutions u o f equation (3.3.1) is deJined similarly Eq. (3.1.8) by the inequality

where o2 is the a priori known variance of errors. Problem 3.3.2a. Given one realization of the approximate random righthand side y , its variance an exactly specijied nondegenerate core K(X,S) and the a priori function 2 , compute the regularized pseudosolution of model

d,

184

Alexander S. Mechenov

(3.3.2)as the argument that minimize the stabilizing junctional in the set of admissible pseudosolutions ku 2

zc = Arg minlc - 21

r=

z$)[, b],on

.

w2

Theorem 3.3.2a. Assume that the conditions of the Problem 3.3.2a are satisfied. Then the variational problem to construct the regularized pseudosolution takes the form

Proof. It is similar to the item 3.1.2 [Metchenov 1977, 19821. 3.3.2.2 Estimator of Regularized Quasisolution on the Solution Error Variance Assumption 3.3.2b. The set of admissible pseudosolutions v is defined similarly 3.1.7 by the inequality

V =

where the value 2 is the realization of the random variable z with mathematical expectation 6 and the variance ? is known in advance. Problem 3.3.213. Given one realization of the approximate random righthand side y" and the a priori function 2 , a variance ?, an exactly specified nondegenerate core K(x,s), compute the regularized quasisolution of model (3.3.2) as the argument minimizing the residuefunctional on the set v

Theorem 3.3.2b. Assume the conditions of the Problem 3.3.2b are satisfied. Then the variational problem to construct the regularized quasisolution takes the form

Pseudosolution of Linear Functional Equations

and the regularized relatively Z quasisolution of the equation (3.3.2) satisfies the Euler equation

Proof. It is similar to the item 3.1.2. Thus it would be possible to count constructed a statistical analog of the regularization method. But in the first, it is variational-statistical, instead of only statistical approach, in the second, in practice it is necessary more oRen to use the ready package of programs of linear and curvilinear regression, and they do not provide a solution of such variational problems. Therefore it is desirable to construct the statistical regularization method on the basis of information about errors of random processes and to set the information as superposition of regression models, application to which the standard methods much more simplify the problem of a computation of estimators. 3.3.3 Statistical Regularization Method for the Fredholm Linear Integral Equations of the First Kind

Definition 3.3.2. Mathematical (probability) expression of concept of the a priori information about unobservable nonrandom@nction T(s)we assume a priori distribution P of a random function z(s), having T(s) as mathematical expectation. As a priori distribution of a required pseudosolution of a Fredholm linear integral equation of the first kind we take the Gauss-Markov distribution of the first order, which can be written in the form

which error norm belongs to the Sobolev space. Having taken advantage of the item 3.2.5, we find that the error norm of a solution is described by the following relation

Alexander S. Mechenov *

Combining with the linear model (3.3.2) of linear integral equation of the first kind in the presence of the measured random errors of the right-hand side, we rewrite the model (3.3.2) in the form:

Passing to finite differences in the two equations of the model (3.3.3), we obtain the following mixed regression-autoregression model of the form:

and applying to the presented model the LSM (the regression analysis), we take possession of the stable estimator of the regularized pseudosolution. At a computation of an estimator of the regularized pseudosolution the values o, v or their ratio are computed by iterative methods either from a residue principle or from a quasisolution principle, or from a functional principle (i.e., so that the RSS was equaled to n+m+ 1). Also on this basis, the method of a statistical regularization is constructed. The given approach realizes a more simple form of mixed regression model, than the Bayesian approach [Zhukovskij & Morozov 19721. Remark 3.3.3. As for obtaining the normal smoothness, it is necessary to use the autoregression processes of the order 4-5 [Gor'kov, Mechenov, etc. 19881, the convenient form of their representation is the computation of their factors through radical product computation of the corresponding algebraic equation (3.2.1).

Pseudosolution of Linear Functional Equations

187

3.4 Linear Integral Equation with the Measured or Realized Core Let for linear integral equations the random fields of errors of the measured or realized core and random processes of the measured right-hand side belong to Hilbert space of square integrable ergodic random functions. This supposition allows to take advantage of ergodic Birhoff-Hinchin theorem which states, that for a determination of mathematical expectation, of variance and of autocorrelation of an ergodic process with probability unit it is possible to be limited only to one realization. In this item we develop the variational statements of problems at the statistical approach in the regularization method of solution of linear integral equations of the first kind with random measurements of the right-hand side and with passive errors in random measurement of the core or with active errors in random representation of the core. 3.4.1 Estimate of the Pseudosolution of Linear Integral Equations of the Second Kind at Passive Measurements in the Core and in the Right-Hand Side

We consider the most popular case when we are homoscedastic errors in the core and in the right-hand side of the Fredholm linear integral equation of the second kind (3.1.2). As this problem is well posed, we construct the standard estimation similar to Chapters 1 and 2. 3.4.1.1 Passive Random Errors in the core Assumption 3.4.la. We use the following linear confluent model of passive experiment

where all errors have normal distributions. Similarly item 3.1 we put the following problem. Problem 3.4.la. Given one realization of the approximate right-hand side y" and the approximate core k"(x,s) of model (3.4.la), compute the pseudosolution minimizing the sum of deviation squares of the approximate input data from the sought data for Eq. (3.1.1) 6 - AKc = q~ :

z =A

{

i n

6

Alexander S. Mechenov

[I? + G 2

inf

K,~:C-x6=9

Theorem 3.4.la. Assume that the conditions of Problem 3.4, l a are satisjed. Then the variational Problem 3.4.la to construct the pseudosolution takes the form

(3.4. la)

and the pseudosolution z," is computedfrom the equation

Proof. Passing from the integral to Darboux sums, that is, making an approximation (without loss of generality, on uniform grids), we replace equation (3.1.1) with the SLAE

where Kth , I t are the matrices such that [Mechenov 19781

and the x- and s-partitions of the interval [a,b] are identical, 9, is a vector such that

rt rh are the solution vectors on the grids t and h respectively. Assuming that 3

the approximate errors are much smaller than measurement errors [Mechenov 19771, and therefore can be incorporated in the input data errors, we consider the variational problem of pseudosolution computing with a known approximate righthand side and the matrix yt, kt/,for the SLAE (approximated as in Eq. (3.1.1'))

Pseudosolution of Linear Functional Equations

{

zP,h = arg min t;h

min { ~ t h p ' p&t;h t -&hSh='pt)

This system has been solved in item 1.4, and its solution exists, is unique (the proof is similar to a corollary 1.4.1) and minimizes the functional

Differentiating, we find that to estimate the solution we need to solve the equation

Passing to the limit as t 4 , h 4 , i.e., passing from Darboux sums back to integrals, we write the expression in functional form, i.e., as the solution of the variational problem. This method is applicable to construct numerically the pseudosolution of linear integral equations of the second kind with random measurement errors in the core and in the right-hand side. 3.4.1.2 Random Active Errors in the core Let us consider the methods for computing the pseudosolution estimate in the presence of random active errors associated with the realization of an exactly specified core and the random passive errors associated with observation of the right-hand side. Assumption 3.4.lb. We use the following linear model of active observations for the Fredholm linear integral equation of the second kind (3.1.1)

where j(x,s) are the L2 random realization errors of exactly specfled core.

190

Alexander S. Mechenov

Exactly specified cores that are realized with an error (an active experiment [Mechenov 19961) a considered here for the first time in the literature. We provide a formal statement of the construction of the pseudosolution. Problem 3.4.lb. Given the approximate right-hand side q", the error variance a 2 , the exactly spec@ed core ~ ( x , s )and , the variance p2 of its realization error for model ( 3 . 1 3 , compute the pseudosolution. Whereas the total error of the right-hand side ii = q" - 6 + AK< = - f i r + 2'

14,

depends on the sought solution, its variance has the form a2+ p2 2 (Corollary 2.1.2). We construct a minimizing functional by the MLM (for simplicity we minimize negative double logarithm of likelihood function, without inessential constant)

Theorem 3.4.lb. Let conditions of the Problem 3.4. l b are satisjed. The solution of the variational problem

zp = arg

for estimation of the pseudosolution exists, it is unique and is computedfrom the equation

Proof. We use the approximation as in 3.4.1.1 and the results of item 2.1.2. Let us now consider linear integral equations of the first kind, which must be solved using the regularization method. 3.4.2 Variational Problems of Estimation of Regularized Pseudosolution and Quasisolution in the Presence of Random Passive Errors in the Core and in the Right-Hand side In practice, in addition to measurement errors in the right-hand side, we have to deal with passive measurement errors in the core or errors associated with the re-

191

Pseudosolution of Linear Functional Equations

alization of the core. We have to distinguish clearly between these two problems. We start with a model of purely passive random observations (i.e., we assume that the core and the right-hand side are observed with random errors). Assumption 3.4.2. We use the following linear model of passive observations for the linear integral equation of the jrst krnd (3.1.7) with L2 random normal distributed measurement errors in the core and in the right-hand side:

3.4.2.1 Regularized Pseudosolution

We develop a method that uses the known variances of random passive errors in the right-hand side and in the core to estimate the pseudosolution (the regularized method of distances [Mechenov 19911). To this end, we define again the set of admissible pseudosolutions. Assumption 3.4.2a. The set of admissible pseudosolutions u of the Equation (3.4.2) is defmed by the inequality

K-K pB:KC=p

where the error variances 2, ,dare known in advance. Problem 3.4.2a. Given one realization of the approximate right-hand side y" and of the approximate nondegenerate core g , where a and p are known values; estimate the regularizedpseudosolution of model (3.4.2): z P = arg inf

IR?~,),

Ccu

w2

where the difference analogue of Sobolev norm is taken. Theorem 3.4.2a. Assume that the conditions of Problem 3.4.2a are satisjled. Then the variational problem to estimate the regularized pseudosolution takes the form

Alexander S. Mechenov

and the regularized pseudosolution of Equation (3.4.2) satisJies the Euler equation

Proof. Passing from the integral to Darboux sums as in item 3.4.1 (i.e. making approximation), we rewrite Equation (3.1.6) in the form

Ignoring the approximation errors, which by assumption are much smaller than the measurement errors [Mechenov 19781, we consider the problem zhp = arg

min E

Its solution has been obtained in [Mechenov 19911, [Mechenov 19881. Noting that the mathematical expectation for the degrees of freedom number of the RSS is equal to n (Theorem 1.4. la), and that the sought value of the pseudosolution is attained on the set boundary, we can write the problem in the form

193 Passing to norms in difference spaces, we can write the expression in integral form. Solving this variational problem by the Lagrange method, we obtain

Pseudosolution of Linear Functional Equations

Differentiating, we find, that the computation of the regularized pseudosolution requires solving the Euler equation simultaneously with the equation for the multiplier A. If the second equation in Eq. (3.4.2a) is not satisfied for any b 0, then the pseudosolution is computed for A+oo [Tanana 198 11. 3.4.2.2 Regularized Quasisolutions Let us construct the solution method that uses the known solution norm (or the known norm of a priori solution errors) in the presence of random passive observation errors in the right-hand side and in the core (a regularized quasisolution). Problem 3.4.2b. Given one realization of the approximate right-hand side y" and the approximate core K of model (3.4.2), a given value j! bounding the set of admissible quasisolutions, error variances d and p2,compute the regularized quasisolution

Theorem 3.4.2b. Assume that the conditions of Problem 3.4.2b are satisjed. Then the variational problem to construct the regularized quasisolution takes the form

Proof. Again passing from the integral to Darboux sums, we repeat the computations of Theorem 3.4.2a and we obtain the variational problem 3.4.2b. Applying the Lagrange method with the multiplier a, we obtain the functional

Alexander S. Mechenov

Computing its variation, we obtain the corresponding Euler equation. The proof of stability of the regularized quasisolution is similar to Theorem 3.1.4. 3.4.3 Statistical Regularization Method of the Solution Estimation of Linear Integral equations of the First Kind in the Presence the Passive Measurements of the Core and of the Right-Hand Side

We consider the most popular case, when the Fredholm linear integral equation of the first kind (3.1) has the homoscedastic errors in the core and in the right-hand side. Regularized solutions were searched initially in Sobolev spaces [Tikhonov 19631. Various authors tried to solve a problem of the probability representation of the a priori information that the solutions appertain to the Sobolev spaces. In [Turchin 1967, 1968, 19691, [Turchin & Nozik 19691, [Turchin, etc. 19701 "the statistical ensemble of smooth functions" as exposition of Sobolev space without construction of probability analog of this space was introduced. According to the Theorem 3.3.2 the Gauss-Markov random functions (or, in a discrete aspect, autoregression processes) belong to probability Sobolev spaces [Sobolev 19501. After the probability analog of Sobolev spaces, construction of a statistical image of regularized algorithm and, further, its discrete analog obviously enough is constructed, using, for example Bayes strategy. Assumption 3.4.3a. We use the following linear confluent integral model of passive experiment with the following a priori model for the sought solution

where all measurement errors have normal distributions and the a priori distribution errors have Gauss-Markov distribution. We enter a priori distribution of the sought estimate of a solution as follows: 1. This random function has the sought exact solution (not observable determined function) of linear integral equation of the first kind as the mathematical expectation;

Pseudosolution of Linear Functional Equations

195

2. Its covariance function sets desirable smoothness of a regularized pseudosolution estimate. On the basis of the Theorem 3.2.5 as a priori distribution, we have a possibility to take the Gauss-Markov distribution. To understand as to construct a statistical estimation, we take advantage of problem of the regularized pseudosolution estimation (item 3.4.2a) and of its solution method. Problem 3.4.3a. Given the approximate functions 7 ,k,as realization y, K, and the realization Z offitnction z having Gauss-Markov distribution, of model (3.4.3a), value o, p, estimates unknown values of the pseudosolution by the MLM. Theorem 3.4.3a. Assume that the conditions of the Problem 3.4.3a are satisBed. Then the conjluent problem for construction of an estimate of a regularized solution takes theform

and the normal equation of a conjluence analysis has the form

where one of unknown variances can be computedfrom the equation

Proof. For the proof it is enough to write out the likelihood function for a difference analogue of an integral equation and for a priori model of the sought solution

Alexander S. Mechenov

and to carry out standard procedure for computing the estimate. Remark 3.4.3. 1. We mark that construction of finite-dimensional analog is simply passage to autoregression processes. Realization of it is possible with the help of standard program FUMILI [Sokolov & Silin 19611. 2. For passive measurements in the core it is possible to consider the most popular case of construction of a regularized estimation when the core and right-hand side are measured and background is present

3.4.4 Estimation of the Autocorrelation Function from the Autoregression Model

Maximum-likelihood estimates of parameters of the autoregression models representing temporal relationship are offered. We consider typical prognostic model for representation of random process y, ,t = 1,...,n; that is, autoregression model with a quantity of explanatory variables which in many applications has the form [Malenvaud 19701

Parameters of this model are usually estimated by the LSM and it was demonstrated, that this estimate is relative bad: is displaced and asymmetric [Malenvaud 19701. However the values y, are random, and according to the established concepts of a classical regression the values y in a matrix Y = [y-l, y~ ,...,y-m] should be nonrandom, that in this case is not so. To correct a situation, we pass to a functional relation

Pseudosolution of Linear Functional Equations

Ey=cp=EP+H6, -

where 8= [cp-l,cp-2 ,...,cp-m] is a matrix constructed from vectors cpVi ,i = 1,m; having shifted on i of time units of value q j = Ey and H = [l,l,...,1lT is a column-vector constructed from units, and, accordingly, to other entry of initial model. Usually used for the prognosis the Eq. (3.4.3) from regression turns to model of a confluent analysis of passive experiment

Parameters of such model should to be estimated, accordingly, by the LDM from a minimum of the following quadratic functional [Mechenov 19911

where y is a realization of random process y. As the error variances are supposed identical (actually they can differ a little) the formula for computing the estimate becomes simpler. The value c? also can be estimated. Frequently the values y are correlated and in this case the problem of a parameter estimation becomes complicated (its solution is explained in item 1.4.2). We mark, that the autoregression model is more often digitization of continuous appearances which are in that case described by an integral equation such as convolution 00

yt

=

1q(t

+ + et ,t e [-a,m],

- s)~(s)ds 6

-00

Yt = Pt

2

+ et ,Eet = 0,Eetketi = Bike ,

where P(s) is an autocorrelation function (more often it, at least, continuous). Therefore for deriving the stable estimates As), it is necessary to take into account its appurtenance to space $1 (that is, to compute a regularized pseudosolution). Applying a statistical regularization, which in this case seems to the most approaching, we come to a parameter estimation of model

Alexander S. Mechenov

where b, are the a priori values for the sought vector of parameters P (can be, in particular, a zero vector). 3.4.5 Random Active Errors in the core of the Linear Integral Equation

Let us construct methods that compute the pseudosolution in the presence of random active errors arising during the realization of the given core and passive errors associated with the observation of the right-hand side. We consider the most popular case when errors in the realization of the core and measurement in the right-hand side are homoscedastic. Assumption 3.4.5. We use following linear influent model of active experimentfor the linear integral equation of thejrst kind b

b

a

a

i (t) = j ~ ( st ),~ ( s ) d+sj j(t ,s)C(s)ds, 2

qt =it +et,t ~ [ a , b ] , ~=O,Eetketi e, =Jiko,

(3.4.5)

k(t,s)=r(t.s)+j ( t , s ) . ~ f ( t . s ) = 0 , ~ ~ ( t i , s ~ ) j ( t ~ . s k ) = [ 6 ~ ] p ~ , where the realization errors j(x,s) in the core and the measurement errors e(t) in the right-hand side have the normal distribution. Let us formulate the construction of the residue-based regularized pseudosolution and the regularized quasisolution. 3.4.5.1 Regularized Pseudosolution Assumption 3.4.5.1. Let be the set of admissible pseudosolutions

Problem 3.4.5a. Given the approximate right-hand side q", the exactly speczjes nondegenerate core ~(x,s)for model (3.4.5), the values o, p, and the a

199

Pseudosolution of Linear Functional Equations

priori function ? , estimate the regularized pseudosolution so that minimizes the square of deviation of this pseudosolution from ? in ble pseudosolutions

$1

on the set of admissi-

Theorem 3.4.5a. Assume that the conditions of Problem 3.4.5a are satis8ed. Then the variational problem for a regularized pseudosolution construction takes the form

and the regularized relatively Y pseudosolution of the equation (3.4.5) is computed from the Euler equation

and it is unique in interval

3.4.5.2 Estimation of Regularized Quasisolutions Problem 3.4.5b. Given the approximate right-hand side q , the exactly s p e c ~ e dnondegenerate core, the variances 2, 2 of model (3.43, the value 3: and the a priori realization F , compute the sought quasisolution minimizing the minus double logarithm of a likelihood of deviation of the approximate input data from the sought data in the restriction: quadrate of the weighed deviation of the pseudosolution from a priori realization is less or equal to m

Alexander S. Mechenov

Theorem 3.4.5b. Assume that the conditions of Problem 3.4.5b are satisjed. Then the variational problem for the regularized quasisolution construction takes the form

zp = Arg

min

5.1?-4:(,) wz

IKC - q12 2

2

2

=my 0 + p

14% 2

+n ln(02 + p2 14);

and the regularized relatively Z quasisolution of the equation (3.4.5) is computed from the Euler equation

and it is unique in interval

Complicated experiments can be considered similarly to Chapter 2. Remark 3.4.5. In Application, summary tables of the elementary cases of the representation of the information are reduced: deterministic errors (Table 3) and random homoscedastic errors (Table 4) for the models considered above. It is shown in Table 3 the models of linear integral equations of the first kind with the deterministic errors and the regularization methods for computing the pseudosolution. It is shown in Table 4 the models of linear integral equations of the first kind with random errors and the regularization methods for computing the estimate.

Pseudosolution of Linear Functional Equations

3.5 Technique of Computations In the given paragraph, the approximation of Fredholm linear integral equation and of Volterra linear integral equation is constructed in view of a computation in further their solutions by the regularization method that keeps all mathematical properties of this method. That is, the corresponding difference model for a regularization method of the integral equation solution is created, and numerical algorithms for this model are developed. 3.5.1 Difference analogue of the Regularization Method 3.5.1.1 Approximation of the Smoothing Functional We consider non-uniform grids

Let's designate [Samarskij 19771

We consider approximation

of the smoothing functional M' [K, 6,y ] from the Eq. (3.1.9') so that

where values of functions K(x,s), <(s), y(x), are taken in the set points of grids Kth=

{K(xi,sj)1

ch=

{ 6(sj)1 Y t= {Y ( ~ i1) 7

202

Alexander S. Mechenov

andLhhis some difference scheme of the Sturrn-Liuville operator, , Ch, Cr are the coefficients of quadrature integration formulas (m and n dimensional diagonal matrices). We consider [Zaikin & Mechenov 1971, 1973b1, [Mechenov 19781 a problem of a choice of quadrature coefficients and approximating coefficients of the difference scheme so that the numerical analog of a regularization method would be identical to a regularization method in finite-dimensional spaces. Definition 3.5.1. We name the conservative difference scheme on a nonuniform grid symmetrical with factor Ch (Ch-symmetrical)if at multiplication Lhhto the matrix C h we receive a symmetric matrix of coeflcients Whh=ChLhh of the difference scheme. Theorem 3.5.1. That approximation of a smoothing functional for a linear integral equation of thejrst kind led to afinctional for a SLAE such, that

is necessary and enough the equality of quadrature factors Ch =

e, et

eh= eh,the

ch

equality and the positiveness of quadrature factors C, = = , and the symmetrical conservative difference scheme Lhhthat approximate the operator L. Proof. Necessity. Let approximation of a flinctional y ] is such, that

MYK,~,

.rh,y, ] = Ma [K.<,y] . For the proof of necessity of quadrature coeffi-

M; [Kth

cients' equality we construct the Euler equation for the functional in Eq. (3.5.1)

and we compare it to the algebraic Euler equation for M"K,<,y] :

For realization of the symmetry condition, the matrix chc,K;,ehKh should be

gY,, Ch = eh= eh,

represented as K ~ K that is fulfilled at K = c ~ ~ K y*= ,

e,

C, = = C, and positivenesses of quadrature coefficients C, . Further, for symmetry of the matrix W = chLhh, the conservative difference scheme Lhh should be Ch-symmetrical.

203

Pseudosolution of Linear Functional Equations

Sufficiency.We consider the functional M " K , ~ , ~for ] the SLAE Kc=y

Let's show, that if to take

where Lhhthere is the Ch-symmetrical conservative difference scheme of order 1, then the functional with such expression for K, y, W approximates smoothing functional M " K , L ~ ] for a linear integral equation of the first kind. For this purpose we check up realization of a condition of the order (k,l) approximation, for example, for the first term of the functional ~ " K , l ; y :]

Remaining terms of the functional M ~ K6,y, ] are checked similarly. For realization of conditions of the k order exactitude on non-uniform grids it is necessary, b

2

that for function d s ) , &) in llzllw,l,= ~ ~ ( ~ z " ( ~ + ~ ( s ) zbelonged ~ ( s ) dtos a

class C [a,b], and the core K(x,s)EC[a,b]at everyone fixed x and on x would be summable with quadrate [Samarskij 19771, [Nikolsky 19791. We consider two most practically applicable special cases for the basic types of integral equations: Fredholm and Volterra. 3.5.1.2 Trapezoid rules and Difference Scheme of the First Type Let Ch = diag(hl,h2,...,h,) , C, = diag($ ,q,---,t,)are quadrature coefficients of integration trapezoid rules. Then the matrix K has the form

K =C*&K* y=

= (fj&.(xi,sj)),

and

the

vector

y

has

the

form

gYI = (fiy(xi)] . To these quadrature coefficients there correspond the stan-

dard Ch-symmetrical conservative difference scheme of the first type, where

204

Alexander S. Mechenov

and, if boundary conditions <(a)=<(b)=O to approximate as follows

that reached symmetry of a matrix of the difference scheme. Such approximation satisfies to conditions of the theorem 3.5.1. The trapezoid rule is the best on a class of continuous functions, as well as the conservative difference scheme. Here approximation of the order O(h) on a non-uniform grid is reached and the exactitude of a solution is equal to 0(h2).The given formula approaches for a Fredholm linear integral equation of the first kind as builds a solution in the same points of a grid better. 3.5.1.3 Formulas of Median Rectangles and Difference Schema of the Second Type Let Ch = diag(h2,- . , Ct = diag(t2,.- .,t,) are quadrature coefficients of the integration formula of median rectangles. Hence, the matrix K is equal to e

K = c~&K& to y =

=

.

4

(hl&K(xi

)

t.

-+,rj

-$)} and the right-hand side vector y is equal

gYt ={h4(xi$1) . To these quadrature coefficients, there correspond -

approximation of the Sturm-Liuville differential operator by the standard symmetrical conservative difference scheme of the second type [Tikhonov & Samarskij 1962, 19631

Pseudosolution of Linear Functional Equations

205

with (keeping the symmetry of matrix) approximation of boundary conditions

Thus symmetry of the difference scheme matrix is reached. This approximation mode also satisfies to the theorem conditions of O(h) order approximation on a non-uniform grid is here again reached and the exactitude of a solution is equal o(h2).The given approximation approaches for the Volterra integral equation of the first kind better as builds a solution on the average between points of the grid points. Really, we consider all over a problem of the derivative computation. 3.5.2 Numerical Derivation by the Regularization Method

We consider a problem of the derivative computation from some function y(x), b]; known in n values X I , X ~..., , X, a non-uniform grid u=xo<xI<...<x~=~ and belonging to space Lz[a,b] (random or deterministic functions). The derivative qs), s~[a,b];from precisely set function d x ) is a solution of the Volterra linear integral equation of the first kind

XE [a,

or a Fredholrn linear integral equation of the first kind b

a = KC = [K(X - s)~(s)ds= a(x), x E[a,b ] , a

where &-S) is the Heviside function

It is known, that it is an ill-posed problem, that is, it is necessary to apply the regularized algorithm to its solution. On the other hand, it is a typical problem of passive experiment of regression type. Here there are two problems: how correctly

206

Alexander S. Mechenov

to construct approximation and to apply a regularization method [Mechenov 19881. Theorem 3.5.2. The equation (3.5.4) approximation by the rectangular formula on a non-unlform grid with approximating Heviside core down-triangle matrix Kh of the form

leads to the result that is equal to numerical derivation. Proof. Really, assuming values 91, p,...,qn set ( P I nonzero) and supposing p = 0 , we receive

el

l2 h1 3 = q 2l , where 912 = 91 2

= -

-2P o ,that is,

2

= P1-Po . hl

7

and so on. In the same way hlC1+.+;ben7 I = P,n~I ,where qn-+ = 2

+"-l

2

,that is, in-; = q n -qn-1 h,

Thus, matrix ChLhh,approximating a differential operator in a regularization method for the given problem, will be the same, as in item 3.5.1.3, that is, will correspond to the rectangle formula. Remark 3.5.2. Similarly the problem of derivation can consider a problem of the numerical solution of Volterra linear integral equation of the first kind with a nonsingular core. The obtained result is easily transferred to this case. 3.5.3 Approximation of the Sturm-Liuville Operator of the n Order

We consider construction of a difference analogue of a smoothing functional with a stabilizing operator of the n order [Tikhonov & Samarskij 19621. As practice shows, .to compute the enough smooth solution it is necessary to use the regularization method of 4-th - 5-th orders [Mechenov 19771. Corollary 3.5.3. That the approximation of a smoothing fimctional the n order for a linear integral equation of thejrst krnd was corresponded to condi-

207 tions of the Theorem 3.5.1, it is necessary and sufJicient the Ch-symmetrical conservative difference scheme Lhhthat approximate operator L("): The proof is similar to item 3.5.1. This corollary also is fulfilled on conservative difference schema of the first and second types. Application of difference schema of the high order exactitude [Tikhonov & Samarskij 1962, 19631 in the given cases is not justified [Zaikin & Mechenov 1973, 1973b, Mechenov 19771.

Pseudosolution of Linear Functional Equations

3.5.4 About the Method of the Solution of the Algebraic Equation of Euler

The methods of approximation explained in the previous items, as well as methods of the estimate construction, explained earlier, frequently base on solutions of the algebraic Euler equation

where L is a symmetric positive definite matrix. We need also the condition of the Lagrange multiplier a computation from principle with quadratic dependence from 6 , and from which the Lagrange multiplier a is calculated by iterative methods. For realization of the iterative process, the numerical solution of Euler equation is necessary. The effective method of its solution is offered in [Voevodin 19691 and consists in the following: 1) We decompose matrix L in product NTN,for example, using a square root method. We designate d=N6, A=HN-' ,then the Euler equation will be noted as

and relations are fulfilled

2) Further, making expansion A= QDR by the rotation method (where Q {nxn} is left orthogonal, R{nxk) is right orthogonal, and D{nxk} is upper two-diagonal), lul= 1 ~ ~the 1 )Euler we receive (designating f=Rd and u=Qy, where Ifa = equation

1 I R ~ , 1,

Alexander S. Mechenov

and for r, = Df, - u ,the Euler equation

Remark 3.5.4. The given method of the numerical solution of the algebraic Euler equation is effective practically for all problems, except for in what matrix L varies completely for each value of Lagrange multiplier (as, for example in the LDM and in the other generalizations of LSM). However, in these cases it is desirable to apply this method as it well enough ensures realization of theoretical properties of these methods. It is shown [Mechenov 19771, that the singular expansion [Wilkinson 19651 is preferable from the point of view of observance of the regularization method properties. But it, naturally, does not possess the same practical advantages as works much longer on time. 3.5.5 About Iterative Methods of the Lagrange Multiplier Computation

The idea of these methods belongs, on seen, [Reinsch 1967, 19711 and [Kiuru & Mechenov 19711, [Zaikin & Mechenov 19711; the most strict approach is explained in [Gordonova & Morozov 19721. We consider the functions depending on unknown value a

Where unknown value a is calculated from one of conditions

As an example it is enough to prove one of them. As the third most common the proof for this case also we consider. Theorem 3.5.4. Function

209

Pseudosolution of Linear Functional Equations

is continuous convex downward function at a e(0,m) ( P a-

with a range

~ ,Pa f ' f).

Proof. We consider the regularized normal equation

Using singular expansion H=KAM where K is left and M is right orthogonal matrices, and A is a diagonal matrix, we receive the equation

As thus all transformations do not change magnitude of quadratic forms,

From here, differentiating on multiplier a, we obtain the statement of the theorem. For the value a computing from principle q(a) = qj2 and which, according to the Theorem 3.5.4, exists and is unique, we use the known Newton method 1 1 [Gordonova & Morozov 19721, the most effective for the equation -= .(a)

3

For the equation q(a) = 3 , the Newton method of computation of the unknown radical has the form

210

Alexander S. Mechenov

Starting value of a can be calculated under the formula [Zaikin & Mechenov 19711

If to put

I

2

Y?(ak-l) z ak-ll~ak-,,it will lead to the simplified formula

demanding computation only bn(ak-l). The same linearized formula was used for the residue principle in the standard program [Kiuru & Mechenov 19711. On the effectiveness, it coincides with the secant method (regula falsi) [Mechenov 19771 which as is known [Ostrowski 19731 in this case is more effective than the Newton method. 3.5.6 An inference of Chapter 3 The given methods were applied to the big class of Messbauer spectroscopy problems [Andrianov, etc. 19881, [Gor'kov & Mechenov 19881, [Gor'kov & Mechenov 19901, [Rejman, etc. 19871, for an estimation of the electron distribution function at energies from various performances of gas discharge [Volkova, etc. 19751, [Volkova, etc. 19761, [Volkova, etc. 1976a1, [Volkova, etc. 1976b1, [Volkova, etc. 19781, [Volkova, etc. 1978a1, for calculation of cross-section of photonuclear reactions from the experimental information [Tikhonov, etc. 19731, for an estimation of principal universal performance of the water-wheel and many others. The basic outcomes of the given chapter are: development of the variational approach for pseudosolution construction of linear integral equations, development of the statistical variational approach in a regularization method of construction of a pseudosolution of integral equations of the first kind with approximately known or inaccurately realized operator; determination of stochastic analog of functions from spaces of the Sobolev: them are widely known Gauss-Markov (called also autoregression) processes, construction of a statistical image of a regularization method of a solution of stochastic integral equations of the first kind; application of the obtained outcomes to an estimation of derivatives and autocorrelation functions.

Pseudosolution of Linear Functional Equations

211

Thus variational methods of construction of a regularized pseudosolution and quasisolution of linear integral equations of the first kind and a pseudosolution of linear integral equations of the second kind with the determined both casual passive and active errors of a core and passive errors of a right member are developed; methods of the representation and the account of the a priori information on a required pseudo-solution with the help of construction of the mixed model; statistical regularization methods on the basis of the account of the a priori information; numerical realization of these methods is constructed.

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Alexander S. Mechenov D.K. Faddeev, V.N. Faddeeva, Computing methods in linear algebra. [inRussian] MoscowLeningrad: Fimatgiz (1963) V.V. Fedorov, Analyze of experiments at presence of errors in definition of inspected variables. Preprint 2. [in Russian] Moscow: Izd.MGU (1968) V.V. Fedorov, Theory of optimum experiment {scheduling of regression experiments). [in Russian] Moscow: Science (1971) V.V. Fedorov, "Regression analysis at presence of errors in definition of a predictor," Problems of cybernetics, Ed. 47,69 (1978) D.W. Forrest, Francis Galton. The life and work of Victorian Genius. London: Paul Elek (1974) R. Frisch, Statistical confluence analysis by means of complete regression systems. Oslo (1934) W. Fuller, "Properties of some estimator for the errors-in-variables model," Ann. Stat., 8. 407 (1980) V.Ya. Galkin, A.S. Mechenov, "Regularization of Linear regression problems," Computational mathematics and Modeling, 13, No. 2. 186-200 (2002) F. Galton, Hereditaly genius. An inquiry into its laws and consequences. London: Macmillan (1869) F. Galton, Inquiries into human Faculty and its development. London: Macmillan. XXII (1883) F. Galton, "Regression towards mediocrity in hereditary stature," Journal of the Royal Anthropological Institute. 15. 246-249 (1885) F. Galton, Natural inheritance.London: Macmillan (1889) F. Galton, "Probability, the foundation of eugenics, " The Herbert Spencer lecture delivered on June 5. Oxford. Clarendon press (1907) F.R. Gantmacher, Theorie des matrices. V. 1: Theorie generale. Paris: Dunod, V . 2: Questions speciales et applications. Paris: Dunod (1966) K.F. Gauss, Theoria motus corporum coelestium in sectioninus copecis solem ambientiunt, Hamburgi (1809) K.F. Gauss, Theoria combinationem observationem erroribus mininis, Traduction francaise par J. Bertran, Paris, (1855) G. Golub, W. Kahan, "Calculating the singular values and pseudoinverse of a matrix," J. SZAM Numeric. Anal., Ser. B, 2,205-224 (1965) V.I. Gordonova, V.A. Morozov, Analysis of numerical algorithms of the choice of paranteter in the method of regularization. Scientific report 161-TZ(507). [in Russian] Moscow: Izd.MGU, (1972). V.P. Gorkov, A S . Mechenov, "Numerical methods for solution of inverse problems of Messbauer spectroscopy," Numerical methods for solution of inverse problems of mathematicalphysics. Moscow: Izd. MGU, 69-74 (1988) V.P. Gorkov, A.S. Mechenov, "Estimation of performances of hyperfine interactions of passage 3Q-1 Q from experimental Messbauer spectra," Materials I l l All-Union meetings on nuclear spectroscopic researches of hyperfine interactions. Moscow: M. MGU, 173179 (1990) A.E. Hoerl, "Application of Ridge analysis to regression problems," Chem. Eng. Progr. No. 58. 54-59 (1962) A.E. Hoerl, R.W. Kennard, "Ridge regression: biased estimation for non-orthogonal problems," Technonzetrics. 12. No. 1. 55-67 (1970) S. Hodges, P. Moore, "Data uncertainties and least squares regressions," Appl. Stat., 21.185 (1972) LM. Islamov, "Asymptotic regularization of an ill-posed problem of identification," ZIz. Vychisl.Matem. i Mat. Fiziki, 28, No.6,815-824 (1988)

,

Pseudosolution of Linear Functional Equations LM. Islamov, "Asymptotically optimum method of a solution of an ill-posed problem of classification," Zh. Vychisl. Matem i Mat. Fiziki, 29, No.6,26-38 (1989) V.K. Ivanov, "About Linear ill-posed problems," Dokl. Akad. Nauk SSSR, 145, No.2, 270272 (1962) V.K. Ivanov, "About ill-posed problems," Matem. sbornik, 61, No.2, 187-199 (1963) A.G. Jagola, "About a choice of parameter of a regularization on the generalized principle of the residual," Dokl. Akad. Nauk USSR, 245, l,37-39 (1979) A.G. Jagola, "Generalize a principle of the residual in reflexive spaces," Dokl. Almd. Nauk SSSR, 249, No. l,71-73 (1979a) A.G. Jagola, "About choice of regularization parameter by solution of ill-posed problems in reflex spaces," Zh. Vychisl. Matem i Mat. Fiziki, 20, No.3, 586-596 (1980) A.G. Jagola, "About solution non-linear ill-posed problems with help of generalized residue method," Dokl. Akad. NaukSSSR, 252, No. 4,810-813 (1980a) M.G. Kendall, A. Stuart, Advanced theory of statistics. Vols. 1-3. London: Griffin (19671969) G. Kim, "About statistical models of research of rounding errors in problems of Linear algebra," Computing methods and programming, Ed. XVIII. Moscow: Izd. MGU, 173-187 (1972) E.M. Kiuru, A.S. Mechenov, Standard program of the solution of integral equations of Fredholm of thefirst kind by the regularization method. Ed. 45. [in Russian] Moscow: Izd. MGU (1971) G.P. Klimov, A.D. Kuz'min, Probability, processes, statistics. Problems with solutions. [in Russian] Moscow: Izd. MGU (1985) A.N. Kolmogorov, Success ofMathematica1Sciences, 1. Ed. 1, 57-70 (1946) A.N. Kolmogorov, Foundations of the Theory of Probabilities. Bronx, New York: Chelsea (1956) A.N. Kolmogorov, S.V. Fomin, Felements of Function Theory and Functional Analysis. Moscow: Nauka (1972) T.C. Koopmans, Linear regression analysis of economic time series, Haarlem (1937) A.E. Korobochkin, AKh. Pergament, "A priori Parchment the information and an exactitude of a solution of Fredholm integral equations of the first kind," Theory and methods of a solution it is ill-posed problems in view and their application. Novosibirsk: VC AN USSR, 122 (1983) A.V. Krianev, "Statistical form of a regularized method of least squares of A.N.Tikhonov," Dokl. Akad. Nauk USSR, 291, No.4,780-785 (1986) L. Lebart, A. Morineau, SPAD - Systeme portable pour l'analyse des donnees. Paris: Cesia (1982) A.L. Legendre, Nouvelle methode pour la determination des orbites de cometes. Paris: Courcier (1806) AS. Leonov, "Method of a minimum pseudoinverse matrix for a solution of ill-posed problems of linear algebra," Dokl. Akad. Nauk USSR, 285, No. l,36-40 (1985). AS. Leonov, "Method of a minimum pseudoinverse matrix," Zh. Vychisl. Matem i Mat. Fiziki, 27, No.8, 1127-1138 (1987) A S . Leonov, "To the theory of a method of a minimum pseudoinverse matrix," Dokl. Akad. Nauk USSR, 314, No. l,89-93 (1990) AS. Leonov, "Method of a minimum pseudoinverse matrix: the theory and numerical realization," Zh. Vychisl. Matem i Mat. Fiziki, 31, No.10, 1427-1443 (1991) Ju.V. Linnik, Method and bases mathematical-statistical theory of handling of observations. [in Russian] Moscow: Fizmatgiz, (1962) D.V. Lindley, "Regression lines and the linear fimctional relationship, J. Roy. Stat. Soc. 9, Supplement, No. 1 and No.2,2 18 (1947)

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Alexander S. Mechenov E. Lloyd, W. Ledermann, Handbook of applicable mathematics. Volume VI: Statistics, Part A, B. Chichester, John Wiley and Sons (1987) A. Madansky, "The fitting of straight lines when both variables are subject to error," J. Anter. Stat. Assoc., 54, 173 (1959) D.W. Marquardt, "An algorithm for least squares estimation of nonlinear parameters," J. soc. and Appl. Math. 11,No 2.431 (1963) P.C. Mahalanobis, "On the generalized distance in statistics, " Proceedings of the National Institute of Science, India, 12,49-55 (1936) E. Malenvaud, Methodes statistiques de l'econometrie. Paris: Dunod (1970) A.A. Markov, Calculus ofprobabilities. 4 ed. Moscow: State publishing house (1924) Mathematical encyclopedia. Head ed. LM. Vinogradov. [in Russian] Moscow: Publishing house Soviet encyclopedia (1985) A.S. Mechenov, "About expansion of a solution of equation of the first kind in a series of functions." Computing methods and programming. Ed. XXI. MOSCOW: Izd.MGU, 165-169 (1973) A.S. Mechenov, Some problems of a numerical solution of linear integral equations of Fredholm of thefirst kind by the regularization method. Scientific report No. 200-TZ(591). [in Russian] Moscow: Izd.MGU, 1-134 (1973a) A.S. Mechenov, Numerical solution of linear integral equations of the first kind by a regularization method. [in Russian] Diss. ... Cand. Physical and Mathematical sciences. Moscow: VINITI (1977) A.S. Mechenov, "About a difference approximation of a regularization method," Handling and interpretation ofphysical experiments, Ed. 6. Moscow: Izd.MGU, 105-1 12 (1978) A.S. Mechenov, "About a stabilizing operator in a regularization method," Republican symposium on methods of a solution of the nonlinear equations and optimization problems. Parnu, June, 5-10 1978. Reports. Tallinn, IK AN ESSR, 67-68 (1978a) AS. Mechenov, "Method of a regularization and problem of a linear regression,"Methods of mathematical modelling, automation of handling of observations and their applications. MOSCOW: M. MGU, 88-92 (1986) A.S. Mechenov, Regularized least square method. [in Russian] Moscow: Izd.MGU (1988) A.S. Mechenov, "About in part approached systems of the linear algebraic equations," Zh. Vychisl. Matem i Mat. Fiziki, 31, No.6, 790-799 (199 1) A.S. Mechenov, "A maximum likelihood approach to linear functional relationship," International Congress of Mathematicians ICM-94 Zurich 3-1 1 August 1994. Abstracts. Short Communications. 153 (1994) AS. Mechenov, "The effect of rounding errors for systems of linear algebraic equations," Comp. Maths Math. Phys, 34, No. 10, 1313-1316 (1995) A.S. Mechenov, "Maximum likelihood approach to parameter estimation for linear functional relationships," Computational Mathematics and Modeling, 8 , No. 2, 187-193 (1997) AS. Mechenov, "On the confluent approach in regression analysis," Computational Mathematics and Modeling, 9, No. 3,203-213 (1998) A.S. Mechenov, "Least Distance Method for a confluent model with homoskedasticity in the matrix columns and in the right-hand side," Computational Mathematics and Modeling, 11, NO. 3,299-304 (2000) AS. Mechenov, "Variational problems for the construction of pseudosolutions of linear integral equations," Computational Mathematics and Modeling, 12, No. 3,279-292 (2001) AS. Metchenov, Econometrie et informatique appliquees. [in French] Antananarivo, Universite de Madagascar (198 1) V.A. Morozov, "About a choice of regularization parameter at the solution of the functional equations by regularization method," Dokl. Akad. Nauk SSSR, 175, No.6,1225-1228 (1967)

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98. V.A. Morozov, "About residue principle at the solution of the operational equations by the regularization method," Zh. Vychisl.Matem i Mat. Fiziki, 8, No. 2,295-309 (1968) 99. V.A. Morozov, Regular methods of a solution of ill-posed problems. [in Russian] Moscow: M. MGU (1974) 100. V.A. Morozov, Methods of Solving incorrectlyposed problems. New York: Springer-Verlag (1984) 101. V.A. Morozov, Regularization method of unstable problems. [in Russian] Moscow: Izd. MGU (1987) 102. V.A. Morozov, A.I. Grebe~mikov,Method of a solution of ill-posed problems in view: algorithmic aspect. [in Russian] Moscow: Izd.MGU (1992) 103. M.V. Murav'eva, "About an optimality and limiting properties Bayes solutions of systems of the linear algebraic equations," Zh. Vychisl.Matem i Mat. Fiziki, 13, No.4,819-828 (1973) 104. S.M. Nikolsky, Quadratureformula. [in Russian] Moscow: Science (1979) 105. A.M. Ostrowskij, Solution of equations and system of equations (in Russian] Moscow: Science (1963) 106. K. Pearson, "On lines and planes of closest fit to systems of points in space," Philosophical Magazine, series 6. No. 2. 559-572 (1901) 107. G. Peters, J.H. Wilkinson, "The least squares problem and pseudoinversion," Comput. J. NO. 13. 309-316 (1970) 108. A.P. Petrov, "About the statistical approach to ill-posed problems of mathematical physics," Transactions of all-Union school of young scientists "Methods of a solution of ill-posed problems and their applications I' (October, 9-18, 1973). Moscow: Izd. MGU, 177-18 1 (1974) 109. A.P. Petrov, A.V. Khovansky, "Estimation of an error of solution of linear problems at presence of errors in operators and right-hand sides of the equations," Zh. Vychisl. Matem i Mat. Fiziki, 14, No. 2. 157-163 (1974) 110. Ju.P. Pyt'ev, "Problem of reduction in experimental researches," Matem. sbornik, 20, No.2, 240-272 (1983) 111. C.H. Reinsch, "Smoothing by spline functions,"J. Numer. Mach. 10. No. 3. 177-183 (1967) 112. C.H. Reinsch, "Smoothing by spline functions," J. Numer. Mach. 16. No. 5.451-454 (1971) 113. S.I. Rejman, V.P. Gorkov, A.S. Mechenov, G.K. Rjasnyj, A.A. Sorokin, "Influence of heat treatment of an amorphous powder of an alloy (Fex Nil, ) P B A1 on the nearest environment of atoms of iron," Physic-chemistry amorphous (glass) metal materials. Moscow: Science, 24-37 (1987) 114. A.A. Samarskij, Theory of dzfleree schema. [in Russian] Moscow: Science (1977) 115. G.A.F. Seber, Multivariate observations, Willey (1984) 116. E.V. Shikin, Vector space and maps. [in Russian] Moscow: Izd.MGU (1987) 117. S.L. Sobolev, Some applications of a functional analysis in mathematical physics. [in Russian] Leningrad, (1950) 118. S.N. Sokolov, I.N. Silin, Determination of a minimum of functionals by the linearization method. Preprint OIIaI, D-810, [in Russian] Dubna (1961) 119. P. Sprent, "A Generalized Least Squares Approach to Linear Functional Relationship," J. Roy. Stat. Soc., 28,278 (1966) 120. C. Stein, "Multiple regression," Contrib. probab. statist. Stanford Univ. Press, 424-443 (1960) 121. O.N. Strand, E.R. Westwater, "Statistical estimation of the numerical solution of a Fredholm integral equations of the first kind, "J. of the Assoc. for Computing Machinery 15, No. 1 (1968) 122. V.P. Tanana, Method of a solution of the operational equations. Moscow: Science (1981) 123. A.N. Tikhonov, "About a solution it is ill-posed problems in view and a regularization method," Dokl. Akad. Nauk SSSR, 151, No.3,501-504 (1963)

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124. A.N. Tikhonov, "About ill-posed problems of linear algebra and steady methods of their solution," Dokl. Akad. Nauk USSR, 163, No.3, 591-594 (1965)

125. A.N. Tikhonov, "About a stability of algorithms for a solution of the degenerated systems of the linear algebraic equations," Zh. Vychisl.Matem i Mat. Fiziki, 5, No.4,763-765 (1965a)

126. A.N. Tikhonov, "About one principle of reciprocity," Dokl. Akad. Nauk USSR, 253, No. 2, 302-308 (1980)

127. A.N. Tikhonov, "About normal solutions of the approached systems of the linear algebraic equations," Dokl. Akad. Nauk USSR, 254, No.3,549-552 (1980a)

128. A.N. Tikhonov, "About the approached systems of the linear algebraic equations," Zh. Vychid Matem i Mat. Fiziki, 20, No.6, 1373-1383 (1980b)

129. A.N. Tikhonov, "About problems with inaccurately set initial information," Dokl. Akad. Nauk USSR, 280, No. 3,559-562 (1985)

130. A.N. Tikhonov, V.Ja. Arsenin, Method of a solution of ill-posed problems. [in Russian] Moscow: Science (1979)

13 1. A.N. Tikhonov, A.V. Goncharskij, V.S. Stepanov, A.G. Jagola, Regularized algorithms and the a priori information. [in Russian] Moscow: Izd. MGU (1971) 132. A.N. Tikhonov, A.A. Samarskij, "Homogeneous difference schema of the high order of an exactitude on non-uniform grids," Zh. Vychisl. Matem i Mat. Fiziki, 1, No.3, 425-440 (1961) 133. A.N. Tikhonov, A.A. Samarskij, "Homogeneous difference schema on non-unifonn grids," Zh. Vychisl. Matem i Mat. Fiziki, 2, No. 5,812-832 (1962) 134. A.N. Tikhonov, A.A. Samarskij, "About homogeneous difference schema of the high order of an exactitude on non-uniform grids," Zh. Vychisl. Matem i Mat. Fiziki, 3, No. 1, 99-108 (1963) 135. A.N. Tikhonov, M.V. Ufimtsev, Statistical handling of outcomes of experiments. [in Russian] Moscow: M. MGU (1988) 136. A.N. Tikhonov, V.G. Shevchenko, P.N. Zaikin, B.S. Ishhanov, A.S. Mechenov, "About calculation of cut of photonuclear responses under the experimental information," Bulletin of the Moscow university, seriesjizika i astronomiia, 14, No.3, 317-325 (1973) 137. V.F. Turchin, "Solution of the Fredholm equation of the first kind in statistical ensemble of smooth functions," Zh. Vychisl. Matem i Mat. Fiziki, 7 ,No.6, 1270-1284 (1967) 138. V.F. Turchin, "Choice of ensemble of smooth functions at a solution of an inverse problenl," Zh. Vychisl. Matem i Mat. Fiziki, 8, No. 1,230-238 (1968) 139. V.F. Turchin, "Solution of the Fredholm equation of the first kind in statistical ensemble of smooth functions," Zh. Vychisl. Matem i Mat. Fiziki, 9, No. 6, 1276-1284 (1969) 140. V.F. Turchin, V.P. Kozlov, MS. Malkevich, "Use of statistics methods for a solution of illposed problems," Successes Phys,. Sciences, 102, No. 3,345-386 (1970) 141. V.F. Turchin, V.Z. Nozik, "Statistical regularization of a solution of ill-posed problems," Physics of an atmosphere and ocean, 5, No.l,27-38 (1969) 142. A.D. Ventsel, Rate of the theoly of random processes. [in Russian] Moscow: Science (1975) 143. V.V. Voevodin, Rounding errors and a stability in direct methods of linear algebra. [in Russian] Moscow: M. MGU (1969) 144. V.V. Voevodin, "On a regularization method," Zh. Vychisl. Matem i Mat. Fiziki, 9, N3,673679 (1969a) 145. L.M. Volkova, A.M. Devjatov, A.S. Mechenov, N.N. Sedov, M.A. Sheriff, "Function of electron distribution on energies by regularization method from probe measurements," Bulletin of the Moscow university, seriesjizika i astronomiia, 16, N3,371-374 (1975) 146. L.M. Volkova, A.M. Devjatov, E.A. Kralkina, A.S. Mechenov, "Definition of a cumulative distribution electron function on energies on intensive spectral lines by a regularization method," Ill-posed inverse problems of atomic physics. Novosibirsk: Institute of exact and applied mechanics, 73-9 1 (1976)

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147. L.M. Volkova, A.M. Devjatov, E.A. Kralkina, A.S. Mechenov, "About a possibility of detection of structure of a cumulative distribution electron function on energies from intensity a continuous spectrum, "Handling and intelpretation of physical experiments, Ed. 5, Moscow: Izd.MGU, 104-108 (1976a) 148. L.M. Volkova, A.M. Devjatov, E.A. Kralkina, A.S. Mechenov, V.F. Fazlaev, M. Sheriff, "Computation of a cumulative function of electrons' distribution on energies in gas discharge by the regularization method, "Handling and interpretation of physical experiments, Ed. 4, Izd.MGU, 88-96 (1976b) MOSCOW: 149. L.M. Volkova, A.M. Devjatov, E.A. Kralkina, A.S. Mechenov, "About a possibility of detection of structure of a cumulative distribution of electron function on energies," Informations of the Siberian division of the Academy of sciences of the USSR, a series of engineering science, No. 3, Ed. 1, 56-61 (1978) 150. L.M. Volkova, A.M. Devjatov, E.A. Kralkina, A.S. Mechenov, "Application of Abel inversion for definition of some performances of discharge plasma, " Abel's Inversion and itseer) generalizations. Novosibirsk: Institute of exact and applied mechanics, 200-21 1 (1978a) 151. I. Vuchkov, L. Boyadjieva, E. Solakov, Prilozhen linear regression analysis. [in Bolgarien] Sofia (1985) 152. L. Vuchkov, L. Boyadjieva, "Error in the factor levels and estimation of regression model parameters," J. Stat. Comp. Simul., 19, 1-12 (1981) 153. G. Wahba, "Practical approximate solutions to linear operator equations when the data are noisy," SIAM J. Numer. Anal., 14, No.4,651-667 (1977) 154. N.A. Weiss, Introductory statistics, New York, (1995) 155. J.H. Wilkinson, Rounding errors in algebraicprocesses, Prentice Ha11 (1963) 156. J.H. Wilkinson, The algebraic eigenvalueproblem, Oxford, Clarendon Press (1965) 157. J.H. Wilkinson, "A priori error analysis of algebraic processes," ICM-66. Abstracts of reports. Moscow. 119-129 (1966) 158. J.H. Wilkinson, "The classical error analysis for solution of linear systems, " J. Inst. Math. Appl. 10. 175-180 (1974) 159. M.K. Winkler, "ALAA Paper No. 69-881," A M Guidance, Control and Flight Mechanics Con$, August 18-20,1969 (1969) 160. P.P. Zabreyko, A.I. Koshelev, M.A. Krasnoselskij, S.G. Mikhlin, L.S. Rakovtshik, V.Ja. Stecenko, Integral equation. [in Russian] Moscow: Science (1968) 161. P.N. Zaikin, A.S. Mechenov, Some problems of a numerical solution of integral equations of the first kind by the regularization method. Scientific report 144-TZ(468). [in Russian] Moscow: Izd.MGU (1971) 162. P.N. Zaikin, A.S. Mechenov, "Some problems of numerical realization of regularized algorithm for linear integral equations of the first kind, " Computing methods and programming, Ed. XXI. MOSCOW: Izd.MGU, 155-164 (1973) 163. P.N. Zaikin, AS. Mechenov, "About a choice of a regularized operator at a solution of the operational equations of the first kind," Some problems of the automated handling and interpretation ofphysical experiments, Ed. 1. Moscow: Izd.MGU, 202-208 (1973a) 164. P.N. Zaikin, A.S. Mechenov, "About numerical realization of a regularization method for solution of linear integral equations of the first kind," Some problems of the automated handling and interpretation of physical experiments, Ed. 2. Moscow: Izd.MGU, 124-140 (1973b) 165. A.I. Zhdanov, "Solution of the ill-posed stochastic linear algebraic equations a regularized method of a maximum likelihood," Zh. Vychisl. Matem i Mat. Fiziki, 28, No. 9, 1420-1424 (1989) 166. .L. Zhukovskij, "Stochastic regularization of systems of algebraic equations," Zh. Vychisl. Matem i Mat. Fiziki, 12, No. 1, 185-191 (1972)

Alexander S. Mechenov 167. E.L. Zhukovskij, "About the generalized solution of systems of linear algebraic equations," Dokl. Akad. Nauk USSR, 232, No. 2,269-270 (1977a) 168. E.L. Zhukovskij, "Method for the degenerated and badly stipulated systems of linear algebraic equations," Zh. Vychisl.Matem i Mat. Fiziki, 17, No.4,814-817 (1977b) 169. E.L. Zhukovskij, V.A. Liptser, "On pseudoinverse Bayes regularization of system of algebraic equations," Zh. Vychisl.Matem i Mat. Fiziki, 12, No. 12,464-465 (1975) 170. E.L. Zhukovskij, V.A. Morozov, "About sequential Bayes regularization of system of algebraic equations," Zh. Vychisl.Matem i Mat. Fiziki, 12, No. 12,464-465 (1972)

APPLICATION Tables of the Models, Quadratic Forms, and Equations All results are shown in small tables that give a total characteristic of an experimental material and methods of their parameter estimation. For the system of linear algebraic equations, summary tables of the representation of the information are reduced for the models considered above. By completely simplifying the description of all the errors occurring in the model, that is, assuming they are homoscedastic and have equal variances, one can construct a summary table of the models, quadratic forms, and equations for estimating the parameters (for computing the pseudosolutions) of all possible combinations of measurements and prescriptions of the input data. Table 1 contains the different models in presence of homoscedastic errors. In Table 2 the models and the quadratic forms in the presence of the heteroscedastic errors are shown. For the Fredholm linear integral equations of the first kind summary tables of the elementary cases of the representation of the information are reduced for the models considered above. It is shown in Table 3 the models of linear integral equations of the ,first kind with the deterministic errors and the regularization methods for computing the pseudosolution. It is shown in Table 4 the models of linear integral equations of the first kind with random errors and the regularization methods for computing the estimate of unknown solution.

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INDEX

A priori distribution 178 A priori information 29, 30, 110 Abbreviations ix Active experiment 93 Active-regression experiment 111, 114 Analysis of variance 3 Approximation 144, 151, 180,210 Arcela theorem 153, 162 Autocorrelation function 196 Autocovariance 167 Autoregression model 196 Autoregression model of the first order 169 Autoregression process 168, 169, 176, 186 Bayesian approach 30, 110, 186 Best linear unbiased estimate 7, 9 Birhoff-Hinchin theorem 187 Bivariate model 78 Bivariate regression 8 1 Bore1 field 171 Canonical expansion 175 Cauchy-Bunyakovskii inequality 154, 156 Chi-square distribution 54 Confluent analysis 36 Confluent equation 40 Confluent model 37,39,64 Confluent-regression analysis 67 Consistent estimate 6 Correlogram 167 Correlation function 175

Covariance matrix 54, 116, 173 Crarner rule 42 Darboux sum 143, 151, 153, 157, 188, 192 Degenerated model 56, 84 Degree of freedoms 54 Diagonal matrix 48,209 Dimension 4 Differential operator 164 Efficient estimate 6 Eigenvalue 48 Ergodic theorem 187 Ergodicity 177 Errors of observation 36 Estimate 6 Estimation 53 Estimator 182 Euclidean norm 170 Euclidean power norm 179 Euclidean sample space 4 Euler equation 34, 104, 148, 151 Fredholm linear integral equation of the first kind 146, 182, 185 Fredholm linear integral equation of the second kind 142, 182, 187 Functional equation of the first kind 2 Functional relationship 2 Galton 3,38, 74, 76 Gauss 6 Gauss process 171

Alexander S. Mechenov

Gauss-Markov distribution 178, 181, 195 Gauss-Markov process 171, 172 Gauss-Markov theorem 7 Gaussian process 173 Heteroscedastic 138 Heviside core 205,206 Hilbert norm 178 Hilbert space 177, 178, 179 Homoscedastic errors 86, 98, 109 Homoscedastic experiment 131 Homoscedastic model 10,65, 113 Homoscedastisity 45, 107, 115, 121, 125, 135 Ill-posed problem 18, 56, 84, 182 Influent analysis 93 Influent equation 95 Influent model 94 Influent-regression analysis 103 Kolmogorov theorem 172 Labels ix, 233 Lagrange method 34, 157 Lagrange multipliers 41, 116, 120, 208 Lagrangian 4 1,60, 124 Least distances method 37 Least squares method 5 Least squares-distances method 68 Lebesgue space 179 Legendre 6,35 Likelihood function 112, 129 Log-likelihood function 100 Mahalanobis distance 6 Markov 7 Markov process 172, 177 Matrix formulation for linear model 4 Maximum likelihood estimates 5, 96

Maximum likelihood method 5, 96, 117, 127 Mixed model 29,64, 108 Mechenov two-stage minimization method 40, 119, 129 Multiple linear regression 2 Multivariate normal distribution 176 Newton method 2 10 Nonlinear analysis 55 Nonlinear model 97 Norm 168 Normal distribution 3, 37, 96 Normal equation 6 Normal process 173 Normal solution 88 Normal vector 115 Orthogonal matrix 209 Orthogonal projection 39 Parameter estimation 3 Parameters 3 Passive-active experiment 117 Passive-active-regression experiment 127 Pearson, Karl 38 Point estimation 5 Positive definite matrix 174 Forecasting using autoregression model 196 Prognosis 197 Program 196 Pseudoinverse 48, 63 Pseudoinverse matrix 33 Pseudosolution 61,62, 142, 150 Quadratic form 60 Quadratic functional 155 Quasiestimate 2 1 Quasisolution 2 1,90, 149, 184 Random function 166

Pseudosolution of Linear Functional Equations

23 1

Random process 166 Random variable 122, 166 Random vector 173 Regression 3 Regression analysis 2 Regression equation 3 Regression line 76 Regression models 2 Regression models passing through origin 52

Two-stage minimization problem 40, 119, 129

Regression parameters 3 Regressor 2 Regularization method 115, 147, 20 1 Regularized distances method 56 Regularized pseudosolution 146, 158, 178, 194, 198 Regularized quasisolution 157, 163, 193, 199 Regularized solution 25 Regularized squares method 26 Residual sum of squares 9, 15, 3 1, 42,48,54, 70 Residue principle 183 Rounding errors 102

Weighed quadratic form 47,70 Well-posed problem 3, 36, 67, 94, 111, 117, 127, 142

Schmidt spectral expansion 156 Simple linear regression 74 Singular models 123, 133, 135 Sobolev norm 170 Sobolev space 164, 165, 166, 194 Stationary process 166 Stochastic model 37,68, 95, 112, 117, 123 Sturm separation theorem 48 Sturm-Liuville equation 164 Sturm-Liuville operator 153, 162, 165,206 Sylvester criterion 174 Taylor series 28 Tikhonov 18,56,84, 142, 183

Unbiased estimate 6 Unbiasedness Uniform distribution 79 Unit vector 178, 179 Variance 12, 53 Variance ratio 53 Volterra linear integral equation 205

GLOSSARY OF SYMBOLS

- Tis distributed under the law; T

X ,y is a transposition of matrix X and of vector y; X - is a column vector composed from rows of the matrix X; det F is a determinant of the matrix F; diag (Il,4 , . ., l n ) is a diagonal (scalar) matrix with elements 1,,1, ,...,1, on a principal diagonal; Ik is an identity matrix of the size kxk; xo=argmin f (x) is a value x, supplying a minimum off (x) (similarly arg inff (x)); lim is a common label of a limit;

EC i=lx2 is a square of Euclidean norm of random variables;

11x11; = ~ 1 x =1 ~

1~11: =

=

2

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is a square of Euclidean norm of nonrandom variables;

11x11;~= ~ 1 x 1 := ExTN-'x is a square of power Euclidean norm of random vari-

ables;

1 1 ~ 1 1 : ~ = ~151: = EgTNdlE, is a square of power Euclidean norm of nonrandom variables; n m

=

k i is the sum of squares of matrix elements for random matrices; t=lh=l

tr F is a trace of matrix F; Ex is the expectation a random variable x; Dx is a variance of random variable x; cov (x, h) is a covariance of random variables x and h; N (a,R) is a multivariate normal distribution with expectation vector a and a covariance matrix R; is a chi-square distribution with k degree of freedoms; dim V is a dimension of the space V; Proj~ b is a projection of a vector b on the set B;

xz

Common Labels of Chapter 1 A=[Z,??,H] is an explaining matrix of functional relations (alpha); ~=[P,8,6] is a full vector of required parameters;

234

Alexander S. Mechenov

m+p+k is a dimension of a full parameter vector; cp is a vector of the response (right-hand side); n is a dimension of a vector of a right-hand side. Labels of the Regression Analysis H is a matrix of theoretical values of a linear functional relation; 6 is a vector of required parameters; k is a dimension of a parameter vector; cp = Hd is a vector of a right-hand side; y is a random vector of observations of a right-hand side; e is errors of observations; X is a covariance matrix of observation errors; L[H] is a linear manifold; L(6) is a likelihood function; y is a realization of a random vector y; Z is a vector of a difference - H6 ,the realization of e;

-

;I is the LSM-estimate of a vector of required parameters 6; f

~d

is the LSM estimate of a vector of a right-hand side; 6 = - Hd is a residue vector; t is a vector; G is a matrix; T is a matrix; A is a matrix; c? is a variance of homoscedastic errors of right-hand side observations; i2 is an estimate of a variance c?; M is a functional; A is a matrix of required parameters; 9=HA is a matrix of right-hand sides; k*r is a dimension of a matrix of required parameters A; Y is a random matrix of observations of a right-hand side; n *r is a dimension of a random matrix of observations; E is errors of observations of a matrix of right-hand sides; I' is a matrix of linear constraints; u is a vector of a right-hand side of linear constraints; 1 is a number of linear constraints; h is a vector of undetermined Lagrange multipliers; K is an inverse matrix; is a LSM-estimate of a vector of required parameters in linear constraints; L is a matrix of passage from a LSM-estimate in linear constraints to a LSMestimate; =

A

a

235

Pseudosolution of Linear Functional Equations

9 is a residual sum of squares; is a vector of normal parameters; d is a vector of testing parameters concerning which the normal vector of parameters is searched; N is a matrix of power norm; r is a rank of the degenerated matrix H.

60

Labels For the Analysis of the Mixed Experiments d is a vector of random required parameters (the a priori information); k' is a dimension of a vector of random required parameters; w is a random vector of errors of observations of the a priori information; 2 is a variance of a random observation error vector of the a priori information; K is a known covariance matrix; H- is the generalized pseudoinverse matrix; H + is a pseudoinverse matrix; A, a=l/Ais a numerical Lagrange multiplier; Labels of the Confluence analysis of Passive Experiment -. a is a matrix of a linear functional relation cp=SP; p is a vector of required parameters of a functional relation; m is a dimension of a vector of required parameters; X is a random matrix of observations E; 2 is a realization of a random matrix of observations X; e is an error of a random matrix of observations X; M is a covariance matrix of a random matrix of observation errors C; T is a covariance matrix between C and e; (A); designates row j of the matrix A;

6 is a vector of unknowns (g,-q T )T ,. Z is a matrix of unknowns [X,-q]; z is a random vector

(5,-y

Tr ;

Z is a random matrix [X,-y]; w is a vector of random errors (_C,-e T )T ,. 2=

-T T (g,-y ) is a vector of realization

= [it,-y] is a matrix of realization Z = [X,-y];

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Z is a random matrix of unknowns X,- ,y R is a covariance matrix of the vector w;

;

236

Alexander S. Mechenov

Y is a covariance matrix of the least distances method;

I? is the matrix of linear constraints constructed from elements of the parameter vector b, zeros and units; 6 is a LDM-estimate of a parameter vector; 6 = - Xb is a residue vector; ,u2 is a variance of homoscedastic errors of an observable matrix of confluent model; - A

y,

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is a vector -P,1 ;

Ji! (a) is a RSS function; %+ is a pseudoinverse in the confluence analysis; A is a minimum eigenvalue of a matrix; E is a matrix totally consisting of units; KAM is a singular expansion; K is an orthogonal matrix; A is a cectangular diagonal matrix; M is an orthogonal matrix; G[g] is a linear manifold; Z[z] is a linear manifold; B is a linear manifold; IC is the ratio of variances; is an expectation of the sum of squares of errors of the vector z; B is a matrix of required parameters of the functional relation 9=EB; rn *r is a dimension of a matrix of required parameters; b is a vector of a priori random parameters; rn' is a dimension of a vector of the a priori information on parameters.

2

Labels of Active And Complicated Experiments of Chapter 2 <E, is an assigned matrix of inspected predictors;

F is a random matrix of realization a; J is an error of a random matrix F; P is a covariance matrix of random errorJ; f;; is a realization of random matrix F; J is a realization of the random matrix of errors J; 8 is a vector of predictor parameters; p is a dimension of a vector of predictor parameters; i=F8 is a random vector of the response of structural relationship; q=F8+ e is random observation vector of a response vector; u=J8+e is a full random error of a right-hand side q; ;ti is one realization of a random vector of observations of the response q;

-

Pseudosolution of Linear Functional Equations

-u = 7 - @€I is one realization of a full random error of a right-hand side;

237

R2 is the double negative logarithm of likelihood function; P2 is a variance of homoscedastic errors of row J; 2 is a MLM-estimate of a vector of predictor parameters; R2(a) is a log-likelihood function; a is a parameter; t is a number of digits in a binary representation of a mantissa; fm (n),fv(n) is the functions depending on a computation method of SLAE solution; m2 is an expectation of the minus double logarithm of a likelihood; A. is a undetermined Lagrange multiplier; 80, 60 is a normal vector; t is a vector a priori random predictor parameters; p' is a dimension of the a priori information vector about predictor parameters. Labels For Chapter 3 K = K ( X , S ) is an exact core of linear integral equation; p(x) is a right-hand side of linear integral equation; L2 [c, 4 is a space of Lebesgue of square summable functions; &s), s E [a, b] is an unknown required function; W$ 1) [a,b] is a Sobolev space of summable with square both functions and their

derivatives; K* = K * ( x ,S ) is a conjugate core to core K = K ( X , S ) ; L is a linear differential operator of Sturm-Liuville of the first order; ilis a numerical parameter; ~ [ zis]a stabilizing functional or otherwise norm of functions in the Sobolev spaces w2(1) [a,b];

an[ z ] is a stabilizing functional or otherwise norm of functions in the Sobolev spaces in)[a,b]; Ln is a differential Sturm-Liuville operator of the order n; 02 is a square of norm of right-hand side errors; z(s) is an a priori known function; M [ Z ] is a smoothing functional; y2 is a square of norm of an exact solution; ,dis a square of norm of an error of the representation of an operator; Khr is a matrix, approximating a core; t, h are the approximation grids;

Alexander S. Mechenov

{f2,7,P) is a probability space; S {o)is a space of simple events; 7 is a a-algebra of its subsets; P is a probability; (X, B) is a measurable space; x is a random variable; F is a measure on (X, B), distribution of an random variable x; F, h..., (A) is a joint distribution of random variables x, h,. .., z; x,(cu) is a random function which at everyone t ET is measurable on w; u(t) = E x (t) is the moment of the first order of random process; K(t,s)=E(x(t)- W(t))(x(s)- 4 s ) ) is the moment of the second order of random process; P is an expectation; p, is an autocorrelation coefficient; p, , q = 0,1,2,..., is a correlogram; T is a set; {R, ,d,t E T) is a metric space; RT is a measurable space; L2 {G,?,P) is a Hilbert space of square integrable random variables; Wz(1) {f2,7,P) is a Sobolev space of square integrable random variables with square integrable derivative of random variables; is a formal norm (without expectation operation) for random variables;

I.14 1.1%

is a formal difference analogue of norm (without expectation operation) for

random variables; ( n ) is a formal Sobolev norm (without operation of expectation) for random I.lWz variables; - ( n ) is a formal difference analogue of Sobolev norm (without operation of exl.lw2 pectation) for random variables; W is a set; Q is an orthogonal matrix; D is a rectangular two-diagonal matrix; R is an orthogonal matrix; K is an orthogonal matrix; L is a rectangular diagonal matrix; M is an orthogonal matrix; a is a parameter of a regularization.