Computational Methods in Sciences and Engineering 2003: Proceedings of the International Conference (Iccmse 2003)

Proceed'ings of the

METHODS in Sciences and Engineering 2003 (ICCMSE 2003)

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Proceedinas of the

METHODS in Sciences and Engineering 2003 (ICCMSE 2003) Kastoria, Greece

September 12 - 16

editor

T E Simos

World Scientific New Jersey London Singapore Hong Kong

Published by World Scientific Publishing Co. Re. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING Proceedings of the International Conference 2003 (ICCMSE 2003) Copyright 0 2003 by World Scientific Publishing Co. Re. Ltd. AN rights reserved. This book, or parts thereol; may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permissionfrom the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-595-9

Printed in Singapore by World Scientific Printers (S)Pte Ltd

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PREFACE FOR THE PROCEEDINGS OF THE INTERNATIONAL CONFERENCE OF COMPUTATIONAL METHODS IN SCIENCES AND ENGINEERING (ICCMSE 2003), SEPTEMBER 12-16, KASTORIA, GREECE T.E. SIMOS Department of Computer Science and Technology, Faculty of Sciences and Technology, University of Peloponnese, GR-221 00 Tripolis, Greece cry3 simos-eclilov@uop. Pr E-mail: tsr'mos(i~tnail.ariadne-t.

It is well known that computational methods originally developed in a given basic science, for example physics, can be applied very effectively in other neighboring sciences, e.g. chemistry, biology etc as well as to engineering. So, it is of great importance the communication of computational scientists and the interchange of ideas on computational methods. This is the main scope of the International Conference of Computational Methods in Sciences and Engineering (ICCMSE), which I have found. This year the International Conference of Computational Methods in Sciences and Engineering 2003 (ICCMSE 2003) is taken place in Kastoria, Greece. We have a lot of submitted papers and the rejected rate (after international peer review) was approximately 50%. I want to thank World Scientific Publishing Company for the excellent cooperation for the publication of Proceedings (Extended Abstracts) of ICCMSE 2003.

Many thanks to Editors-in-Chief and Publishers of the journals that accept to publish selected full paper of ICCMSE 2003. The selection will be based on the international peer review of at least two independent reviewers.

I want also to thank: 0

The Scientific Committee:

vii

...

Vlll

Prof. H. ¥, Royal Institute of Technology, Sweden Prof. H. Arabnia, The University of Georgia, USA Prof. J. Vigo-Aguiar, University of Salamanca, Salamanca, Spain Prof. D. Belkic, Karolinska Institute, Stockholm, Sweden Prof. K. Belkic, University of Southern California, USA Prof. E. Brandas, University of Uppsala, Uppsala, Sweden Prof. J.C. Butcher, The University of Auckland, New Zealand Prof. A.Q.M. Khaliq, Western Illinois University, USA Prof. G. Maroulis, University of Patras, Patras, Greece Prof. R.M. Nieminen, Helsinki University of Technology, Finland Prof. S. Wilson, Rutherford Appleton Laboratory, UK Prof. J. Xu, Pennsylvania State University, USA for their help and their important support. I must note here that it is a great honour for me that the above leaders on computational Sciences have accepted to participate Scientific Committee of ICCMSE 2003. The invited speakers for their acceptance to give keynote lectures on Computational Sciences. The Chair of the Organizing Committee Dr. Zacharoula Kalogiratou for the support and great efforts for the success of the organization of ICCMSE 2003. The Organizing Committee: Mr. Th. Monovasilis, Mr. D. Sakas, Dr. I. Sinatkas, Dr. E. Siskos, Dr. S. Spartalis, Mr. Th. Themelis, Mr. K. Tselios, Dr. N. Tsounis for their help and support. Special thanks for the Secretary of ICCMSE 2003 Mrs. Eleni RalliSimou for her excellent job. The Technological Educational Institute of Western Macedonia, Kastoria Campus and its Director Dr. N. Tsounis, for the hospitality. The Region of Western Macedonia, Greece, for their financial support. Dr. T.E. Simos, Active Member of the European Academy of Sciences and Arts Invited Chair of ICCMSE 2003 Editor

TABLE OF CONTENTS

1.

P. Abad, 1-3 Componentsfor Time Serie Receiver Clock Oflset in GPS Solutions

2.

4-8 E. Afjei, M. H. Arbab, Magnetostatic Field Analysis by Employing Absorbing Boundary Condition

3.

9-1 1 Z. Akdogan, M. Demirci and 0. Sh. Mukhtarov, Sturm-Liouville Problems with Eigendependent Boundary and Transmissions Conditions

4.

T. M. Alkhamis, 12-14 Computational Methodfor Unconstrained Optimization Functions with Noise

5.

M. Al-Refai, 15 Convergence Analysis for an Iterative Method for Solving Nonlinear Parabolic Systems

6.

G. M. Amiraliyev, 16-21 Unform Numerical Method for a Quasilinear System with Boundary Layer

7.

22-23 Z. A. Anastassi and T. E. Simos, A Family of Optimized Runge-Kutta Methods with Five Stages and Fourth Orderfor IVPS with Oscillating Solutions

8.

24-27 R Anguelov, P. Kama and JM-S Lubuma, Nonstandard Theta-Method and Related Discrete Schemes for the Reaction-Dirusion Equation

9.

28-32 0. Angulo, J. C. Lopez-Marcos, Numerical Integration of a Size-Structured Cell Population Model in an Environment of Changing Substrate Concentration

IX

X

10.

1. Arregui, J. J. Cendan and C. Vazquez, 33-37 A Duality Method for the Compressible Reynolds Equation. Application to Simulation of Readwrite Process in Magnetic Storage Devices

11.

F. Balibrea, J. L. C. Guirao and F. L. Pelayo, 38-46 An Environment for Computing Topological Entropy for Skew-Product Transformations

12.

P. G. Bagos, Th. D. Liakopoulos and S. J. Hamodrakas, 47-55 Maximum Likelihood and Conditional Maximum Likelihood Learning Algorithms for Hidden Markov Models with Labeled Data-Application to Transmembrane Protein Topology Prediction

13.

M. K. Banda, 56-59 Variants of Relaxed Schemes and two-Dimensional Gas Dynamaics

14.

S. C. Basak, B. D.Cute and D. Mills, 60 Predicting Toxicity of Chemicals Using Chemodescriptors and Biodescriptors: an Integrated Approach

15.

S. R. Basu, Measuring Economic Methodological Tools

Well-Being

and

Governance:

61-65 Some

16.

F. A. Batzias, N. P. Nikolaou, A. S. Kakos and 1. Mihailides, 66-73 Modelling the Natural Gas Consumption in a Changing Environment

17.

F. A. Batzias, A. S. Kakos and N. P. Nikolaou, 74-82 Computer Aided Dimensional Analysis for Knowledge Management in Chemical Engineering Processes

18.

L. Bayon, J. M. Grau, M. M. Ruiz and P. Suarez, 83-86 New Developments on Equivalent Thermal in Hydrothermal Optimization. An Algorithm of Approximation

19.

87-89 Unique Virtues of the Pad& Approximant for High-Resolution Signal Processing

20.

Y. S. Boutalis and 0. I. Kosmidou, 90-92 A Feedback Linearization Technique by Using Neural Networks: Application to Bioprocess control

Di. Belkif

xi

21.

J. C. Butcher, Some Numerical Methods for Stiff Problems

22.

E. Camouzis, R. Devault, G. Papaschinopoulos,

93-97

98

a + yxN.-I + S x N - , Period Two Trichotomy on xN+, = xN

+xN-2

23.

P. Carbonniere, D. Begue, A. Dargelos and C. Pouchan, 99-103 Least Squares Fits of Potentials by Using Energy and Gradient Data: Vibrational Anharmonic Spectra for H2Co@om DFT Calculations

24.

G. Castellano, A. M. Fanelli and C. Mencar, Deriving Prediction Intervals for Neurofuzzy Networks

25.

M. M. Cerimele, D. Mansutti and F. Pistella, 110-113 Marangoni Efsects in a Horizontal Solidjkation Process in Microgravity

26.

C. S. Chew, K. S. Yeo and C. Shu, 114-117 Simulation of Incompressible Flows Across Moving Bodies Using Meshless Finite Difrerencing

27.

A. Chortaras, Y. Guo, M. M. Ghanem, F. 0. Bunnin, 118-121 Automatic Generation of Software Components for Real Options Modelling

28.

V. N. Christofilakis, Ch. Alexopoulos, 122-125 Modeling the State and Behavior of an Enzyme Using UML- an Object Oriented Approach

29.

K. Daoulas and V. G. Mavrantzas, 126-132 Atomistic Monte Carlo Simulation Studies of Polymer Melts Grafted on Solid Substrates

30.

N. J. Daras, 133-137 Markov 's Property and Generalized Pad&-TypeApproximants

31.

M. Darbandi, K. Mazaheri-Body, S. Vakilipour, 138-143 A Pressure Weighted Upwinding Scheme for Calculating Flows on Unstructured Grids

104-109

xii

32.

K. Dosios, K. Paparrizos, N. Samaras and A. Sifaleras, 144-147 An Eflcient Modijkation of the Primal-Dual Ttwo Paths Simplex Algorithm

33.

V. Drakopoulos, 148-151 Comparing Sequential Visualisation Methodsfor the Mandelbrot Set

34.

152-155 I. Z. Emiris and Th. G. Nikitopoulos, Structured Matrix Perturbationsfor Molecular Conformations

35.

G. Cerruela Garcia, I. Luque Rub, M. A. Gomez-Nieto, 156-159 A New Algorithm to Obtain All Maximum Common Subgraphs in Molecular Graphs Using Binary Arithmetic and Constraints Satisfaction Model

36.

M. Christodoulakis, C. S. Iliopoulos, Kunsoo Park, J. S. Sim, 160-165 Implementing Approximate Regularities Extended Abstract

37.

166-170 R.Clifford and M. Sergot, Distributed Suff? Trees and Their Application to Large-Scale Genomic Analysis

38.

A. Episkopakis, D. Nikolopoulos, K. Arvanitis, N. Dimitropoulos, 171-174 G. Panayiotakis, D. Cavouras and I. Kandarakis, Modeling the Detective Quantum Eflciency of Scintillators Used in Medical Imaging Radiation Detectors

39.

S. C. Farantos, 175-178 Bihrcation Phenomena in Molecular VibrationalSpectroscopy

40.

179-182 A. Gaitanis, M. R. Freedman and N. M. Spyrou, Verijkation of a Simple Techniquefor the Removal of Motion Artefacts in Electrical Impedance Epigastrography Signals

41.

P. J. Garcia-Nieto, 183-189 Numerical Simulation of Scavenging of an Urban Aerosol by Filtration Taking Into Account the Presence of Coagulation, Condensation, and GravitationalSettling

42.

E. E. Gdoutos, A. A. Giannakopoulos and D. A. Zacharopoulos, 190-191 Stress Analysis and Failure Mechanisms of Compose Materilas with Debonded Interfaces

...

Xlll

43.

D. Glotsos, P. Spyridonos, P. Petalas and G. Nikiforidis, D. Cavouras, P. Ravazoula, P. Dadioti and I. Lekka, 192-195 Support Vector Machines for ClassiJcation of Histopathological Images of Brain Tumour Astrocytomas

44.

D. Greenspan, N-Body Modelling

45.

A. P. Grinko, M. M. Karpuk, 197-204 About One Approach to the Minimization of the Errors of the Tutoring of the Neuron Networks

46.

F. Grondin and G. Mounajed, A. Ben Hamida and H. Dumontet,

196

205-208

Digital Concrete: A Multi-Scale Approachfor the Concrete Behavior 47.

S. Guangyi, T. Kawabe, K. Toraichi, K. Katagishi, 209-2 12 A New Approach to Discrete Approximation of a Continuous-Time System Model Based on Spline Function

48.

J. L. Guisado, F. Jimenez-Morales, J. M. Guerra, 213-216 Application of Shannon 's Entropy to ClassiJLEmergent Behaviors in a Simulation of Laser Dynamics

49.

G. Hanna and J. Roumeliotis, Collocation and Fredholm Equations of the First Kind

217-222

50.

A. Haskopoulos and G. Maroulis,

223-227

Intermolecular Interactions of

(H,O),

51.

U.Hohm, L. Zarkova, 228-235 Accurate Thermophsical Properties of Neat Globular Gases and their Binary Mixtures Determined by Means of an Isotropic TemperatureDepandent Potential

52.

R Hoppe and W. Litvinov and T. Rahman, 236-241 Modelling and Computation of Axially Symmetric Flows of Electrorheological Fluids

53.

D. T. Hristopoulos, Simulations of Spart an Random Fields

242-247

xiv

54.

L. S. Iliadis, S. H. Spartalis, 248-249 Fundamental Fuzzy Relation Concepts of a D.S.S. for the Estimation of Natural Disasters’ Risk (The Case of a Trapezoidal Membership Function)

55.

S. Itoh, M. Igami, 250-253 A Hybrid Molecular Dynamics Simulation Methodfor Solids

56.

I. G. Ivanov and L. G. Taseva, The Contract Gas Market with a Linear Supply Function

254-257

57.

A. Kaczanowski, K. Malan and K. Kulakowski Hysteresis Loop of a Nanoscopic Magnetic Array

258-261

58.

Z. Kalogiratou, Th. Monovasilis, Th. Simos, 262-267 Numerical Solution of the Two- Dimensional Time Independent Schrodinger Equation with Exponential-Fitting Methods

59.

I. Kalatzis, N. Piliouras, E. Ventouras and I. Kandarakis, C. C. Papageorgiou and A. D. Rabavilas, D. Cavouras, 268-271 Probabilistic Neural Network Versus Cubic Least-Squares MinimumDistance in Classijjing EEG Signals

60.

I. Kalatzis and N. Piliouras, D. Pappas, E. Ventouras and D. Cavouras, 272-276 Probabilistic Neural Network Classifier Versus Multilayer Perceptron ClassiJier in Discriminating Brain Spect Images of Patients with Diabetesfiom Normal Controls

61.

T. E. Karakasidis, A. B. Liakopoulos, N. S. Cholevas, 277-280

Parallel Molecular Dynamics Simulation of Lennard-Jones Liquids on a Small Beowulf Cluster 62.

T. E. Karakasidis,

281-284

Vibrational Properties of NiO(lI0) Surface by Molecular Dynamics Simulation 63.

P. Karamanis and G. Maroulis, Electric Properties of Substituted Diacetylenes

64.

N. Karatsis and G. Maroulis, 289-291 Molecular Structure and Electric Polarizability in Sodium Chloride Clusters

285-288

xv

65.

M. M. Karpuk, 292-296 About the Possibility of Applying the Neuron Networksfor Determining the Parameters of Uniaxial Films on the Basis of the Ellipsometric measurements

66.

S. H. Kashani, A Fuzzy Logic Paradigmfor Industrial Economics Analysis

67.

H. Katsuragi and H. Honjo, 298-301 Monotonic Scaling of the KPZ Growth with Quenched Disorder

68.

H. Kaya, M. Kaplan, H. Saygin, 302-305 A Recursive Algorithm for Finding HDMR Terms for Sensitivity Analysis

69.

P. Kolorenc, J. Horacek, K. Houfek and M. C. Zek, G. Mil’Nikov H. Nakamura, 306-308 Calculation of Vibrational Excitation of Diatomic Molecules Below Dissociative Attachment Threshold

70.

K. Konstantinidis and I. Andreadis, 309-310 On the Use of Color Histograms for content Based Image Retrieval in Various Color Spaces

71.

A. M. Kosmas, 311-315 Theoretical Structural and Relative Stability Studies of Isometric and of xooy Peroxides Conformeric Forms

297

(X= H , CH, ,CI, Br ,I , Y = CI, Br ) 72.

W. J. Kowalski, J. Nowak and M. Konior, 316-322 Modeling of Chiral Separations in Chromatography by Means of Molecular Mechanics

73.

M. I. Krivoruchenko, E. Alessio, V. Frappietro and L. J. Streckert, 323-326 Probability Distributions of Volatility in Financial Time Series

74.

M. Kunik, S. Qamar and G. Warnecke, Kinetic Solution of the Boltzmann-Peierls Equation

75.

P.V. Kyratsis, D. A. Panagiotopoulos, D. V. Kakogiannis, 333-336

327-332

Computer Aided Engineeringfor Theoretical Studies of Vehicle Active Suspension Systems

xvi

337-339

76.

M. Lambiris, Ch. Tsitouras and K. Evmorfopoulos, Four-Step, Two-Stage, Sixth-Order, P-Stable Methodrr

77.

340-345 G. Lappas and V. Ambrosiadou, Binary and Multicategory ClassificationAccuracy of the LSA Machine

78.

E. C. Laskari, G. C. Meletiou, D. K. Tasoulis, M. N. Vrahatis, 346-349 Data Mining and Cryptology

79.

Ming-Gong Lee, 350-359 Application of Automatic Diflerentiation in Numerical Solution of a Flexible Mechanism

80.

360-364 T. Levitina and E. J. Brandas, Numerical Quadrature Performed on the Generalized Prolate Spheroidal Functions

81.

T. Levitina and E. Brandas, 365 Multitaper Techniques and Filter Diagonalisation -a Comparison

82.

M.H.X. Liang & B. Wetton, T.G. Myers, 366-376 Combined Air and Rivulet Flow and Application to Fuel Cells

83.

377-381 M. Liapi, K. Alketas Ougrinis, The Transmutation of the Architectural Synthesis. Morphing Procedures Through the Adaptation of Information Technologv

84.

382-386 Y. Li, Shao-Ming Yu and P. Chen, A Parallel Adaptive Finite Volume Methodfor Nanoscale Double Gates Mosfets Simulation

85.

Y. Li, 387-390 An Iterative Method for Single and Vertically Stacked Semiconductor Quantum Dots Simulation

86.

391-394 J. A. Lopez and F. J. Marco, M. J. Martinez, Proposal of a New Computational Method for the Analysis of the Systematic Dsfferences in Star Catalogues

87.

M. S. Magdon - Maksymowicz, A. Dydejczyk, P. Gronek and A. Z. Maksymowicz, 395-399 Simulation of the Switching Curve in AntiferromagneticIsing Model

xvii

88.

G. Maroulis, Electric HyperpolarizabilityCalculations

400-404

89.

F. Martinez, A. Guillamon and J. J. Martinez, Segmentation Natural Speech Using Fractal Dimension

405-408

90.

J. Mateu and J.A. Lopez, 409-412 Cluster Modelsfor Spatial Point Processes with Applications

91.

T. Mavromoustakos, P. Zoumpoulakis, M. Zervou, I. Kyrikou, A. Kapou, N. Benetis, 413-417 The Use of ComputationalAnalysis to Design Novel Drugs

92.

E. Miletics, 418-426 Energy ConservativeAlgorithmfor Numerical Solution of ODES Initial Value Problems

93.

B. F. Minaev and H. Agren, Enzymatic Spin Catalysis Involving 0,

94.

N. Moir, 432-435 A New Class of Methodsfor Solving Ordinary Diferential Equations

95.

G. Molnhrka, 436-445 implicit Extension of Taylor Series Methodfor Initial Value Problems

96.

446 450 Th. Monovasilis, Z. Kalogiratou, T. E. Simos, Exponential-Fitting Symplectic Methods for the Numerical Integration of the Schrodinger Equation

97.

H. Nakatsuji, Structure of the Exact Wave Function: Progress Report

451-453

98.

S. Nikolif and N. Trinajstib, Complexity of Molecules

454-456

99.

D. Nikolopoulos, P. Liaparinos, S. Tsantis, D. Cavouras and I. 457 460 Kandarakis, G. Panayiotakis, Radiation Detection Eficiency Evaluation of YAP;CE Scintillator by Monte-Carlo Methods

427-431

-

-

xviii

100.

L. A. A. Nikolopoulos, 461-465 B-Splines: A Powerful and Flexible Numerical Basisfor the Continuum Spectrum of the Schrodinger Equation. An Application to Hydrogenic Atomic Systems

101.

L. A. A. Nikolopoulos, A Finite Element Approachfor the Dirac Radial Equation

102.

470-473 I. Ntzoufras, A. Katsis, D. Karlis, A Bayesian Statistical Modeling for the Distribution of Insurance Counts

103.

N. Orfanoudakis, H. Hatziapostolou, E. Mastorakos, E. Sardi, K. 474-478 Krallis, N. Vlachakis, S. Mavromatis, Design, evaluation measurements and CFD Modeling of a Small Swirl Stabilised Laboratory Burner

104.

479-483 Y. Panagis, E. Theodoridis, K. Tsichlas, Data Structuring Application for String Problems in Biological Sequences

105.

484-489 N. G . Pavlidis, K. E. Parsopoulos and M. N. Vrahatis, Computing Nash Equilibria Through Particle Swarm Optimization

106.

D. G. Pavlou, N. V. Vlachakis, M. G. Pavlou, V. N. Vlachakis, M. Kouskouti, I. Statharas, 490-491 Foundamental Solution of the Cracked Dissimilar Elastic Space

107.

G . Papakaliatakis, D. Karalekas, Study of Fracture in SIC/AL Composites

108.

G. Papakaliatakis,

494-495 Computational Study of the Crack Extension Initiation in a Solid Propellant Plate with a Circular Hole

109.

496-499 S. H. Park, J. H. Kim, Nodal Stress Recovery and Error Estimation Based on Variation of Mapping Function

110.

K. Perdikuri, C. Makris, A. Tsakalidis, Discovering Regularities in Biosequences: Applications

466-469

492-493

500-503 Challenges and

xix

111.

I. Petrounias, A. Tseng, P. Chountas, Constraint Based Web Mining

504-511

112.

I. Petrounias and A. Assaid, Temporal Web Log Mining Using Olap Techniques

512-519

113.

L. Pogliani, 520 Introducing Complete Graphs in Molecular ConnectivityStudies

114.

M. Sekkal-Rahal, D.C. Kleb and P. Bleckmann, 52 1 Structures and Energies of p-Neocarrabiose in Vaccum and in Aqueous Solution 522-525

115.

H. Rarnos, J. Vigo-Aguiar, Variable Step-Size Stormer Methods

116.

H. Ramos, J. Vigo-Aguiar, 526-529 A Note on the Selection of the Step Size in the Variable Step-Size Stormer Method

117.

A. R k o , 530-533 Birej-ingences: A Challengefor Both Theory and Experiment

118.

J. Roca J. R., J. Roca, J. Martinez and F. J. Martinez, F. J. Gil and J. A. Alvarez-Gomez 534-537 Feasibility of Closed-Loop Target Controlled Infusion of Intravenous Anaesthesia

119.

P. Roubides, 538-541 The Fundamental Solution Method for Elliptic Boundary Value Problems

120.

J. Roumeliotis, Axisymmetric Rigid Bodies in Creeping Flow

121.

546-549 N. RUSSO, T. Marino, E. Sicilia and M. Toscano, Past, Present and Future Challenge of Density Functional Theory Based in Molecular Sciences

122.

550-554 T. Rusu and M. Pinteala, V. Bulacovschi, Artzjkial Intelligence Methods Used in the Investigation of Polymers Properties

542-545

xx

123.

555-556 D. Sakas and T. E. Simos Symmetric Multistep Methods with Minimal Phase-Lag for the approximate Solution of Orbital Problems

124.

J. K. Sakellaris, 557-560 Finite Element Analysis for Weakly Coupled Magneto - ThermoMechanical Phenomena in Shell Structures

125.

S. Sanchez and R. Criado, C. Vega, 561-566 A Generator of Pseudo-Random Numbers Sequences with Maximum Period

126.

P. Sasavat, N. Gindy, G. F. Xie and A. T. Bozdana 567-570 Near Force-Balanced Cutting: Key to Increase Productivity in Machining

127.

L. P. Schulz, 571-584 Symmetry Formation Principles of the Chemical Computer Software

128.

A. Sharma,

585-586

The Generalised Mass-Energv Equation hE = AC2hM;Its Mathematical Justification and Application in General Physics and Cosmology 129.

Shenghua Shi and Atsuo Kuki, 587-592 A Simple Approach to a Multi-Objective Design with Constraints in Compound Selectionfor Drug Discovery

130.

S. V. Shepel, S. Paolucci, 593-597 Finite Element Level Set Formulationsfor Modelling Multiphase Flows

131.

K. Sivagurunathan, P. Chountas, E. El-Darzi, Representation h Modelling of Electronic Patient Records

132.

Y. V. Skorov, B. J. R. Davidsson, G. N. Markeleov, 604-610 Consistent Kinetic Model of Innermost Cometary Atmosphere and Boundary Layers of Cometary Nucleus

133.

P. Spyridonos, P. Petalas, D. Glotsos, G. Nikiforidis, D. Cavouras, P. Ravazoula, 611-614 Comparative Evaluation of Support Vector Machines and Probabilistic Neural Networkrs in SuperJicial Bladder Cancer Classification

598-603

xxi

134.

J. P. Suarez and P. Abad, A. Plaza, M. A. Padron, 615-618 ComputationalAspects of the Refinement of 3 0 Complex Meshes

135.

Chen-Yin Suen, 619-621 The Impact of Graphics Calculator on Mathematics Education in Asia

136.

A. J. Thakkar, Density Functionals for Moments of Distribution

622-625

the Electron Momentum

137.

P. Theocharakis, I. Kalatzis and N. Piliouras, N. Dimitropoulos, E. Ventouras and D. Cavouras, 626-630 Relationship Between Carotid Plaque Composition and Embolization Risk Assessed by Computer Processing of Ultrasound Images

138.

E. S. Tentis, D. P. Margaris, D. G. Papanikas, 631-633 Transient Simulation of Large Scale Gas Transmission Networks Using an Adaptive Method of Lines

139.

D. Tomtsis, V. Kodogiannis, E. Wadge, Optical PH Measurement using Chromatic Modulation

140.

S. Tsantis, I. Kalatzis, N. Piliouras, D. Cavouras, N. Dimitropoulos, G. Nikiforidis, 639-642 Computer-Aided Characterization of Thyroid Nodules by Image Analysis Methods

141.

S. Tsantis, D. Cavouras, N. Dimitropoulos, G. Nikiforidis, 643-646 Denoising Sonographic Images of Thyroid Nodules Via Singularity Detection Employing the Wavelet Transform Modulus Maxima

142.

K. Tselios and T. E. Simos, 647-649 Runge-Kutta Methods with Minimal Dispersion and Dissipation for Problems Arisingfrom ComputationalAcoustics

143.

S. Tsitmidelis, M. V. Koutras, V. Zissimopoulos, 650 ReliabiIity Bounds Improvement Via Cut Set or Path Set Rearrangements

144.

J. M. Ugalde, 651-655 The Electron-Pair Density and the Modeling of the Spherically Averaged Exchange-Correlation Hole

634-638

xxii

145.

E. Varnvakopoulos, G. A. Evangelakis, D. G. Papageorgiou, 656-659 Solidjication of Pb PRE-Covered Cu(1 I I) Surface

146.

660-662 J. A. Vera and A. Vigueras, Stability of an Equilibrium Solution for a Gyrostat About an Oscillating Point

147.

G. D. Verros, 663-666 Computer Aided Estimation of Molecular Weight and Long Chain Branching Distribution in Free Radical Polymerization

148.

J. sgo-Aguiar, H, Rarnos, 667-669 VS-VO Numerov Method for the Numerical Solution of the Schrodinger Equation

149.

670-672 M. De’ Michieli Vitturi, F. Beux, Nonlinear Pressure and Temperature Waves Propagation in Fluid Saturated Rock

150.

673-677 E. Wadge, V. Kodogiannis, D. Tomtsis, Neuro-Fuzzy Ellipsoid Basis Function Multiple Classifier for Diagnostic of Urinary Tract Infections

151.

G. Wei and N. Mousseau, P. Derreurnaux, 678-681 Protein Folding Simulations Using the Activation-Relaxation Technique

152.

S. Wilson, On the Systematic Construction of Molecular Basis Sets

153.

687-691 Z. Xiong and N. C. Bacalis, Generalization of Laguerre Orbitals Toward an Accurate, Concise and Practical Analytic Atomic Wave Function

154.

S. Zimeras, F. Georgiakodis, 692-694 Bayesian Models for Medical Image Biology Using Monte Carlo Markov Chains Techniques

682-686

COMPONENTS FOR TIME SERIE RECEIVER CLOCK OFFSET IN GPS SOLUTIONS

P.ABAD University of Las Palmas de Gran Canaria Department of Cartography and Graphic Engineering Department Las Palmas de Gran Canarias, 3501 7, SPAIN E-mail:[email protected]

In the equations systems generated for the GPS satellite observations, when the positional parameters are held fixed or constrained the clock receiver solutions are significantly affected, and this variable absorbs the non controlled effects. In this work, for the clock receiver solutions viewed a time serie , we analyze the components to detect periodicity in the no controlled effects, and built new mathematical models for control them.

1. Motivation and Introduction A great variety of corrections contribute to the quality of results when using the GPS observations [l]. All these effects must be considered for positioning, even for pseudorange positioning at the meter precision level. The effects (first kind effects) are: The special and general relativity, instrumental delays, atmospheric delays, satellite and receiver clock offsets and multipath effect. When we talk about position with precision of a few cms. in the correction model for Precise Point Positioning (PPP) it is important to take into account some second kind of effects that may not have been considered in other processing modes. The effects are the satellite antenna offsets and phase wind-up correction (Satellite attitude effects), Earth orientation parameters (EOP), solid Earth tides and ocean loading (site displacements effects), and the last, precise ephemeris [2]. 2. Two-Correctionsmodels to solve the equations systems The simultaneous observation of four or more satellites of known position from an unknown place allows us to build an equation system using the distance formula. There are two kind of equations depending on the satellite signal: phase(@) equations and pseudo-range(P) equations. The GPS observation equation following a mechanistic model, see for example [3], can be written as follows: 1

2

@,'(t)= p,"( t )- cdTs( t ) + cdt, ( t ) + NA +

-

+ ETropo. + ERelu + EInsi

+ EAnt + EWind + EE(idc + EOcean

+ EMull. + + EEop +

P;' ( t )= p,"( t )- CdT' ( t ) + cdt, ( t ) + + EIono. + ETropo. + ERela + EIn.sl

+

+

Where: p; is the pseudo-range, dr,(r) is the receiver clock offset from GPS time, d76(t) is the satellite clock offset from GPS time, c is the vacuum speed of light, A is the wavelength, N is the carrier phase ambiguity, E , are the effects mentioned above, and E is the random error distributed as N(0,d). Usually the first kind of effects are removed but normally the second kind are omitted. The simplified mathematical model can be written:

@: ( t )= p,"( t )- cdTs ( t )+ cdt, ( t )+ NA + E Prs(t)= p ~ ( t ) - c d T ' ( t ) + c d t , ( t ) + ~ The least squares solution with a a-priori weighted constraints P to the parameters is given by S=-(P+ATPA)-'ATPW,, so that the estimated parameters are:

2=X,+S When the positional parameters are held fixed or constrained the clock solutions are significantly affected, and this variable absorbs the non controlled effects.

3.

Conclusions

The different components are quantified and classified by order of magnitude. Depending on the value of these components will be taken into account in the mathematical correction model.

3

References 1.

Global Positioning System, Papers published in Navigation, v01.- 1, 1980.

2. J. Kouba, P. HCroux, Precise Point Positioning Using IGS Orbit and Clock Products, Geodetic Survey Division, Ottawa, Ontario. September 2000. 3.

G. Strang and K. Borre, Linear algebra geodesy and GPS, WellesleyCambridge Press ,1997.

MAGNITOSTATIC ANALYSIS BY EMPLOYING ABSORBING BOUNDARY CONDITION E. AFJEI AND M. H. ARBAB Depi. Elecirical & Computer Engineering Shahid Beheshti University E-mail: afjei @yahoo.corn

The need for absorbing boundary condition and coordinate stretching arises when one wishes to simulate the extension to infinity on a finite domain of computation for a problem. This paper develops the proper absorbing boundary operators for an elliptical partial differential equation for a magnetostatic problem. It then uses finite difference technique with the founded absorbing boundary condition in order to investigate its effect on the solution of the problem. In this procedure a finite distance is considered instead of the original infinite distance. Applying proper boundary conditions at that distance one can emulate the open boundary conditions. The absorbing boundary condition have been employed extensively to hyperbolic problems in which the solutions move at finite speed are limited in duration by the return of outwardly propagating feature of the solution. In this problem we consider an elliptic magnetostatic problem and then try to develop absorbing boundary condition formulas of different orders and finally applying them to the problem using the finite difference technique.

Problem Description & Development Consider the following two-dimensional Problem shown in Fig. 2 where bounded domain which is a current carrying conductor with a highly permeable material for the exterior LIT.

R, is a simply connected

The governing equations are

V2A=-~

onR

and

V2A = 0 The boundary conditions are

4

on

RT.

5 on the interface between L? and nT and A, = 0 at the outer boundary very far distance. Due to symmetry in 8 direction, this problem can be reduced to a 1-dimensional problem in r direction. The known solution for the exterior region of this problem is written as

N A , ( r ) = (-)In2n

c1

(4)

r

where CI is a constant and determined by the far distance radius at which, the boundary condition A,=O is applied. The asymptotic expansion of (4) results in

Expanding ( 5 ) results in

a a a A Z ( r ) = a o+L+++<+..r r r

a, a a u ( r ) = +-+++++... r r r

(7)

It an be seen that

dv dr

dv dr

-

v r

a, r2

-+-=--(a,+-

1 r3

2a, - -... 3a3 r3 r4

2a3 + ...) r

(9)

6

1

is a series expansion that vanes as 7 and has eliminated the coefficient a,. We

r

define a first-order operator for the absorbing boundary condition to be

d dr

B, = (-+-) In terms of

1 r

v, d dr

B, (v) = (it can be seen that V

1 + -)v

r

= AZ( r )since Iimr+ A, ( r ) = a,

for our problem it is required

lim

~~~

A, ( r ) = 0

then, a. =O The first order absorbing boundary operator as a function of the potential, A,, then becomes

it can be seen that

1

valid to the order of 7

r

and higher-order absorbing boundary condition can also be developed by the same procedure.

7

Numerical Analysis Equations 1 and 2 are solved using two types of outer boundary conditions namely, Az=O and Absorbing are used at different radius in the high permeable material. In the analysis, current is 5A and relative permeability considered to be 6500. The radius of current carrying coil is 10 cm. Figs. 3 , 4, and 5 are the results of the magnetic vector potential and magnetic field density vs. radius when the first order absorbing boundary condition and also Az=O at different radius is employed. The results for the vector potential show lower constant values when absorbing boundary condition is applied, especially closer to the current carrying conductor. At smaller radius the effect of absorbing boundary is more pronounced. The magnetic field density is the same in all cases, since it deals with the curl of A, .

Fig. 1 Non-uniform mesh

Fig. 2 Bounded closed domain

8

1 a

0

0.005

0.01

0.015

0.02

Radlus(m)

Fig.3. A- Magnetic vector potential vs. radius B - Magnetic field density vs. radius (BC at r-2cm)

Fig. 4 .A- Magnetic vector potential vs. radius B - Magnetic field density vs. radius (BC at r=4cm)

0

0.01

0.04

0.06

0.08

0.1

0

Rmdlu.(n)

0.025

0.05

0.075

0.1

Ridlui (m)

I

Fig.5. A- Magnetic vector potential vs. radius B- Magnetic field density vs. radius (BC at r=8cm)

STURM-LIOUVILLE PROBLEMS WITH EIGENDEPENDENT BOUNDARY AND TRANSMISSIONS CONDITIONS

Z A K D O ~ A N ,M.DEMIRCI AND O.SH.MUKHTAROV Gaziosmanpaga University, Faculty of Science-Art, Department of Mathematics, 60100 Tokat, Turkey E-mail: [email protected], [email protected] and [email protected]

It is well-known that the sturmian theory is an important aid in solving many problems in mathematical physics. Therefore this theory is one of the most actual and extensively developing fields in spectral analysis of boundary-value problems. Basically it has been investigated boundaryvalue problems which consist of ordinary differential equations with continuous coefficients and end-point boundary conditions. The purpose of this work is to extend some fundamental spectral properties of regular Sturm-Liouville problems to special kind discontinuous boundary value problem, which consist of a Sturm-Liouville equation together with eigendependent boundary and transmission conditions. Namely we shall consider one discontinuous eigenvalue problem which consist of Sturm-Liouville equation.

+

7u := -u’! q(z)u= xu, z E [a,c) u (c,b]

(1)

with boundary conditions Ll(U)

: = x ((u;u(a)- (u’2u’(a))- (a1u(a)- azu’(a))= 0

Lz(u) : = x (Piu(b) - P;u’(b))

+ ( P l U ( b ) - PZU’(b)) = 0

(2) (3)

and transmission conditions

L3(u): = u(c + 0 ) - u(c - 0) = 0

(4) L4(u) : = u/(c 0) - u’(c - 0 ) X S l U ( C ) = 0 (5) where X is complex eigenparameter, q(z)continuous in [a,c) and (c, b] and (i = 1,2) are real has a finite limits q(fz) := lim q(z) ; ai,ai,/?i,,8;,61, x-fc numbers; we assume that 61 > 0 .

+

+

9

10

In the Hilbert space L2 [a,b]@ C3 of vectors F := (f(z),fl,fz, f3) we define an equivalent inner product by b

a

[ a)

where p1 = aim - alak > 0, p2 = Pip2 - P1P; > 0 and linear operator in this space by the domain of definition

D(A):={F=

f (x)

f is absolutely continuous in [a,b] , f’ is absolutely continuous in [a,c) and (c,b] , f’(&c) = lim f’(z) , ~f E L2 [ah]f l = Ni(f), x+fc

f2 =

N8f) f3 = MXf) 3

and action low

A F = (7f,Na(f),Nb(f),Mc(f)) where N,(f)=

).(fi

Plf(b) - Pzf‘@),

- %#(a), NL(f)= a:f(a) - Ohf’(a), Nb(f)=

Wf)= P X b ) - Pif’@),M4.f) = f’(c + 0) - f‘(c

-

O ) , Mi(f) = -S,f(c)

so that the problem (1)-(5) can be considered as the eigenvalue problem of this operator. By using this Hilbert space formulation and own Special techniques we investigate some spectral properties of the considered eigenvalue problem (1)-(5). In particular we obtain asymptotic approximate formulas for eigenvalues and corresponding eigenfunctions. In contrast to previous works, the derivative of eigenfunctions of our problem may have discontinuity at the inner point z = c. For example, for the case1 # 0 , ah # 0 we found the next asymptotic formulas for eigenfunctions and corresponding eigenvalues b-c

c-a

(n+i)+O

[A]

(6)

11

This kind of eigenfunctions may arise in spectral problems of the theory of heat and mass transfer, in diffraction problems and varied assortment physical transfer problems. The main references

References 1. C.T. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions,Proc.Roy.Soc.Edin. 77A 293-308(1977). 2. 0. Sh. Mukhtarov, M. Kandemir, Asymptotic behaviour of eigenvalues for the discontinuous boundary-value problem with functional-transmission conditions, Acta Mathematica Scientia 22 B(3) 335-345(2002). 3. S. Yakubov and Y . Yakubov, Abel basis of root functions of regular boundary value problems, Math. Nuchr. 197 157-187(1999).

COMPUTATIONAL METHOD FOR UNCONSTRAINED OPTIMIZATION FUNCTIONS WITH NOISE T A L L M. ALKI-IAMIS Department of Statistics and Operations Research Kuwait Universiv, P.O. Box 5969, Safat, Kuwait E-mail: alkhamis(lii,kucOl.kuniv.edu.hv

1. Guidelines Unconstrained nonlinear optimization problem UNOP arise in science and engineering when the goal is to find a solution, expressed as a vector of variables, that minimizes or maximizes some function that acts as a measure of the merit of the solution. A standard UNOP can be defined as follows:

min f(x) X

+

where x E Rn and f : R n R. In the literature of nonlinear optimization, the Hooke and Jeeves algorithm is widely used as a pattern search procedure to optimize a nonlinear functions that are not necessarily continuous nor differentiable. The HJ pattern search algorithm consists of one variable at a time exploratory moves along the individual coordinate directions in the neighborhood of a base point solution to determine an appropriate direction of search (pattern). Following the exploratory search, a series of pattern moves are made to accelerate the search in the direction determined in the exploratory search. Exploratory searches and pattern moves are repeated until a termination criterion is met. Most work on HJ assume implicitly that to each solution point x, the corresponding function value f(x) can be computed accurately. In many situations of real life application function f(x) is either does not exist in a closed form or too complicated to be calculated analytically. An example where function L does not exist in a closed form is arising in the area of nonMarkovian, queuing networks, where any of the factors such as non-exponential service times, multiple classes of customers, limited buffer sizes or transient behavior is significant then system measure of performance such as waiting time in the system does not have a closed form. An example where function L is too complicated to be calculated analytically arising in the area of engineering

12

13 optimization where the measure of performance usually defined on the solution of complex systems of equations including implicit equations, ordinary differential equations, and partial differential equations. In this work, we are considering the case where f(X) can only be evaluated via Monte Carlo simulation i.e f(X) is observed with a random error. It is assumed that at each time when we want to determine f(X), a

-

value f (X) can be observed, which is obtained from -

random noise:

f(x)

=

f(x)+

't,

f(x) by superposition of

where 7, are independent random L

variables with mean zero and variance Gx. Thus our optimization problem becomes min f(x) = E[T(x)]

x E R n and our objective is to seek the global

X

optimal solution X*

E R n ,where

f(x*) = E[f(x*)] <

E[T(x)] = f(x)

'd x E R n

Steps of Hooke and Jeeves Algorithm (Bazara et. al. [ 13): a formal statement for a minimization problem is given below Initialization step. Let d,, d,

,..., d,

be the coordinate directions. Choose a

scalar & > 0 to be used for terminating the algorithm. Furthermore, choose an initial step size, A 2 0 , and an acceleration factor, cx > 0. Choose a starting point X I ,let y 1 = X I , let k = i = 1, and go to the main step. Main Step

1.1 Evaluate function f a t point y i and point ( y;

+ A d i ),

= yi

+ A d i and go to step 2, else go

1.2 Evaluate function f a t point ( y i - A d ; ) ,

if f(yi -Ad; ) < f(yi) set

if f(yi to step 1.2.

+ A d i ) < f(yi) set y i + l

y i + l = y i - Adi and go to step 2, else set y i + l = y i and go to step 2.

14

i < n, set i = i + 1, and repeat step 1. Otherwise, go to step 3 if f ( y n + l ) < f ( xk) , andgo to step4 if f(yn+l) 2 f ( x k ) .

2. If

4. If

A I E ,stop; Xk is the solution. Otherwise, replace A by A 12. Let

Y1 = x k , Xk+l = x k , s e t k = k + l , s e t i = l ,an d g o to s te p I . As it can be seen above, for the HJ algorithms to work it needs to evaluate the objective function value accurately. Here we are considering situation where the measure of performance of the objective function can only be evaluated via Monte Car10 simulation. Due to the nature of simulation, the outputs of the experiments are stochastic. This means that the value of the objective function obtained from the simulation experiments belong to some probability distribution. Thus a modification to the HJ algorithm to handle the stochastic noise is very important to reduce the chances that the search technique is not being misguided by stochastic noise. In the simulation literature, ranking and selection (RS) [2] procedures have often been recommended for solving optimization problem via simulation. RS procedures are statistical methods specifically developed to select the best system from a set of competing systems. Provided certain assumptions are met, these methods usually guarantee that the probability of correct selection will be at least some user-specified value. The statistical methods of RS are applicable when comparisons among a finite and typically small number of systems are required. In this paper we present a computational method that combines HJ and RS for solving unconstrained stochastic optimization problems. We present empirical results that illustrate the performance of the proposed computational method.

References [l] Bazaraa, M. S., Sherali, H. D., Shetty, C. M., Nonlinear Programming Theory and Algorithms (2ndEdition), John Wiley and Sons, New York, 1993. [2] Law, A. M. ,Kelton, W. D., Simulation Modeling and analy~is(2"~ Edition), McGraw-Hill, Inc., New York, 1991.

CONVERGENCE ANALYSIS FOR AN ITERATIVE METHOD FOR SOLVING NONLINEAR PARABOLIC SYSTEMS

MOHAMMED AL-REFAI Department of Mathematics and Statistics, Jordan University of Science & Technology P.O. Boa: 3030, I d i d 22110, JORDAN E-mail: m- [email protected]

Nonlinear parabolic systems of partial differential equations are considered. In a recent work, we have proposed a new iterative method based on the eigenfunction expansion to integrate these systems. In this paper, we prove the convergence of the method on bounded time intervals under certain condition that can be more easily to satisfy. We then show that the solution obtained by the new method will converge to the exact solution for a problem in combustion theory. Moreover, we determine the number of iterations needed to obtain a solution with a predetermined level of accuracy. It is expected that the convergence analysis can be used for similar systems of time dependence.

15

UNIFORM NUMERICAL METHOD FOR A QUASILINEAR SYSTEM WITH BOUNDARY LAYER

G. M. AMIRALIYEV 100. Yil University, Faculty of Art and Science, Department of Mathematics, 65080 Van, TURKEY E-mail: gamirali2000@yahoo. com Keywords: Singular perturbation, Non-uniform mesh, Difference scheme, Uniform convergence

1. Introduction

This paper is concerned with the numerical solution for singular perturbation system of two coupled ordinary differential equations with first and second orders and with initial and boundary conditions, respectively. Thus we will consider the initial boundary value problem of the form.

u2(0)= A,

2~2(1) = B,

(1-4)

where 0 < E < 1 is a small parameter, u*,A , B are given constants. f1(x,u1,UZ),f2(2,u1,ug)are given smooth functions satisfying certain regularity conditions. These conditions will be specified whenever necessarily. In addition assume

16

17

Finite difference scheme on a special non-uniform mesh, whose solution converges pointwise independently of the singular perturbation parameter is constructed and analyzed. Numerical results supporting the theory are presented. The above type initial boundary value problems arise in the nonlinear vibration theory of solid mechanics 5,8, in fluid mechanics to study nonsteady filtration in a porous medium and other physical models. The problem (1.1)-(1.4) exhibits a boundary layer of small thickness at left end. It is well known that a difference schemes on a uniform mesh are not suitable to nonlinear singularly perturbed problems as a special fine mesh is required in boundary layer region and comparatively much coarser mesh elsewhere. Ideally, the mesh should be adapted to the features of the exact solution using an adaptive grid generation technique. This approach is now widely used for numerical solution of differential equations with steep, continuous solutions (see, for more recent work, e.g. monographs 4 , 6 ) . In the present paper, we have derived a method based on using finite elements with piecewise constant and piecewise linear basis functions and appropriate quadrature formulae with remainder term in integral form. In the boundary layer, we introduce the non-uniform mesh, which is constructed by using the estimates of derivatives of the exact solution and the analysis of the local truncation error. This technique is similar to those used in l , 2 , 3. 2. Analytical Results

Here we give a useful asymptotic estimations of the exact solution of (1.1)(1.4), that are needed in later sections.

Lemma 2.1. Suppose the conditions (1.5) and (1.6) are satisfied. Then

{

Iuy(z)I 5 c 1 +&-I€!-?

I

},

0 5 ~ 5 1 ,k = 0 , 1 ,

IuF)(x) I C, 0 5 z 5 1,

k = 0 ,1 , 2 ,

(2.1)

(2.2)

provided that Iafk/azI, I a f k / a u k I , Idfk/au3-kl I C , k = 1,2, where C is positive generic constant independent of E (also mesh parameter in OUT following discussion about numerical solution).

18

3. Discretization and Non-uniform Mesh

Let W N be any non-uniform mesh on [0,1]: WN

= (0

< 5 1 < ... < X N - 1 < 1 } ,

and W$ = W N U { X N = 1 } , WN = W N U (50 = 0 ,X N = 1). For each i 2 1 we set the stepsize hi = zi - zi-1. Before describing our numerical method, we introduce some notation for the mesh functions:

for any function g , 9f,i = (9i - 92-1)/hi,

gz,i = (gi+l - gz)/hi+l,

1 92,i = (9i+l - 9i)/hi, ti, = -(hi 2 + hi+l), 9f&i = (&,i

-

%,i)/hi.

We propose the following differencescheme for approximating (1.1)-1.4): EYlf

+ fl(X,Yl,Y2)

= 0,

2 E w;,

(3.1)

(3.4)

Remark 3.1. Under the above assumption on the data of problem (1.1)(1.4), it is possible to prove, for (3.1)-(3.4), the existence and uniqueness result. The difference scheme (3.1)-(3.4), in order to be &-uniformconvergent, we will use the fitted form of W N . This is a special non-uniform mesh, ) a which is condensed in the boundary layer. According to (2.1), u ~ ( xhas boundary layer at z = 0 of order a,'aIlna( thickness. The fitted special non-uniform mesh W N on the interval [0,1] is formed by dividing the interval into two subintervals [0,a], [a,11, where

19

The interval [O,B] is contained in the boundary layer. Here we use a fine mesh. The interval [a,l]contains the smooth part of the solution. There we use a coarse mesh. The corresponding mesh points are

if B

(3.6)

, i = 0,..., N/2,

(3.7)

+ 1,..., N; h = 2(1 - a)/N.

(3.8)

< 1/2, zi = -a;'eln

if

1

, i = 0, ..., N/2,

zi E [O,B] : zi= -a; Eln 1 - (1- E ) l [ N 2i

$1

[l - (1 - e - s )

= 1/2;

xi E [cr,l] : zi = B + (i - N/2)h,i = N/2

Here, without loss of generality, we assume that N is even. The special mesh W N , given by (3.6)-(3.8) denote by & N . 4. Uniform Convergence

Theorem 4.1. Let ( U I , U ~ } be the solution of ( l . l ) - ( l . 4 ) and {y1,y2} be the solution of (3.1)-(3.4). T h e n the following &-uniform error estimate holds IlYk(.)

k = 1,2.

- ~ k ( 4 1 1 c ( 3 N )< CN-',

5. Numerical Results We consider the test problem (1.1)-(1.4) with the data fi(z,ul,u2) = 2u1 - e-"l

U*

= 1,

-

tanh(u2)

+ z2 - 1,

z E (0,1],

B = 0.

A=0,

The numerical solution is obtained by using the following quasilinearization technique

-ygi + y p ) + (ez/2 We take

a1

-

2)y?-')

= 0,

z E 2,;

= 2 in (3.5). The initial guess is chosen as

y y = 1 - z,

5

E

5;.

=

1 , 2...

20

A numerical method is &-uniform of order (PI,p2) on a mesh LJN if

where pk are constants independent of E and N . Approximations t o p k , the &-uniform rates of convergence, are determined using the double mesh method as follows pk

= In (r,"/riN)/

ln2,

IC

= 1,2

where

The resulting errors r y and r? and the corresponding numbers pl and p2 after five iterations are listed in following Table 1 and Table 2. Table 1

N = 32

N=16

&

7'1

lo-' loW4

0.0090035 0.0091113 0.0091123 loW8I 0.0091124 I

I

I

Pl

T2

TI

7'2

0.0048555 0.0049131 0.0049137 0.0049137

0.0001242 0.0000458 0.0000466 0.0000467 I

0.0000294 0.0000112 0.0000116 0.0000117 1

I I

I

1

P2

0.8908734 0.8910329 0.8910372 0.8910277 I

I

2.0787771 2.0318489 2.0064673 1.9997697

Table 2

N

&

lo-'

1

I

N

= 32 7'2

7'1

7'2

0.0048555 0.0049131 0.0049137 0.0049137

0.0022525 0.0025550 0.0025553 0.0025553

0.0000590 0.0000113 0.0000118 0.0000118

0.0000144 0.0000027 0.0000029 0.0000029

1

I

PI

= 64

7'1

I

I

P2

0.9433048 0.9433007 0.9433411 I 0.9433075

I

2.0346461 2.0652915 2.0146840 2.0164671

1

References 1. G. M. Amirali(yev), 1988, On the numerical solution of the system of Boussinesque with boundary layers, USSR Modelling in Mechanics, 3(5), 3-14. (Russian)

21

2. I. P. Boglaev, 1984, Approximate solution of a nonlinear boundary value problem with a small parameter for the highest-order derivative, USSR Comput. Math. and Math. Physics, 25, 30-39. (Russian) 3. I. P. Boglaev, V. V. Sirotkin, 1993, The solution of singularly perturbed Huge Problems via domain decomposition, Computers Math. Appl., 25(9), 31-42. 4. F. A. Farrell, A. F. Hegarty, J. J. H. Miller, E. O’Riordan, G. I. Shishkin, 2000, Robust Computational Techniques for Boundary Layers, CRC Press, Boca Raton. 5. I. F. Morozov, 1967, The nonlinear vibrations of thin plates taking account of inertia of the rotation, DAN USSR 176(3). (Russian) 6. H. G. Roos, M. Stynes, L. Tobiska, 1996, Numerical Methods for Singularly Perturbed Differential Equations, Springer, Berlin. 7. M. D. Rosenberg, 1952, O n non-steady filtration of partially gaseous fluid in a porous medium, Izv. AN USSR, Tech., 10. (Russian) 8. G. M. Vorovich, 1977, O n some direct methods in the theory of nonlinear vibrations of sloping shell, Izv. AN USSR, Tech., 21. (Russian)

A FAMILY OF OPTIMIZED RUNGE-KUTTA METHODS WITH FIVE STAGES AND FOURTH ORDER FOR IVPS WITH OSCILLATING SOLUTIONS

Z.A. ANASTASSI AND T.E. SIMOS *+ Department of Computer Science and Technology Faculty of Sciences and Technology, University of Peloponnese GR-22 100 Pipolis, GREECE

In this paper we present a family of explicit Runge-Kutta methods of 4th algebraic order with 5 stages for the efficient solution of problems with oscillating solutions. The main purpose is t o achieve better results by minimizing the dispersion and dissipation of the method, while maintaining constant algebraic order. In order t o produce the equations necessary for the 4th algebraic order, we will use the tree theory. The tree theory offers a convenient way of generating these equations as well as the equations of every algebraic order. Some of the fundamental definitions and theorems of the tree theory will be presented. High algebraic order is essential for every Runge-Kutta method since it increases the order of the principal error, thus increasing the rate that the error reduces while the step-length reduces. However further increasing of the algebraic order costs in terms of the number of the coefficients needed t o satisfy the equations, while the benefit isn’t that significant. Instead we can keep these coefficients free for increasing the order of properties such as dispersion and dissipation. Although dispersion (or phase-lag) was introduced for cyclic orbit, *Active Member of the European Academy of Sciences and Arts tcorresponding author. Please use the following address for all correspondence: Dr. T.E. Simos, 26 Menelaou Street, Amfithea - Paleon Faliron, GR-17564 Athens, GREECE, Tel: 0030 210 94 20 091, E-mail: [email protected] 22

23

Runge-Kutta methods with high phase-lag order are more efficient in many other problem types than methods with lower phase-lag order and higher algebraic order with the same number of stages. They are even more effective in problems with oscillating solutions. We will try to maximize the order of both dispersion and dissipation, as well as equalizing them to zero. In the first case we produce methods with constant coefficients, in opposite to the second case where the methods have variable coefficients. Methods with variable coefficients are usually more effective, thanks to the infinite order of either dispersion or dissipation, especially when the solution is a linear combination of trigonometric functions. However they depend on the problem’s frequency and the step-length. Fortunately there are ways of determining the dominant frequency of problems with oscillating solutions. One of them is by giving the frequency a value close to the spectral radius of the coefficient of the dependant variable in the general form of the problem, although they are often better even when we don’t know a good approximation of the frequency. The methods are compared with each other as well as with the most efficient among already known methods, while solving various problems from the literature. The comparison’s purpose is to show the importance of dispersion and dissipation in oscillating problems and also to show the efficiency of the produced methods.

NONSTANDARD THETA-METHOD AND RELATED DISCRETE SCHEMES FOR THE REACTION-DIFFUSION EQUATION

R. ANGUELOV, P KAMA AND JM-S LUBUMA* Department of Mathematics and Applied Mathematics University of Pretoria Pretoria 0002 (South Africa) Let us consider the following initial-value problem for an autonomous system of n differential equations in n unknowns:

The &method

of order 1 or 2 (if 0 = 1/2) is often used for the numerical solution of (1). However, unless 6 = 1/2, this method is not elementary stable in the sense that its fixed-points do not display the linear stability properties of the fixed-points of the differential equation. hrthermore, in the particular case of linear constant coefficient stiff systems, the constraint 1/2 5 8 5 1, which excludes the explicit forward Euler method, is essential for the method (2) to be A-stable. Our aim is to design a qualitatively stable scheme and to investigate its impact on efficient numerical solution of the one dimensional reactiondiffusion equation au at

-=

8% -+T(U). ax=

(3)

To achieve the above for (l),we use two strategies from Mickens’ nonstandard finite difference approach ’. Firstly, assuming that Eq. (1) has a (nonzero) finite number of fixed-points jj, all being hyperbolic, we require *corresponding author: e-mail: j1ubumaQscientia.up.ac.za;fax: +27-12-4203893; phone: +27-12-4202222 24

25

the properties of the fixed-points to be directly reflected in the new version of (2). The properties will be captured by a fixed number Q

L m={l41

where X traces all the eigenvalues of the Jacobian matrices J ( f ) ( j j )off at all the fixed-points. Notice that the choice of the number q is not so critical if the system is non-stiff. In practice, one may take q = max{llJ(f)(jj)lloo; Vjj fixed-points of (1)) where Il.lloo is the matrix norm associated with the supremum norm on Rn. Secondly, we renormalize the denominator of the discrete derivative in (2) through a real-valued function 4, defined for z > 0 , such that

4 ( t )= t + O ( z 2 )as z

--f

0 and 0 < +(z) < 1.

(4)

(A typical example is $ ( z ) = 1 - exp(-2)). Our non-standard &method for (1) is

4

We show that (5) has the same order as the standard method (2). In particular, when 0 = 0 (forward Euler), we obtain explicit error estimates. We prove a result on the elementary stability of the new method, irrespective of the value of the parameter 0 E [0,1]. We also establish some absolute elementary stability properties pertinent to stiffness. As for the correlation of the scheme (5) to the numerical solution of the partial differential equation (3), we observe that the latter has the stationary equation

d2u -+?-(ti) dx2

= 0,

which, in terms of first integral equation, is equivalent to the conservation u)along the trajectory u,that is: V(x,u(x)) = constant. of the energy V(x, With 'p a function satisfying the same conditions as q5 in (4),let us associate

a nonstandard finite difference scheme for ( 6 ) , which is equivalent to the conservation of the discrete energy, i.e., VA,(U,) = V~~(u,-l),Vm 2 1. A general procedure for deriving energy preserving difference schemes is discussed in and from where it follows that one could take Y('llrn+l, urn, urn-1)= Purn - P

%n+l +urn

3

+ ~ r n - 1urn

26

in the case of the Fisher equation, i.e., r(u) = Pu(1-u)in (3). On the other hand, the space-independent equation of (3) is similar to (1). Therefore, we may, with appropriate adaptation, consider its non-standard 0 analogue method ( 5 ) . Assembling this scheme with (7), we obtain the following nonstandard finite difference scheme for (3):

e-hkY1,

uk+l m

7

uk+l m-l>

k + (1- 0 ) Y ( U km + l , uk,um-l).

(8)

It is shown that the scheme (8) preserves the principle of conservation of energy and the linear stability property of the equation (3) in the limit cases of time and space independent variables, respectively. It is also shown that, under a suitable functional relation between the step sizes At and A x , the scheme (8)has the positivity and boundedness property 0 5 uk 5 1 whenever the solution of (3) satisfies the physical property 0 5 u(x,t ) 5 1. Our further scheme for (3) is in the context of variational analysis and spectral methods '. If the space-domain in (3) is the interval (0,27r), then this problem admits the variational formulation: u(x,O)= uo(x),u(.,t)E V for any t > 0 and

0=

127r

*@(x)dx dt

+

1

27r

du(x,t)d@(x) dx dx dx

____-

27r

r(u(x,t))@(x)dx,

(9) for any test function Y E V where V is the subspace of the Sobolev space H 1 ( 0 , 2 7 r ) that consists of periodic functions with period 27~.For the numerical treatment of (9), we couple a spectral discretization in the x-variable (based on the Fourier system (ezmz)mEz)with the non-standard &method in the t-variable. If u",.) E VM := span{eZmz;Iml 5 M } , for M E N, denotes an approximation of u(., t k ) , we obtain the following fully-discrete spectral-nonstandard-theta method: for all Y E V M ,

-

127r

0r[uZ1(x)]+ (1 - 0)r[u~(x)]@(x)dx,

(10)

where q~ is given by the eigenvalues of the Sturm-Liouville problem associated with (3). (We restrict ourselves to a linear reaction r ( u ) ) . We prove a result on rates of convergence, in terms of powers of 1/M and At,for the scheme (10) and we provide numerical experiments.

27

References 1. R. Anguelov and J.M-S Lubuma, Contributions to the mathematics of the nonstandard finite difference method and applications, Numerical Methods for Partial Differential Equations 17 (2001), 518-543. 2. C. Canuto, M.Y. Hussani, A. Quarteroni and T.A. Zang, Spectral methods in fluid dynamics, Springer-Verlag, Berlin, 1988. 3. R.E. Mickens, Nonstandard finite difference models of differential equations, World Scientific, Singapore, 1994.

NUMERICAL INTEGRATION OF A SIZESTRUCTURED CELL POPULATION MODEL IN AN ENVIRONMENT OF CHANGING SUBSTRATE CONCENTRATION

0. ANGULO Dpto. Matema'tica Aplicada a la Te'cnica, Escuela Universitaria Polite'cnica, Universidad de Valladolid. C/ Fco. Mendirabal, 1, 4'7014 Valladolid, Spain E-mail: [email protected]

c.

J. LOPEZ-MARCOS Dpto. Matema'tica Aplicada y Computacidn, Facultad de Ciencias, Universidad de Valladolid, C/ Prado de la Magdalena, s/n, 4'7005 Valladolid, Spain E-mail: [email protected]

We formulate a new numerical method based on integration along characteristics curves t o solve a size-structured cell population model in an environment with an evolutionary resource concentration. Numerical simulations are also reported in order t o show that the approximations converge to the solution of the continuous problem. Also, we show their fine behaviour to study the dynamics of the considered problem.

In the present work, we study a nonlinear size-structured population model that describes the dynamics of a cell population in an environment with an evolutionary resource concentration (Ramkrishna 5). The model assumes that cells are placed on a continuously stirred tank reactor (CSTR), that no cells death occurs and that they grow in one stage, i.e. there is no cell-cycle structure in the mathematical model. This problem consists on a 28

29

partial integro-differential equation (with nonlocal terms) ut

+ ( g ( x ,S ( t ) )u)x = -r(x, +2

S ( t ) )4 5 , t ) - D u(x ,4

LX-

F(a,S ( t ) )P ( x ,a,S ( t ) )4

g ,

4 da,

(1)

< x < xmax, t > 0, that represents the balance law of the population; an initial condition for the state distribution function,

xmin

~ ( 2 , o )= 4(x),

xmin

Ix I Zmaz;

(2)

the appropiate boundary conditions,

g ( x : , S ( t ) ) u ( x ,=tO) ,

x=xmin,xmax,

O I t ,

(3)

that, from a biological point of view, can be thought as defining the boundary of the physiological state space (see 4, and the next initial value problem, d -s(t) = D (Sf - S ( t ) )dt

/

Xmaz

q(a,S ( t ) )u(a,t )da, 0

5 t ; S(0) = So,

Xmin

(4) that describes the dynamics of the abiotic environment. Equation (1) includes information about growth, division and birth at the single-cell level. The first term of the right-hand side of the equation means the lost of cells with physiological state x due to division leading to the birth of the smaller cells. The second term is the dilution term describing the rate by which cells exit the reactor. And the last term represents the rate of birth of cells with physiological state x originating from the division of all bigger cells. On the other hand, Eq. (4)includes information about nutrient uptake at the single-cell level, where the first term of the right-hand side of the equation describes the difference between the inlet and outlet rate from the reactor and the integral term represents the rate of loss of substrate leading to cell growth. Functions u(z,t ) and S ( t ) represent, respectively, the density distribution of the cell culture with size (or another structural property of the single cell) x and the concentration of the available substrate, at time t. The processes involved in Eqs. (1) and (4)are mathematically described by a set of functions known as intrinsic physiological state functions which, in general, depend on the physiological state of the cell x and the state of the abiotic environment S. The growth process is represented by the function g(x,S) which denotes the rate of increase in size of each cell. The cell division and birth ones are described by the division rate r ( x ,S) and the partition

30

probability density function P ( x ,y, S ) , respectively. The later function represents the partition of the mother cell with size y into two daughter cells with size x and y - x. It must satisfy the next conditions

IX-

P(z,y,S)dx=l,

P(x,y,S)=O,

s>y.

(5)

Xmin

Finally, the nutrient consumption one is characterized by the function q(x,S ) which represents the single-cell rate of consumption of the substrate. In order to conclude the description of the terms involved in the problem (1)-(4), the growth of the cells on a CSTR introduces two constants: D ,the dilution rate, and Sf,the concentration of the substance of the abiotic environment at the stream feed of the reactor. Notice that the coupling between (1) and (4) occurs through the dependence of all the physiological state functions on the concentration of the substrates and that this coupling is the source of nonlinearity in the model. In general, we cannot obtain explicitly the solution of the problem (1)(4) then we propose a numerical scheme to integrate it. This method integrates the model along the characteristic curves, these schemes are quite suitable because they use the dynamical behaviour of the population in order to develop the discretization of the model. Next, we describe the theoretical background of the discretization. We rewrite the partial differential equation by means of r*(z,S ) = r(x,S ) + gx(x,S ) D , and (1) can be transform into

+

'Iltk,

t ) + g(z, S ( t ) )%(Z, t ) = 2

lxmaz qU, s(t))

P ( x ,0,s(t)) u(U,t )do

s(t>) 'Il(x,t),

- r*b,

(6)

< x < Xmax, t > 0. Next, we denote by z(t;to,zo)the characteristic curve of the equation (1) that takes the value xo at the time instant to. This characteristic curve is the solution of the following initial value problem xmin

I % ( t o ;t o ,

(7) 20) = xo.

Then, we define the function

31

which satisfies the following initial value problem

Therefore, the partial differential equation (6) reduces to a family of ordinary differential equations. Recall, that we have to integrate three types of problems: the equation that defines the characteristic curves (7), those that obtain the solution of the problem (9) and the one that describe the dynamics of the abiotic environment (4). Note that problems (7), (9) and (4) are coupled. The numerical solution of Eqs (4), (7) and (9) has been performed using the authors experience in the integration of similar size-structured problems, Ref. 1, 2, 3. This type of models usually presents some properties that difficults the development of numerical methods. One of the most significant features is the shape of the growth funtion that usually takes the value zero in somewhere of the space domain. This aspect makes that characteristic curves methods presents some problems in the construction of the spatial grid. In order to diminish this difficulty we have chosen the development of a characteristic curves scheme on a regular grid. Other important feature is the existence of nonlocal terms that we have to approach with suitable quadrature rules, therefore, we need the values of the numerical solution at all the grid points in the level we are working. Besides, in order to integrate equations (4), (7) and (9) we have employed RungeKutta methods. These schemes are composed by different stages that needs unknown values at time levels that are not, in general, a node of the grid defined on the time interval. To circumvent this problem we compute this values using others computational techniques. On the other hand, we have to carry out the numerical integration of the three equations at the same time because all of them are coupled. We have carried out a significant numerical experimentation with different test problems in which are included different features: the use of different growth functions; batch growth or growth on a CSTR; unequal or equal partition (the integral term of (l),2

lICmaz

qu,S ) ~ ( zu,, S ) u(u,t )do,is

replaced by 4 I'(2 z, S ) u(2 z, t ) ) . This experimentation shows that the numerical approximations converge to the solution of the continuous problem.

32

Acknowledgements The authors were supported in part by project DGESIC-DGES BFM200201250 and project Junta de Castilla y Le6n and Uni6n Europea F.S.E. VA002/01.

References 1. 0. Angulo and J.C. L6pez-Marcos. Numerical schemes for size structured population equations. Math. Biosci., 157:16%188, 1999. 2. 0. Angulo and J.C. L6pez-Marcos. Numerical integration of nonlinear s i z e structured population equations. Ecol. Model., 133:3-14, 2000. 3. 0. Angulo and J.C. L6pez-Marcos. Numerical integration of autonomous and nonautonomous nonlinear sizestructured population models. Math. Biosci., 177-178:3%71, 2002. 4. N.V. Mantzaris, P. Daoutidis and F. Srienc. Numerical solution of multivariable cell popultion balance models: I. Finite difference methods Comp. Chem. Eng., 22:1411-1440, 2001. 5. D. Ramkrishna. Statistical models of cell populations. Adu. Biochem. Engng., 1l:l-47, 1979.

A DUALITY METHOD FOR THE COMPRESSIBLE REYNOLDS EQUATION. APPLICATION TO SIMULATION OF READ/WRITE PROCESS IN MAGNETIC STORAGE DEVICES

c. VAZQUEZ Dep. de Matemdticas, Universidade da Coruiia, Campus Elviiia s/n, 15071-A Coruca, Spaan

I. ARREGUI, J.J CENDAN AND

In magnetic storage devices, heads are designed so that a thin air film (air bearing) is generated between the head and the magnetic storage device in the readlwrite process. In this way, the head-device contact only takes place at the initial and final moment. Thus, once a velocity value is reached, the air film is build up so that the hydrodynamic load balances the external load. Hydrodynamic and elastohydrodynamic lubrication theories govern this kind of processes. In the case of hard disk devices the air hydrodynamic displacement is governed by a nonlinear compressible Reynolds equation and the elastic effects are neglected. In tapes and floppy disks (flexible storage media) the elastohydrodynamic model consists of a coupled system based on the compressible Reynolds equation for air pressure and a rod model for the tape deflection. More precisely, in the case of flexible media, the coupled problem providing the air pressure, p , in the thin film and the head-media gap, h, is posed in terms of equations:

dii ii(0) = G(L)= -(O) dY 33

=

dii -(L) = 0 , dY

(5)

34

where V denotes the tape velocity, X the particle mean-free length, p , the ambient pressure and p the air viscosity. Moreover, T , p, E and I r e p resent the tension, the density, Young modulus and the inertia moment, respectively. The ends of the tape are placed at y = 0 and y = 1 while the edges of the head are localized at y = 11 and y = XZ. The coupled feature results from the definition of the gap, = ii - 8, b being the head geometry, and from the pressure acting as a normal force on the flexible tape (2). The notation KC holds for the characteristic function for the set C. The mathematical analysis for the previous model is presented in and finite difference schemes are used in for numerical simulation. The analysis starts from the change of unknowns and variables:

x

= lOOy, p = @ / p a , u = lo%,

h = lo%, 6 = 1068,

which leads to system:

du u(0)= u(L)= -(O) dx

=

du -(L)

dx

=0

,

where a = (dp,)/(pV), P = (ep,)/(6 x 106pV), r] = E I / ( T - pV2) and K = 102p,/(T - p V 2 ) . From this new scaling, the hard-disk model can easily be recovered. More precisely, in this case only the boundary value problem with equation (6) for the air pressure is considered for E = 1/(6pV), a = 6X@,, ,B = 1 and h a given function, as stated in Jai '. In this paper we propose new techniques to deal with the main two difficulties associated to the nonlinear Reynolds equation (6): the convection dominated feature ( € a= 10W2,€0M for typical device values) and the nonlinear diffusion term. Thus, for the first one we propose the numerical method of characteristics adapted to stationary problems, and for the nonlinear term we propose a duality method for maximal monotone operators.

35

First, we consider the variational formulation for (6): Find p E V1 = {'p E H1(L1,L ~ ) / (=P1 on x = L1, x = L2} such that

+

L2

(ah2*d x

E

L1

+ ph3p*)d x 9 dx dx v'p E H&,

= 0, L2).

(11)

Next, by using the method of characteristics for steady problems as it has already been used in lubrication problems en ', we consider the approximation:

where x k ( x )= x the algorithm:

-

k , k being a positive small parameter. So, we propose

For n = 0,1,2,. . . , given p n , find pn+' E V1 such that:

Next, in order to apply Bermlidez-Moreno duality method for the nonlinear diffusive term, we introduce the maximal monotone operator: s2 if s 0 if s

2 0, 5 0.

so that the variational formulation to obtain pn+' remains:

Then, following 2 , we introduce the parameter w > 0 to obtain pn+' as the limit of the sequence {p?"}, which is computed by the algorithm: 0

0

Initialize p:+' Compute p?:

to pn. as the solution of the linear problem:

36

k’ 0

(hpn)o X‘ydx -

dx

Update the auxiliary variable O by

where f,” denotes de Yosida approximation of the operator f - w I with parameter X > 0, I being the identity function. For convergence purposes 2 , the choice X = ( 2 ~has) already ~ ~ been considered. In this way, from subdifferential calculus, we deduce:

fG(2w)(S)=

2ws - 2w2 (-0.5 -2ws

+ -)/,

if s 5 0, if s 2 0.

(14)

For the spatial discretization of the linear problem (13), piecewise linear finite elements have been used, combined with adequate Gauss formulae for numerical quadrature. In order to validate the good performance of the method, a first test with an additional source term associated to the analytical solution p ( x ) = -x2+x+1 has been done. Next, in order to compute solutions with steepest gradients, more realistic tests from mechanical viewpoint, several tests proposed by Jai for his special finite difference method have been reproduced with our method, leading to satisfactory results. Moreover, the optimal value for w predicted by Par&-Macias-Castro in terms of convergence has been experimentally observed. Moreover, in order to numerically solve the elastohydrodynamic coupled problem associated to flexible storage devices, we propose a fixed point iteration between the hydrodynamic model (6) and the elastic one (7), updating the gap with (8). In this setting we propose a cubic Hermite finite element discretization for (7), combined with different alternatives to solve the discrete obstacle problem due to the presence of the head. Finally, the numerical simulation results for several problems with real data are presented.

References 1. G. Bayada, M. Chambat, C. VBzquez, Characteristics method for the formulation and computation af a free boundary cavitation problem, Journal of Computational and Applied Mathematics., 98, 191-212(1998). 2. A. Bermfidez, C. Moreno, Duality methods for solving variational inequalities, Comp. Math. with Appl., 7 , 43-58(1981).

37

3. B. Bhushan, 'Pribology and Mechanics of Magnetic Storage Devices, Springer, New York, 1996. 4. A. Friedman, B. Hu, Head-media interaction in magnetic recording, Arch. Rational Mech. Anal., 140,79-lOl(1997). 5. M. Jai, Homogenization and two-scale convergence of the compressible Reynolds lubrication equation modelling the flying characteristics of a rough magnetic head over a rough rigid-disk surface, Math. Mod. Num. Anal., 29, 199-233(1995). 6. C. Par&, J. Macias, M. Castro, Duality methods with authomatic choice of parameters, Numer. Math., 89, 161-189(2001).

A N ENVIRONMENT FOR COMPUTING TOPOLOGICAL ENTROPY FOR SKEW-PRODUCT TRANSFORMATIONS

FRANCISCO BALIBREA Department of Mathematics, University of Murcia, 30100 - Murcia. SPAIN e-mail: balibreaC2um.e.s JUAN L. G. GUIRAO AND FERNANDO L. PELAYO * Departments of Mathematics 63 Computer Science, University of Castilla-La Manchu, 16071 - Cuenca. SPAIN e-mail: [email protected], FernandoL. [email protected] The main objective of this paper is t o introduce a specific software package for computing topological entropy of two-dimensional skew-product transformations. The Kneading Theory establishes the necessary background for this study is some particular cases. This is not only a hard but also a computationally expensive problem. The CAS Mathernatica provides a suitable environment for implementing the algorithm associated t o the rigorous computation of the topological entropy of the considered systems. Keywords: Symbolic Mathematical Computing, Kneading Theory, Skewproducts Maps, Topological Entropy, Dynamical Systems, Mathematica. Topics: Computational Sciences. Nonlinear dynamic systems models are ubiquitous throughout the sciences and engineering. Efficient computational analysis of specific dynamical systems is often a critical component for the successful completion of a research or design project. We live in a dynamical world. For many reasons, we want t o understand these dynamics: t o predict the weather, to prevent heart attacks and limit spread of infections diseases, t o control agricultural pests, to farsee the consequences on man’s activities on the global climate and the impact of climate changes that might result from these activities, t o design both more reliable and more efficient machines, and so on. *corresponding author 38

39

Questions about how processes evolve and change in time are really important and of broad usefulness. Indeed in various domains of science many situations can be, at least approximately, modelled in a very simple way via difference equations of the form x,+1=

f(x,),

n = 0 , 1 , 2 ,...

where f is a particular function which define a dynamical system. The notion of dynamical system is the mathematical formalization of the more general concept of a deterministic process. The future state of many Physical, Chemical, Biological, Engineering, Economical and even Social Systems can be predicted, to a certain extent, by knowing the present state, 20, and the law governing its evolution, f. Because of that, this work could be a useful object for researchers in these areas who use dynamical systems as model tools in their studies. We are not concerned with the problem of how to find such f. Generally, an experimental function f is approximated by an explicitly defined function depending on parameters that are subsequently determined by means of statistical methods. These dynamical systems are defined by rules of transformations which are needed in order to determine how points in a state space evolve when time elapses. Time can either be discrete or continuous. The traces of points as they move in discrete or continuous time are called trajectories. Dynamical systems theory seeks a comprehensive description of the geometric structures arising from these trajectories. In most of cases such trajectories are difficult to describe, even is not possible, when it occurs for many of them, the system behaves in a very complicated way. The scenario: Let I = [0,1] be the compact unit interval of the real line. We consider skew-product transformations on the unit square, that is, surjective continuous maps from I 2 into itself of the form F : (x,y) -+ (f(x),g(x, y)) ( F E Ca(12)).In this setting, the maps f and g are respectively called the basis and the fiber map of F . For every x E I , the maps gx defined by gx(y) = g(x, y) form a system of one-dimensional mappings depending continuously on x. More details on this kind of maps can be found in These types of systems have a lot of applications in pure mathematics (e.g. in the study of geodesic flows on Riemannian surfaces of constant 16317,576.

40

negative curvature, strange attractors, certain polynomial endomorphisms of en,...)and in other sciences (e.g., all known examples of systems in engineering, physic and economic with strange nonchaotic attractors having the skew-product form, see for instance 23). One of the most important tools for measure the complexity of a dynamical system is the concept of the topological entropy introduced by Bowen in 1971, '. When the map F has zero topological entropy ( h ( F )= 0), this means that the behaviour of the system is not difficult from a dynamical point of view. In some sense, in the previous case we can say that the system is not chaotic. Whereas, positive entropy means complicated dynamic. In this setting, two interesting problems can be stated, on one hand how to characterize maps with zero entropy, i.e., maps with a simple dynamics 15, and on the other hand how to compute explicitly the positive entropy. This paper deals with the second problem for some types of maps on C A( I 2 ) . Firstly, we remark that the mathematical problem of computing the topological entropy for a map F E Ca(12)remains open and far from being solve. In 'l, is developed a kneading theory which solves the problem under some restrictions. Let F E C A ( I ~be ) a map holding the following properties:

4.5

-

b

1-

(1) the basis map f is uni-modal with critical point c, see figure 1 for m=1, (2) for some integer p , the critical point cis pperiodic for f (i.e., fp(c) = c and fj(c) # c for 0 5 j < p where f k denotes the composition of f with itself k times),

41

(3) for some integer m, the map (y) = g p where g p = g (zp-1,g ( ~ ~ - 2 ., ..,g ( X I ,g (50,y)) ...)) is m-modal with critical points (c1, c~,...,cm}, see Fig. 1, (4) for every i E 1,2, ..., m, there exists an integer qi such that the critical point ci is qi-periodic for g p .

+

Then, the topological entropy of F is h ( F ) = h(f) h(gp). The previous result is based on the construction of a symbolical dynamic associated to the critical points which generates a Markov partition of the unit square 12. Two transition matrix are obtained and by the sum of the logarithms of their spectral radius the exactly topological entropy is calculated. The main idea of the program is to compute the periodic order of the critical points of the basis map f , once this is done, function gp is defined and its critical points will be computed, in the same way the periodic order of the critical points of gp has to be computed. Although the location of critical points process can be not too hard (because of the existence of analytical -thanks to symbolic mathematical computing power of Mathematicclr and numerical methods), the computation of the periodic order of such points could become computationally expensive process, especially when the critical points have been obtained via numerical methods. Afterwards, a Markov partition for both functions, f and gp, is defined and their transition matrix will be computed. From the characteristic polynomials associated to these transitions matrix, is necessary to calculate their spectral radius and finally the polynomial entropy of the two-dimensional skew-product transformation is obtained by applying the main result presented in 21 . In order to accomplish our goal we have chosen the Computer Algebra System C.A.S. Mathematica, 35, because it is not only a mathematical assistant which allows us to make both symbolic and numerical analysis, but also, it provides us with a Functional Programming Language. These two characteristics give rise t o a desirable environment, to deal with any problems which need 0 0 0

to handle with data in a symbolic way to compute numerical analysis algorithms t o execute the above tasks by means of a functional language which is extremely next to the theoretical results that make possible this analysis

42

This C.A.S. has been widely used by the authors, e.g. looking for and identifying bifurcations 30, or studying stability of continuous systems 'Of Next, the result obtained from the program execution when the input is the below function F is shown:

F (x, y) = (f (x) ,g (zly)) = (1 - 1.76~',x - 0.823~') CRITICAL POINTS O F f: -0.7589 3-periodic - XI = -0.7589,~2 = -0.0135,~3= 0.9997

ASSOCIATED gp: - g p (y) = 0.9997 - 0.823(-0.0135

- 0.823(-0.7589

- 0.823~')')~

CRITICAL POINTS OF gp: -0.0018 5-periodic - yl = -0.0018,~2 = 0.8041,~3= -0.5795,~4 = 0.3396,~5=

0.6899 0

MARKOV PARTITION FOR THE PERIOD 15 ORBIT OF

0.8

-%

0.8

-

&

0.4 -

v, 0.2

-

0 - yl

-0.2

-

4.4

-

-0.6-

v3

,x, 4.8

I

4.6

, -0.4

, 4.2

x.

x2,

0

0.2

0.4

0.8

0.8

THE CHARACTERISTIC POLYNOMIALS: - PA, - PA,

( t )= 1 - t - t 2 ( t )= 1 - - t' -k t3 - t4

1

43

0

THE SPECTRAL RADIUS: M 1.6183 - A, = l / t , = - A, == l / t , M 1.5128

0

TOPOLOGICAL ENTROPY: - h(F)=h(Fp)= l ~ ~ ( A ) ~ l ~ ~ ( A , ~ A y ) ~ l ~ ~ ( X , ) +

0.8952

An interesting application: We consider Cournot maps on 12, i.e., continuous transformations of the form $(x,y ) = (q51(y),q5z(x)). There exists a very well-known economic production process called Cournot duopoly which is mathematically modelled by discrete dynamical systems based on Cournot maps. “There exist two companies producing a n identical good. I n each step of the process, the responses of the firms in terms of the production , are simultaneous and they depend o n the production of the rival firm in the last step”. In order t o have information about the economical situation previously described, these discrete models have been studied from different points of view with the aim of describing their dynamical behaviour, It has been proved that the topological entropy is the key for studying the complexity of this kind of systems. Specifically in lo, a topological characterization for it was given. This result is exactly the generalization for Cournot maps, on one hand of the one-dimensional Misiurewicz’s theorem (see 22, ( 1 ) @ ( 2 ) ) and on other hand of certain results proved by Sharkovsky in the sixties (see 26, ( 1 ) @ 7712125.

*

(4)). By Sn(.),Per(.), h(.),UR(.), Rec(.) and AP(.) we respectively denote a stratification set, the set of periods of periodic points, the topological entropy and the sets of uniformly recurrent, recurrent and almost periodic points. Misiurewicz & Sharkovsky’s Theorem: Let 4 E C ( I ) . The following properties are equivalent: (3)

(1) h ( 4 ) = 0 , (2) the period of any periodic point is power of two, (3) W 4 ) = W + ) , (4) AP(4) = {x E I : lim q52”(x)= x}. n+m

44

Assume now, that 4(x,y) = ($l(y), 42(x)) is a Cournot map and we are interested in analyzing the complexity of the system modelled by 4, i.e, we want to calculate h(4). Consider the second iterated of 4, It can be observed that q52 is a triangular map. Assume that 42 holds the restrictions of our environment (this is a natural assumption since in the most of Cournot models the reaction functions $1 and 4 2 are multimodal with periodic critical points for having convex utility sets), thus our environment can compute h(42). Now, for a discrete dynamical system generated by a map $I of a compact metric space into itself, it is true that for every integer n 2 1, h($In)= n . h($I)(see '). Then, it can be applied to the case of the Cournot map, n = 2, and h(q52)= 2 . h(4) is obtained. Therefore, we are able to compute the topological entropy of the triangular map 42 and as a corollary we obtain an explicit calculation of the topological entropy of the Cournot map 4 as a half of the topological entropy of 42.

References 1. 2.

3.

4.

5.

6.

7.

8.

V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, Springer Verlag, 1983. F. Balibrea and J.C. Valverde, Bifurcations Under Non-degenerated Conditions of Higher Degree and a New Simple Proof of the Hopf Bifurcation Theorem, J. Math. Anal. Appl. 237, 93-105(1999). F. Balibrea and J.C. Valverde, Structural Stability Under Conditions of Non-Hyperbolicity, Comput. Math. Appl. 41,757-768(2000). F. Balibrea and J.C. Valverde, Extreme Degenerations for the Simplest Generic Bifurcations and New Transversally Conditions, Discrete and Continuous T i m e Dynamical Systems Special Volume added to Volume 6, 22-30(2000). F. Balibrea and J.L. Garcia and J.I. Muiioz, Description of w-limit sets of a triangular map on 12, Far East Journal of Dynamical Systems 3(1), 87-201(2001). F. Balibrea and J.L. Garcia and J.I. Muiioz, A Triangular map on I 2 whose w-limit Sets are all Compact Intervals of ( 0 ) x I , Discrete and Continuous Dynamical Systems 8(4),983-994(2002). G.I. Bischi and L. Gardini, Cycles and Bifurcations in Duopoly Games, Chaos, Solitions and Fractals 14,139-150(2001). L.S. Block and W.A. Coppel, Dynamics in One Dimension, Lecture Notes in Math., Springer, 1993

45

9. 10. 11. 12. 13. 14. 15.

16. 17. 18. 19. 20.

21. 22. 23. 24. 25. 26.

27. 28. 29.

30.

R. Bowen, Entropy for Group Endomorphism and Homogeneous Spaces, Transactions American Mathematical Society 153,401-414( 1971). J.S. Canovas and A. Linero, Topological Dynamics Classification of Duopoly Games, Chaos, Solitions and Fkactals 12,1259-1266(2001). J. Carr, Applications of Center Manifold Theory, Springer Verlag, 1981. R.A. Dana and L. Montrucchio, Dynamical Complexity in Duopoly Games, Journal of Economical Theory 40, 40-56( 1986). J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, 1983. J. Hale and H. KO&, Dynamics and Bifurcations, Springer Verlag, 1991. Z. KoEan, The Problem of Classification of Triangular maps with Zero Topological Entropy, Annales Mathematicae Silesianae 13, 181192( 1999). S.F. Kolyada, On Dynamics of Triangular Maps of the Square, Ergodic Theory and Dynamical Systems 12,749-768(1992). S.F. Kolyada and L. Snoha, On w-limit Sets of Triangular Maps, Real Analysis Exchange 18(1), 115-130(1992/93). Y .A. Kuznetsov, Elements of Applied Bifurcation Theory, Springer Verlag, 1995. J. Marsden and M. McCracken, Hopf Bifurcation and Its Applications, Springer Verlag, 1976. J.A. Martinez and F.L. Pelayo and J.J. Miralles and J.C. Valverde, Stability of Continuous System by Routh-Hurwitz and Mathematica, Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2002), Alicante, Spain, 20-23 September, 2002. D. Mendes and J. Sousa Ramos, Kneading Theory for Triangular Maps, Submitted t o Discrete and Continuous Dynamical Systems, 2003. M. Misiurewicz and W. Szlenk, Entropy of Piecewise Monotone M a p pings, Studia Mathematica 67, 45-36( 1980). S.S. Negi and R. Ramaswamy, A Plethora of Strange non-chaotic Attractors, Pramana Journal of Physics 56( l), 47-56(2001). Z. Nitecki, Differentiable Dynamics, M.I.T. Press, 1971. T. Puu, Chaos in Duopoly Pricing, Chaos, Solitions and Fractals 1, 573-581( 1991). A.N. Sharkovsky and S.F. Kolyada and A.G. Sivak and V.V. Fedorenko, Dynamics of One-dimensional Maps, Dordrecht : Kluwer Academic Publishers, 1997). J. Sijbrand, Properties of center manifolds, 7kans. Amer. Math. SOC. 289,431-469(1985). J. Smital, On Functions and Functional Equations, Adam Hilger, 1988. J.C. Valverde, Teoria de Bifurcaciones Locales de S.D.D. Generalizada, Departamento de MatemBticas de la Universidad de Murcia, Master Thesis, 1998. J.C. Valverde and F.L. Pelayo and J.J. Miralles and J.A. Martinez,

46

31. 32. 33. 34. 35.

Symbolic Mathematical Computing of Bifurcations in Dynamical Systems, Proceedings of the International Conference on Computational and Mathematical Methods in Science and Engineering (CMMSE 2002), Alicante, Spain, 20-23 September, 2002. S. Wagon, Mathematica in action, Springer-Telos, 1998. D.C. Whitley, Discrete Dynamical Systenls in Dimensions One and Two, Bull. London Math. Soc. 15, 177-217(1983). S. Wigginns, Introduction to Applied Nonlinear Systems and Chaos, Springer-Verlag, 1990. S. Wolfram, Mathematica, a System for Doing Mathematics by Computer, Addison-Wesley, 1991. S. Wolfram, The Mathematica Book, Wolfram Media, 1996.

MAXIMUM LIKELIHOOD AND CONDITIONAL MAXIMUM LIKELIHOOD LEARNING ALGORITHMS FOR HIDDEN MARKOV MODELS WITH LABELED DATA-APPLICATION TO TRANSMEMBRANE PROTEIN TOPOLOGY PREDICTION P. G. BAGOS, TH. D. LIAKOPOULOS AND S. J. HAMODRAKAS Department of Cell Biology and Biophysics, Faculty of Biology, University of Athens Panepistimiopolis,Athens 15701, Greece E-mail: [email protected] Hidden Markov Models (HMMs) have been widely used in applications in computational biology, during the last few years. In this paper we are reviewing the main algorithms proposed in the literature for training and decoding a HMM with labeled sequences, in the context of the topology prediction of bacterial integral membrane proteins. We evaluate the Maximum Likelihood algorithms traditionally used for the training of a Hidden Markov Model, against the less commonly used Conditional Maximum Likelihoodbased algorithms and, after combining results previously obtained in the literature, we propose a new variant for Maximum Likelihood training. We compare the convergence rates of each algorithm showing the advantages and disadvantages of each method in the context of the problem at hand. Finally, we evaluate the predictive performance of each approach, using state of the art algorithms proposed for Hidden Markov Model decoding and mention the appropriateness of each one.

1. Introduction 1.1. Hidden Markov Models

Hidden Markov Models (HMMs) are probabilistic models suitable for a wide range of pattern recognition applications. Initially developed for speech recognition [I], during the last few years they became very popular in molecular biology for protein modeling and gene finding [ 2 ] . A Hidden Markov Model [2] is composed of a set of states. Two states k, I are connected by means of the transition probabilities akl, forming a lst order Markovian process. Assuming a protein sequence x of length L denoted as: x = x , , x , ).”) x , , where the xi’s are the 20 amino acids, we usually denote the “path” (i.e. the sequence of states) ending up to a particular position of the amino acid sequence (the sequence of symbols), by n. Each state k is associated with an emission 47

48

probability ek(xJ, which is the probability of a particular symbol xi to be emitted by that state. 1.2 Maximum Likelihood training

Traditionally, the parameters of a Hidden Markov Model are optimized according to the Maximum Likelihood criterion [ 11,

dM

= argmaxP(x e

18)

(1)

The total probability of a sequence given the model is computed by summation over all possible paths through the model: P ( x IS) = C P ( X , Z 18) 77

This quantity is calculated using a dynamic programming algorithm known as the forward algorithm [2], or alternatively by the similar backward algorithm. The logarithm of P(xl0) is the log-likelihood of the sequence. For numerical stability reasons the usual approach is to maximize the log-likelihood (or equivalently minimize the negative log-likelihood). A widely used algorithm for this task is the efficient Baum-Welch algorithm (also known as forwardbackward) [3], which is a special case of the Expectation-Maximization (EM) algorithm, proposed for Maximum Likelihood (ML) estimation for incomplete data [4]. The algorithm, updates iteratively the model parameters (emission and transition probabilities), by assigning to them their expectations, computed with the use of forward and backward algorithms. Convergence to at least a local maximum of the likelihood is guaranteed. Baldi and Chauvin [ 5 ] , were the first to propose a gradient descent method capable of the same task, which offers a number of advantages over the Baum-Welch algorithm, including smoothness and on-line training abilities. 2.

Labeled sequences and learning algorithms

2.1. Labeled Sequences

In molecular biology there is often a need to train models with a large number of parameters (states), in order to explicitly model different “classes” of data. Examples can be found in the topology prediction of transmembrane proteins [6] or in gene finding [7,8]. To accomplish that, one has to train separate submodels corresponding to the different “classes” (i.e. to train one model for the transmembrane regions, another for the cytoplasmic loops and a third one for the extracellular loops), and then combine them using the appropriate transitions. To overcome this complication, Krogh suggested the use of labeled sequences

49

[9]. Thus, each amino acid sequence x is accompanied by a sequence of labels y for each position i in the sequence: Y = Y , Y2 YL Krogh proposed a simple modified version of the forward and backward algorithms [9], incorporating the concept of labeled data. The likelihood to be maximized in such situations is the joint probability of the sequence and the labeling given the model. P( x, y I 0) = p (x,Y," I 0) = fJ (%Z1 0) 7

7...7

c x

c

Zen,

The simple idea behind this approach is that summation has to be done only over those paths I7, that are in agreement with the labels y. Consequently, one has to declare a new probability distribution, in addition to the transition and emission probabilities, the probability dk(c) of a state k having a label c. In almost all biological applications this probability is just a delta-function, since a particular state is not allowed to match more than one label. The expectations for the emission and transition probabilities are computed with the modified forward and backward algorithms [9]. Consequently, it is straightforward to derive the modified Baum-Welch algorithm. This is one of the methods that we consider in this study. 2.2. Discriminative training With the use of labeled sequences, Krogh also derived a learning algorithm that maximizes the probability of the labeling [9], instead of the probability of the sequences. This procedure is referred to as the Conditional Maximum Likelihood (CML) criterion and the parameters of the model are optimized so that:

The denominator in the right hand side of Eq. (4) is the likelihood computed allowing all paths in Eq. (2), whereas the numerator is the likelihood considering only paths consistent with the labels, Eq. (3). Once again, turning to logarithms is more convenient. The maximization procedure cannot be performed with the Baum-Welch algorithm and an incremental algorithm has been proposed instead [9]. In this work we employed a maximization procedure, using a gradientdescent method as proposed in [ lo].

50

2.3 Maximum Likelihood (ML) gradient descent method for labeled sequences Starting from the gradient-descentML learning algorithm proposed by Baldi and Chauvin [ 5 ] , and incorporating the idea of labeled sequences, we derived a similar ML gradient-descent learning algorithm suitable for labeled sequences. The derivatives of the negative logarithm of the likelihood in Eq. (3), (denoted by 1) with respect to the emission and transition probabilities can be expressed in terms of the forward and backward variables. For example, the derivative with respect to a transition probability a k l is:

-= --

(5)

8% a k l ’ whereas, the derivative with respect to an emission probability is:

ae - --E, ( b ) ( b ) - ‘k ( b ) The quantity Akl is the expected times a transition happens from state k to state 1 whereas &(b) is the expected times a symbol b is emitted from state k. These quantities [2], are calculated according to: aek

wherefk(i) and b k ( i ) are the forward and backward variables respectively, el(xi+l) and a k l are the emission and transition probabilities respectively and & ( Y i + / ) is the delta function indicating the agreement of the current state with the labeling. By calculating the derivatives of the log-likelihood with respect to a generic parameter 0, we proceed with gradient-descent and iteratively update these parameters according to:

where q is the learning rate. To avoid the risk of obtaining negative estimates, we used a proper parameter transformation and performed gradient-descent optimization on the new variables, as proposed in [lo]. For example, for the transition probabilities, we used the softmax transformation:

51

which yields the following update formula:

The gradients with respect to the new variables zkl can be expressed entirely in terms of the expected counts and the transition probabilities at the previous iteration.

Substituting now Eq. (1 1) into Eq. (lo), we get an expression entirely in terms of the model parameters and their expectations. The same holds for the emission probabilities. The procedure described above was also used for the CML training described in section 2.2, for minimizing the negative loglikelihood in Eq (4). Employing this gradient-based optimization method we were able to use some known tricks for escaping from local minima of the negative log-likelihood, such as the use of a momentum parameter. 3.

Transmembrane protein topology prediction

3.1. Data sets and topology of the model To evaluate the performance of the training techniques described above, we developed two different HMMs: one for the prediction of the transmembrane/Istrands and the topology of the bacterial /I-barrel outer membrane proteins [ 1l l , and another for the prediction of the transmembrane a-helices of the bacterial ahelical membrane proteins. Transmembrane protein topology prediction is a task of great interest in molecular biology since transmembrane proteins perform a wide variety of important biological roles, comprising approximately 25-30% of all fully sequenced genomes [121. On the other hand, only a small fraction of them is of known structure and thus there is the need to develop computational tools capable of recognizing membrane proteins when screening fully sequenced genomes. Both models are cyclic (Figure l), with states belonging in three classes, consisting of a total of 70 and 114 states for the outer membrane proteins and the a-helical membrane proteins, respectively. In the case of a-helical membrane proteins the three classes correspond to transmembrane (TM), cytoplasmic (IN) and periplasmic (OUT) segments, whereas for /I-barrel outer

52 membrane proteins to transmembrane (TM), periplasmic (IN) and extracellular (OUT) segments. Each model was constructed on the basis of expert knowledge and designed specifically for capturing the structural features of the corresponding class of proteins. For training we used two non-redundant sets of proteins with experimentally determined topologies. The outer membrane protein set consists of 14 proteins derived from the Protein Data Bank [13], whereas the a-helical membrane protein set, of 110 bacteria membrane proteins taken from the TMPDB 1141.

5gure 1. General topology of the models. Dashed rectangles represent the three different classes (IN, inner; TM, transmembrane; OUT, outer). The two models have major differences in the number of states and their connectivity within each class.

The training was performed in batch mode (off-line) with the use of the modified Baum-Welch algorithm, the gradient method described in section 2.3, and the gradient method for Conditional Maximum Likelihood proposed by Krogh [lo]. For the decoding process we used the well-known Viterbi algorithm, which finds the most probable-state path [1,2], and the N-best decoding algorithm [ 15,161, which aims at finding the most probable labeling".

All algorithms used in this work, were implemented by the authors using the JAVA programming language, on a dual Pentium 111 866 Mhz workstation under Linux. a

53

3.2. Results and Discussion From the results of this study, presented in Tables 1 and 2, the first obvious conclusion is somewhat expected, and is relevant to the performance of the prediction algorithms on the two distinct structural classes of transmembrane proteins. In particular, the overall performance of the prediction methods is significantly higher for ec-helical membrane proteins than for /?-barrel outer membrane proteins. This finding is due to the presence of a clearer pattern accounting for the transmembrane regions in the case of a-helical membrane proteins, which consists of a stretch of 15-30 consecutive highly hydrophobic residues. Table la. Measures of predictive performance for the ^-barrel outer membrane proteins. Gradient

Baum-Welch

CML

Prior Uniform Prior Uniform Prior Uniform Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best 02

0.739 0.804 0.781 0.824 0.771 0.802 0.786 0.823 0.772 0.801 0.693

Ca

0.481 0.603 0.564 0.643 0.542 0.599 0.569 0.639 0.546 0.597 0.393 0.559

TP

121

163

146

FP

1

4

1

FN

93

51

68

176 5 38

139

163

154

175

1

4

5

5

139 1

75

51

60

39

75

0.78

161

95

144

2 53

2

3

119

70

Table 2". Measures of predictive performance for the a-helical membrane proteins. Gradient

Baum-Welch Prior

Prior

Uniform

CML

Uniform

Prior

Uniform

Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best Viterbi N-best

Q2

0.875 0.888 0.876 0.888

Ca

0.703 0.735 0.703 0.735 0.715 0.735 0.725 0.733 0.604

TP

496

547

FP

57

32

FN

97

46

498 58 96

543 31 50

0.88

0.888 0.884 0.887 0.838 0.853 0.725 0.823 547

537

63

32

64

32

51

62

46

56

46

212

531

547

381

0.61

408 9 185

0.264 0.505 94 44 499

332 7 261

The second observation is the clear advantage of the N-best decoding algorithm, which significantly outperforms the predicting performance of the Viterbi algorithm. This is caused probably by the fact that in applications with a

Q2: Total fraction of correctly predicted residues in a two state mode (transmembrane vs. non transmembrane). Ca: Mathew's correlation coefficient. TP, FP, FN: True positives, False Positives and False Negatives transmembrane segments respectively.

54

labeled data, the total likelihood of different paths sharing the same labeling is significantly higher than the likelihood of the best single path of states. Clearly, N-best decoding should be preferred since the cost of its computational demands is negligible compared to its increased predictive performance. B

-1

-wanw-)

Figure 2. Evolution of the negative log-likelihoods: (A) a-helical membrane protein set initialized by uniform probabilities, (B) a-helical membrane protein set initialized by prior distribution, (C) p-barrel outer membrane protein set initialized by uniform probabilities, and (D) /?-barrel outer membrane protein set initialized by prior distribution.

When referring to the convergence rates of the Baum-Welch algorithm and the gradient-based method for Maximum Likelihood estimation (Figure 2), we observed that for such applications the Baum-Welch algorithm achieves faster convergence. However, we should mention that in cases where the training started with uniform transition and emission probabilities (all parameters were set equal), both ML algorithms reached a local maximum of the likelihood, with a better predictive performance. On the other hand, when the training started from some fixed prior probabilities for the parameters, the algorithms reached a maximum significantly higher than in the former case, but with decreased predictive Performance. Presumably, this is caused by overfitting since it is known that iterative methods such as EM and gradient-descent strongly depend on the initial values of the parameters. This is more obvious in the case of outer membrane proteins where the number of examples in the training set was limited.

55 Comparing the prediction accuracies for each training method, we also observed that discriminative training (CML) does not clearly outperform either the traditional ML training via the Baum-Welch algorithm, or the gradient method proposed in this work. This is expected, as a consequence of the fact that discriminative algorithms are sensitive on data mislabeling [lo]. In such applications the data are inherently incorrectly labeled, since it is known that even in proteins with structure known at atomic resolution, the exact boundaries of the membrane regions are not well determined [6,12]. It is also worth noting that CML training requires twice as much computational time as that for ML training, since it requires two complete passes of the forward and backward algorithms. It seems that CML training might be advantageous in cases like gene finding [7,8], where mislabeling of the data is not expected (exons-introns), and also in cases of hybrid methods where other methods are not suitable [ 101.

References 1. Rabiner L. R. Proc IEEE 77 (2), 257-286 (1989). 2. Durbin R., Eddy S., Krogh A., and Mitchison G. Cambridge University Press, (1998). 3. Baum L.E. Inequalities 3, 1-8 (1972). 4. Dempster, A. P., Laird, N. M. and Rubin, D. B. J Roy Stat Soc B 39, 1-38 (1977). 5. Baldi, P. and Chauvin Y. Neural Comput 6(2), 305-3 16 (1994). 6. Krogh, A., Larsson, B., von Heijne, G. and Sonnhammer, E. L. J A401 Biol 305,567-580 (2001). 7. Henderson, J., Salzberg, S., and Fasman, K. J Comput Biol 4(2), 127-141 (1997). 8. Krogh, A., Saira, I. M., and Haussler, D., Nucleic Acids Res 22 4768-4778 (1994). 9. Krogh, A. Proc 12th Int ConfPatt Recog 140-144 (1994). 10. Krogh, A., and Riis S., Neural Comput 11,541-563 (1999). 11. Schulz, G. E., Biochim Biophys Actu 1565,308- 3 17 (2002). 12. Von Heijne, G., Quart Rev Biophys 32(4), 285-307 (1999). 13. Berman, H.M., Westbrook, J., Feng, Z., Gilliland, G., Bhat, T.N., Weissig, H., Shindyalov, I.N., Bourne, P.E., Nucleic Acids Res 28,235-242 (2000). 14. Ikeda, M., Arai, M., Okuno, T. and Shimizu, T., Nucleic Acids Res 31(1), 406-409 (2003). 15. Schwartz, R. and Chow Y. L. Proc IEEE Int ConfAcoust, Speech, Sig Proc, 1,Sl-84 (1990). 16. Krogh, A. Proc 5th Int Conf Intel Sys Mol Bioll79- 186 (1997).

VARIANTS OF RELAXED SCHEMES AND TWO-DIMENSIONAL GAS DYNAMICS *

MAPUNDI K. BANDA Fachbereich Mathematik, Schlossgartenstr. 7, 64289 Darmstadt, Germany

E-mail: bandaC2mathematik.t.u-darrnstadt.de

We present a relaxed scheme with more precise information about local speeds of propagation and a multidimensional construction of the cell averages. Hence the physical domain of dependence is simulated correctly and high resolution is maintained by a genuinely multidimensional piecewise nonoscillatory reconstruction. Relaxation schemes have advantages that include high resolution, simplicity and explicitly no (approximate) Riemann solvers and characteristic decomposition is necessary. Performance of the scheme is illustrated by tests on two-dimensional Euler equations of gas dynamics.

1. Introduction Given an open bounded domain R c R2 and a time interval [O,T],in this paper higher order relaxed schemes are used to approximate a 2-dimensional N x N hyperbolic system of conservation laws:

a

-U at

a a + -F(U) + -G(U) ax

dY

= 0, U E RN

The relaxation system proposed by Jin and Xin [S] is:

au av aw at ax a y

-+-+-=(I,

*This work is supported by deutscher forschungsgemeinschaft grant kl 1105/9. 56

57

-

where E > 0 is the relaxation rate. The matrices A and B are appropriate 0, the solution of diagonal matrices. In the zero relaxation limit, E Eq. (2) approaches the solution of the original system Eq. (1) provided a subcharacteristic condition [6] holds. 1.1. Space Discretization

For the space discretization of Eq. (2), we cover 52 with rectangular cells Ci,j := [xi-+,xi++] x [yj-l,yj++] of uniform sizes Ax and Ay. The cells, Ci,j, are centred at (xi = iAx,yj = j a y ) . We use the notation: wi*;,j(t) := w(xi*;,Yj,t), wi,j*;cq := w(xi,Yj*$,t),

to denote the point-values and the approximate cell-average of the function w at (xi+;, yj,t), (xi, y j * + , t ) , and (xi, yj,t), respectively. Point-values Ui+4 , j , Uij++ , Vi++ ,j , and Wi,j+ ;are defined by upwind schemes and used in defining divided differences for discretizing Eq. (2). The scheme realised is referred to as a relaxing scheme and in the limit E 0 as a relaxed scheme.

-

2. Extensions to Relaxed Schemes

We consider variants in the x-direction only. Those in the y-direction follow analogously. 2.1. More Accurate Speeds of Propagation

In the upwind schemes, piecewise polynomial reconstructions of approximate solutions on each cell introduce possible discontinuities at the interface points. These discontinuities propagate with right- and left-sided local speeds, which, in the genuinely nonlinear or linearly degenerate case, are estimated by

-

dF

au

-

.I), A

%+I ,I. := min{Al(-(y+L, 2 3

dF ~ ( ~ ( ~ ~ + J L o } ~

respectively. Here, A1 < . . . < AN are the N eigenvalues of the Jacobian 273 2 3 and u:++,~:= p:++l,j(xi++,j) h e r e p are and ti;+. 2 13. := p?.(zi+l,.) piecewise polynomials used in the reconstruction of component u of U at time step n.

58

2 . 2 . Multidimensional Schemes

To obtain a second-order multidimensional scheme cell averages are integrated using a trapezoidal rule [2]. We approximate U(z, y, t”) by a linear polynomial reconstruction on each cell Cij. The scheme is now genuinely multidimensional since we add cross-diagonal directions to the Cartesian directions utilised in local speeds. 3. Numerical Tests

We would like t o approximate a solution to the 2-D Euler Equations for gas dynamics: m

2[!]+;[ dt n

p:p]+-&[

n

vp( E ~+ ; P) p]=o,

(3)

U(E + P) where p, u, v, rn = pu, n = pv, p, and E are the density, velocity in xdirection, velocity in y-direction, momentum in z-direction, momentum in y-direction, pressure, and energy, respectively. The equation of state for a polytropic gas is given by p = (y - l ) . ( E - : l l ~ 1 1 ~ ) , where y = 1.4. Radially Symmetric Riemann-Problem This is a test which we used to check the conservation of radial symmetry [4]. The computational geometry is a box R = [-0.5,0.5] x [-0.5,0.5]. An equidistant grid in both the z- and y-direction with 100 x 100 cells was used. The following are the initial conditions:

with (pl,pl,ul,v ~ =) (2.0, ~ 15,0,0)T and (pT,pT,ur,v,)T = (1.0, l.O,O,O)T and x = (z,y) is the space variable. Boundaries were maintained at ( p , p , ~ , v ) ~ ( x= , t )( p r , ~ , , ~ r , v , . ) TFigure . 1 (left) is a contour plot of density at t = 0.13 in which 15 equidistributed contour lines were used. “Double Sod Tube Problem” In this test case a “Double Sod Tube” Riemann problem [l]is considered. The same geometry as above is used with an equidistant grid in both the z- and y-directions with 200 x 200 cells. The following is the initial profile: p(x,O) = 0.1 if z y < 0, 1 otherwise p(x,O)= 0.1 if zy < 0, 1 otherwise u(x,O) = 0;

59

0.5

0.4

O

-0.1

-0.1

4.2

-0.2

-0.3

-0.3 -0.5 -0.4

4.4 4.5

-0.6

I 4.4

-0.2

0

0.2

0.4

0.6

Figure 1. Density Profiles for radial symmetric problem (left) at t = 0.13 and “Double Sod Tube” (right) at t = 0.2.

where x = (5,y) are space variables. The approximate solution at t = 0.2 is presented in figure (1) (right) plotted with 15 equidistributed contours. For both the above tests CFL = 0.475. These profiles are resolved without any input on the elementary waves involved, beyond the characteristic speeds. These preliminary results above compare favourably with the results in [4, 11. For time integration we used a second-order Runge-Kutta scheme [5]. 4. Summary

We have presented modifications to the relaxed schemes for conservation laws and tested it in two-dimensional gas dynamics. More comprehensive tests and comparisons with exisiting schemes applied to gas dynamics are underway.

References 1. D. Aregba-Driollet and R. Natalini, SIAM J. Numer. Anal. 37, 1973-2004 (2000). 2. M.K. Banda, Relaxation Schemes for Multidimensional Conservation Laws in preparation. 3. M.K. Banda, A High Resolution Relaxation Scheme with more Accurate Local Speeds of Propagation - in preparation. 4. M. Brio, A.R. Zakharan, and G.M. Webb, J . Comp. Phys. 1 6 7 , 177-195 (2001). 5. S. Gottlieb, C.W. Shu, E. Tadmor, S I A M Rev. 43,89-112 (2001). 6. S. Jin and Z.P. Xin, Comm. Pure Appl. Math. 48, 235-277 (1995). 7. H.J. Schroll, J. Sc. Comp.17, 599-607 (2002).

PREDICTING TOXICITY OF CHEMICALS USING CHEMODESCRIPTORSAND BIODESCRIPTORS: AN INTEGRATED APPROACH SUBHASH C. BASAK, BRIAN D. GUTE, AND DENISE MLLS Natural Resources Research Institute, University of Minnesota Duluth, 5013 Miller Trunk Highway, Duluth, Minnesota 55811, USA

Modern lifestyle is highly dependent on the routine use of thousands of chemicals. Many of these chemicals have toxic effects which are brought about through a myriad of biochemical mechanisms. A combination of functional (biochemical) criteria and some aspects of the molecular structure have been used for the hazard assessment of pollutants. In recent years, quantitative descriptors of molecular structure, usually called chemodescriptors, are being used for the prediction of biochemical/ toxicological properties of chemicals. In the post-genomic era, mechanisms of action of hazardous chemicals can be characterized by techniques like proteomics and quantitative descriptors derived from proteomics maps, called biodescriptors. This presentation will discuss the relative utility of chemiodescriptors and proteomics-based biodescriptors in the development of integrated QSARs (I-QSARs) in predicting activity/ toxicity of chemicals.

60

MEASURING ECONOMIC WELL-BEING AND GOVERNANCE: SOME METHODOLOGICAL TOOLS SUDIP RANJAN BASU Graduate Institute of International StudiesiKJHEI InternationalEconomicsDepartment and Econometrics Department Universityof Geneva, Switzerland E-mail: sirdip hasu@,hotrnail.com.hasul@,hei.imiee.ch

Over the last decade, there has been a continuos attempt to provide some suitable methodologies for the measurement of ‘Human Development’, ‘Human Poverty’, ‘Gender Developoment’, ‘Governance’, ‘Corruption’, ‘Happiness’, etc., among the International Organisations (e.g., UNDP, World Bank), and in the academic profession (e.g., ICRG, BERI, FH etc.) . In all these studies the single most important element is to quantify some of the subjective or abstract variableslelements into a single quantitative value, so that the countries or the regions could be placed in the rank ordering on the basis of the values of such indices. In this paper we attempt to provide a new method to compute the economic well being (a measure of over all economic development) and governance (a measure of institutional quality) in the economylsociety. The principle economic rationale of the present paper is to attempt explore the dynamics of the economic well being with that of the other socio-economic dimensions, i.e., the quality of institutions and decentralisation for invigorating the economic well being. In exploring this inter linkage between the economic well being and that of quality of institutional arrangements, we attempt to develop a new measure of the economic well being, and that of the quality of institution and then to search for their potential relationship in this present study. Initially the paper attempts quantify the economic well being for the society. The economic well being is a multidimensional concept, and is abstract in nature. We therefore suggest a methodology to compute the Economic Well Being Index (EWBI), combining several variables, which incorporates the socio-economic dimensions of the quality of life. These indicators are related to human resources, to availability of health services, to utilisation of technology, to availability of physical infiastructure, and to the flow of financial services in the economy (measured as: income per capita, literacy rate, enrolment ratio, infant mortality rate, life expectancy, hospital beds, electricity consumption, bank branches, telephone lines, road length, railway routes, intensity of

61

62 cropping, fertilizer consumption). This computed well being index intends to capture the various levels of economic prosperity in the society. We attempt to compute the economic well being index in the following way: as the included indicators are mutually interdependent, we use factor analysis to compute this EWBI. The Factor Analysis (FA) technique is a powerful tool, which is used to reduce the number of influencing indicators, to detect structure in the relationships among indicators that is to classify variables according to their effect on the variables of interest. The underlying assumptions of factor analysis are that there exist a number of unobserved ‘factors’ that account for the correlation among the observed indicators, and because of this relation, the unobserved factors can be inferred from the observed indicators. Thus, the base model can be written as, a j=l

Now rewriting the above equation in a linear FA model yields:

XI =AIL+ A 2 f 2+ .......... +Aqfq+el

1

X, =

+ A2f2+ .......... + A2qfq+ e2

X,

+ AP2f2+ ..........+ Apqfq+ ep

= APIA

We then estimate factor scores in the FA model as below: for a given

f j the i th extracted factor score, denoted by q,, is given by = p , X , , + ........ + p p X , p,where p, ,p , ,.......p pare referred to as regression

factor

e,

coefficients and Xi,, Xi*, .....Xi, are the p observed indicators, for the *

th

observations. We define the Economic Well Being Index (EWBI) as a weighed average of the factor scores, where the weights A’s, are the eigenvalues of the correlation matrix. Thus: I

-

EWBI” =

CTA: Fj Aj ,

s = 1,2 ,.....S(states I regions)

Y

J

Then we quantitatively measure the quality of good governance. The quality of good governance is a quite multidimensional concept, likewise EWBI, and it should be examined along with different socio-economic dimensions. The good governance measure is captured through the ability of the governments to provide basic law and order, to provide social services to build up human capital, to provide physical infrastructure or to properly maintain the economic and administrative management that is to keep up high level of efficiency in its activities. In the present analysis, our governance measure is related to four

63 dimensions, namely, peace and stability; people's sensibility; social equality; and management of government (measured as: crime rates, riots, industrial disputes and strikes, government debt-income ratio, and gini inequality measure). We postulate that Quality of Governance Index (QGOI) is a latent variable, which is supposed to be linearly dependent on a set of observable indicators plus a disturbance term capturing error. Let

QGOI = a + PIXl + ......... + PkXk + e,where XI, ,......X,is set of indicators that are used to capture this QGOI, so

x,

that the total variation in the QGOI is composed of two orthogonal parts: a) variation due to set of indicators, and b) variation due to error. We propose to replace the set of indicators by an equal number of their principal components (PC), so that 100% of variation in indicators is accounted for by their PCs. We now solve the determinental equation,

IR -

= 0 for

A,

th

where R is a K X K matrix ;this provides a K degree polynomial equation in and hence K roots. These roots are called eigenvalues of R. Let

A us arrange A in descending order of magnitude, as A,)A2)..........) A k . Now , corresponding to each value of A ,we solve the matrix equation (R - u)cr= 0 for the K x 1 eigenvectors a'a = 1.

a , subject to the condition that

Then we write the characteristic vectors as

which correspond to 2 = Al = .........., i z k respectively. Hence we obtain the principal components as:

I

4 = a , , X l + .........+ a I k X k

P2 = a,,X, + ......... + a2kXk Pk = a,,X, + .........+ akkXk

Thus we compute all these PCs using elements of successive eigenvectors corresponding to eigenvalues, ,?,,12,. ........Ak respectively. We now estimate the QGOI as weighted average of the PCs, thus:

.

64

QGOIs =

qn, + Ga2 ...........+ P k A k n, +A, + ............ + A + a

eigenvalues

of

,where the weights are the

the

correlation matrix R and ill = Var pl ,......iz, = var pk,and s = 1,2 ,.....S(states/regions) As noted in the literature that the issue of decentralisation is also crucial in percolating the h i t s of development to the people at the grass root level. We use the simple measure of financial decentralisation to capture this argument into our model. Here we focus only on the aspect of financial decentralisation as measured by the assignment of expenditure functions and revenue sources to sub-national levels of governments. In the economy the devolution is supposed to provide more efficient and equitable service delivery, to enhance revenue mobilisation, to promote participation, and to improve political stability in the society. The Financial Decentralisation Index (FIDI) is defined as:

assignment and compensation to local bodies FIDI” = ( 1 governments total revenue expenditure *loo, s = 1,2,...S(states/regions) Then we ran the Panel data regression to the dataset to obtain more eflcient estimates, by pooling the time series and cross section data. We use the model in the following framework: initially, we look at the pooled ordinary least square model of estimation. The pooled model contains observations on N units of observations (cross-section units), over the T time points (time points). The purpose is to estimate a standard regression model of the form:

Ypoi,= a + flxi,+ e, ei,

assumption the

.

are

where i = 42,...N and

iid over

iand t,

.

t = 1,2,....T by i.e., E(e,) = Oand

2

var(e,) = cre . Then we use the Fixed Effect estimation model, specification for the individual state specific effects is given by

y

FE

it

= ai+ p

I

.

x + e, i = 1,.....N ; t = 1.....T .

The Random Effects

l x k kxl

model has been rejected on the basis of Hausman test statistic. So, we use statehegional panel by estimating the following regression: E WBI,, = a p QGOI ,t QGOI,, + p FIDIst FIDI,, 6, E,, ,

+

+ +

where

EWBI,, is the dependent variable for state/ region s i n period t , QGOI and FIDI are independent variables, with 6, are fixed or random effects, and E , are error terms. The econometric analysis of Panel data estimate

65

after controlling for other variables indicates that the estimated parameters are statistically different from zero. Our basic model specification therefore shows a systematic strong positive link between economic well being and the institutional arrangements. The dataset for this study is compiled for 16 major states of India over four points of time (1970s, 1980s, 1990s, and 1997-latest).

MODELLING THE NATURAL GAS CONSUMPTION IN A CHANGING ENVIRONMENT F. A. BATZIAS, N. P. NIKOLAOU, A. S. K A K O S m I . MICHAILIDES Department of Industrial Management and Technology, University of Piraeus, 80 Karaoli & Dimitriou St., 185 34 Piraeus, Greece, E-mail:[email protected] A composite function was used successfully for modelling the Natural Gas (NG) consumption in 16 European energy markets. Background of the model is a logistic function where the upper limit is also a logistic function of time, with secondary parameters determined either endogenously together with the rest primary parameters or exogenously in a sample space of the energy market. Fitting of this ‘double logistic’ dynamic model to NG consumption data of the period 1980-2000 gave better Standard Errors of Estimate (SEES) for ten energy markets in comparison with the linear, the expone-ntiavasymptotic and the static logistic model. Supplementary results obtained by statistical analysis of answers selected by circulating a questionnaire in the wider area of Attica in Greece, led to the conclusion that income/welfare, residential place and information play an important role as regards the intention of inhabitants to adopt the NG alternative.

1. Introduction Natural Gas (NG) has been actually introduced to most European national energy markets a long time after their maturation, which was based on other energy sources (basically coal, oil, hydro- and nuclear energy). In several energy markets, it is expected that the adoption (or dissemination or diffusion or spread) of NG will follow the familiar pattern of an S-growth curve, which has been observed in many cases of technological substitution; although the validity of such a model may not be finally confirmed, it is widely preferred, at first instance or as a rough approximation, by several researchers, because it implies a rational mechanism of interaction between individuals in society: the rate of adoption is proportional to all possible connections between adopters and nonadopters at any instant of time, or mathematically dy/dt = by(K-y), which gives by integration the well-known simple logistic function y = K / [l + m exp(-bt)], where y is the number of adopters cumulatively calculated at time t, K is the upper limit of the energy market share to be covered by NG, b is the proportionality factor in the differential form or the exponential parameter in the

66

67 integrated form of the model (i.e. a rate constant), m is the relative ‘vertical’ range of the logistic curve measured in yo-units: m = (K-yo)/yo, y=yo at t=O. If the rational mechanism of diffusion suggests that the role of adopters is not significant (e.g. when the diffusion of information depends mainly on a central node transmitting the corresponding message without significant interaction between the receivers) then the logistic model is reduced to dy/dt =b(k-y), which gives by integration the well-known simple exponentiallasymptotic function y = K - (K - yo) exp(-bt). Several versions of the logistic function have been developed so far in various scientific disciplines (like Physical Chemistry, Engineering, Biology and Economics) and certain attempts have been made to design generalized models that incorporate most of these versions as special cases [l, 21. The aim of the present work is (a) to develop a dynamic model for NG consumption, (b) to test its validity with data corresponding to European energy markets, (c) to suggest alternative approaches for parameter values estimation and (d) to show a way for influencing these parameter-values by means of methods used in normative economics in order to accelerate NG adoption in countries like Greece, that are actually late arrivals in the area of NG utilization. 2.

Methodology

The K-value is very important because it determines both, the upper limit and the rate of adoption, as it is evident from the differential form. One of the criticisms most often heard in relation with the applicability of the logistic model concerns this parameter, as it seems unreasonable to assume that its value is constant over the long period of time necessary to provide adequate data for reliable parameter value estimation by means of time series analysis. For the needs of the present study, we have introduced a K-parameter, which is also a function of time and has a different semantic content: it refers to possible adopters at time t (a new parameter, variable over time) rather than to expected adopters at t + 00 (the old parameter, constant over time). As the set of possible adopters, who determine the driving force (K-y) is formed and continuously transformed within the same mechanism of information dissemination accompanied by rational thinking with a tendency for imitation, it is reasonable to assume a logistic model to simulate the dependence of K on time: K = KO/[ l + m exp(-bot)]. Consequently, the following double logistic model (LL) is derived: y = W[l+mo exp(-bot)]/[ 1 + m exp(-bt)]

(1)

This dynamic model does not have the constraint of symmetry, a disadvantage of the static logistic function according to most researchers (see

68 e.g. [3]), although it keeps the limit of this simple form, i.e. y-&, dy/dt--tO for t - m and y+O, dy/dt-0 for t+-Oo. Moreover, (1) maintains simplicity, is recognizable and offers appropriate background for embedding parametervalues exogenously determined; the latter advantage is very important as we can simulate the expected behaviour of the energy market by using a representative sample of the population under consideration to investigate the individuals’ intention at time t to adopt NG (in the near future) as a function of economic and social parameters/characteristics. Turner et al. [4] had inserted a logistic model for K in a Bemoulli-type differential equation representing the rate of population growth. Their integrated model is complicated in comparison with (l), includes an additional exponential parameter, implying decrease of the degrees of f?eedom, and its sum of squares of errors is further increased because of the necessity to improve the short-term predictability by forcing the fitted curve to go exactly through the latest datapoint. This model was used for forecasting the growth of U.S. population, not quite successfully as the upper limit was estimated to be 1751 million inhabitants. It is worthwhile noting that in the same work the suggested integrated model is in error while the cited parameter values do not confirm the quoted y-estimates.

3. Implementation and Results We have applied the LL equation (1) to model the NG adoption in the energy market of 16 European Countries for the period 1980-2000 (see Table 1). Most results indicate a better fitting in comparison with the results obtained by the linear, the exponential/asymptotic and the simple logistic model (LN, EX, LG, respectively). The criterion used for comparison is the Standard Error of Estimate SEE = [ c ( y i - j i ) * i ( n - p ) I”* where j , is the estimated, by means of nonlinear regression, value of yi (i=l, 2, ..., n), p is the number of parameters used for regression and n-p is the number of degrees of freedom; in the case of regression of a logistic model of any kind, a fixed kernel is introduced, implying a corresponding decrease of the degrees of freedom. Three different algorithmic procedures were used for non-linear regression (a) to avoid convergence to local optima and (b) to cross check the SEE-values. The flexibility/adaptability of the dynamic model in comparison with the other models is shown in the diagrams of Figures 1-2. Due to its flexibility, there is no need to obtain data quite beyond a fixed inflection point, which is 1/2, 1/3, l/e of the upper limit for the simple logistic, the Floyd and the Gompertz models, respectively, for parameter values estimation by regression; neither further assumptions on expected values of the

69 dependent variable or smoothing of data are necessary, as it is the case when a limited number of data is available (see e.g. [5]). Table 1. SEE-values (dry NG 109 ft3) for best fitting of the four models under consideration (minimal values are quoted in gray). Model Code Country EX LN LL LG Austria 33.41 40.21 39.08 165.68 160.42 543.22 Belgium 15.92 17.14 Denmark 22.48 17.85 7.55 Finland 17.37 France 1134.10 1103.84 #59.08 2047.51 2364.38 2283.13 Germany 4.25 Greece 54.48 0.87 7.78 Ireland """""5.42 32.98 Italy 1458.39 1417.60 ""143064 1056.65 932.57 Netherlands 960.73 62.12 62.06 Norway 65.88 21.89 21.96 ~ 2L40 Portugal 387.81 593.81 Spain 577.94 0.28 10.48 3.21 Sweden 1.93 Switzerland 1.88 134 r 6761.17 United Kingdom 1529.01 7783.37

lljllHH

j;vi;:;iiijj

jl tiiil '^llitl!

70

600

Spain

Data - -LL 0

nz500 U a

;Al

..

.... P.... . ..

- 400 0

c)

Q

5 300 (b

C

0" 200

6 100 0 ,380

1985

1990 Year

1995

2000

Figure 1. Fitting of models to data in the case of Spain; the superiority of the LL model is evident, especially near the ends; an extrapolation for short term forecasting based on the simple logistic LG might lead to significant overestimation of NG consumption.

160

-m

2

g

140 120 100

'Jp

5

80

t

20

0

o 1980

I985

1990

Year

1995

2000

Figure 2. Fitting of models to data in the case of Finland; only the LL model is capable to 'catch' a local fluctuation, like the one observed near the start of the time period.

71 4.

Supplementary Information by Statistical Analysis

The parameter values K,,. m. bo of the function K = f(t) can be estimated exogenously by means of a questionnaire addressed each year to a sample of population selectedlstructured to be a representative one for the market under consideration. Income level and residence area of the families selected to constitute the representative sample are expected to influence their attitude as it is shown in Table 2, where the cited statistical results refer to a questionnaire answered by 74 persons living in a very high income suburb of Athens (H) and by an equal number of persons living in a very low income suburb of Piraeus (L) within the same urban area of Attica in Greece. The first line (HALF) refers to the question “do you consider seriously the possibility of adopting NG for heatinglcooking needs within the next three years on the grounds that such an adoption might cut your electricity bill to half?”. The second line (ENVR) refers to the same question by informing also about the expected environmental benefits at local and national level in the case of adoption. The third line (RESR) refers to the same question by further adding information on the benefits of replacing lignite by NG from the point of view of saving natural resources for ourselves and the future generations, implying rational decision making, better living and independence. The fourth line (DUBL) refers to the same question (with all additions so far) with a change of the economic incentive to “. .. on the grounds that without such an adoption your electricity bill is expected to double within the same time period due to dramatic increase of k w h - price which is already low for a EU member country?”. The symbols X2, C, Cr, r, D stand for the chi-square statistic, the absolute coefficient of contingency, the relative C as a fraction of C, the tetrachoric correlation of attributes and the difference between X2 and X2, where X2, is the critical value of X2 at confidence level 99.5% (found in statistical Tables with percentile values for the chi-square distribution), respectively. Evidently the results are significant at this confidence level as in all cases D>O or X2 > X2,, i.e. the null hypothesis Ho that there is no difference as regards the intention (I) of each group to adopt the NG alternative is rejected. In other words, the residents of the low income suburb (L) exhibit a significantly stronger intention to adopt the NG alternative in comparison with the residents of the high income suburb (H), with the following percentages in favour of adoption: 29/74 = 39.2%, 56/74 = 75.7%, 62/74 = 83.8%, 65/74 = 87.8%, against 7/74 = 9.5%, 14/74 = 18.9%, 17/74 = 23.0%, 36/74 = 48.6%, for HALF, ENVR, RESR, DUBL, respectively. Nevertheless, the situation is not so clear in the case of middle income suburbs and the determination of the dependence of (I) on income distribution,

72

welfare level and adequacy of information (cross-section analysis in each year) is necessary to obtain information useful to desigdimplement energy policy instruments (economic incentives and direct regulations) for diffusion hastening. This might prove an effective way to overcome social inertia and cope with the energy ‘paradox’, i.e. the very gradual/slow diffusion of apparently costeffective and environmentally more friendly technologies ([6]). Table 2. Statistical analysis of results based on the chi-square test.

Code HALF

ENVR RESR DUBL

Values without Correction I c 1cr( r 17.77 0.33 0.46 0.35 9.89 47.82 0.49 0.70 0.57 39.94 54.98 0.52 0.74 0.61 47.10 26.22 0.39 0.55 0.42 18.34

x’I

Values with Yates Correction c IcrI r I D 16.19 0.32 0.44 0.33 8.31 45.57 0.49 0.69 0.55 37.69 52.56 0.51 0.72 0.60 44.68 24.44 0.38 0.53 0.41 16.56

I D Ix’I

5. Conclusions and Recommendations The NG consumption in several European energy markets can be modelled by the logistic function and in most cases better results are obtained when the upper limit of the energy market share to be covered by NG is also considered as a function of time. When this function is also a logistic, the composite dynamic model, named herein double logistic (LL), appears to (a) maintain simplicity, (b) include meaningful and recognizable parameters and (c) offer appropriate background for embedding parameter-values exogenously determined; the latter advantage is very important as we can simulate the expected behaviour of the energy market by using a representative sample of the population under consideration to investigate the individuals’ intention to adopt NG as a function of economic and social parametershharacteristics. Statistical analysis we performed on data collected through a questionnaire proved that residents on low-income suburbs (which are also environmentally degraded) exhibit stronger intention for adopting the NG alternative in comparison with residents of highincome suburbs. By combining the results of NG consumption modelling (time series analysis) with the statistical results from processing the answers to a questionnaire circulated within a representative sample (cross-section analysis) we can offer incentives that facilitate the adoption of the NG alternative.

Acknowledgments

The authors kindly acknowledge financial support provided by the Research Centre of the University of Piraeus.

73 References 1. A. Tsoularis, Res. Lett. In$ Math. Sci., 2,23 (2001). 2. C. Skiadas, Tech. Forecasting andSoc. Change, 27,39 (1985). 3. C . Skiadas, Tech. Forecasting andSoc. Change, 30,313 (1986). 4. M . E. Turner Jr., B. A. Blumenstein, and J. L. Sebaugh, Biometrics, 25,577 (1969). 5. J. Siemek, S. Nagy, and S. Rychlicki, AppIiedEnergy, in press, (2003). 6. A. B. Jaffe, and R. N. Stavins, Resource and Energy Economics, 16, 91 (1994).

COMPUTER AIDED DIMENSIONAL ANALYSIS FOR KNOWLEDGE MANAGEMENT IN CHEMICAL ENGINEERING PROCESSES F. A. BATZIAS, A. S. KAKOS AND N. P. NIKOLAOU Department of Industrial Management and Technology, Universip of Piraeus, 80 Karaoli & Dimitriou St., 185 34 Piraeus. Greece E-mail: [email protected]

An algorithmic procedure has been designedldeveloped for Computer Aided Dimensional Analysis (DA) of chemical engineering processes. The main purpose of this software is to construct'select the best combination of dimensionless groups describing adequately a process under certain criteria. The creationloperation of an Ontological Knowledge Base ( O m ) plays a central role in this procedure as it provides, inter alia the means for filteringlreducing the dimensionless groups obtained by solving the system of dimensional equations according to the Buckingham n Theorem. The successful implementation of this software is thoroughly presented step by step in a case of mass transfer (liquid drops moving in immiscible liquids).

1.

Introduction

According to the classical work of Johnstone and Thring [ 13, Dimensional Analysis (DA) is a technique for expressing the behaviour of a physical system in terms of the minimum number of independent variables and in a form that is unaffected by changes in the magnitude of the units of measurements. The physical Variables, Parameters, and Dimensional Constants (VPCs) are arranged in Dimensionless Groups consisting of ratios of the VPCs, which mostly characterize the system under consideration; these groups constitute the new variables in the dimensionless equation of state of the system. In case that the system is a chemical engineering process, the equation of state is used either (i) for scaling down an industrial process (already in operation or in the stage of design) to optimize its (real or expected) operation or solve relevant problems, or (ii) for scaling up a process synthesized in laboratory scale within an R & D programme. The number of dimensionless groups, which constitute the equation of state, is (n - m),where n is the number of VPCs and m is the number of Primary Quantities used to define dimensionally the VPCs, according to the well known Buckingham's Il Theorem (see, e.g. [ 11). The proof of this Theorem, by means of Linear Algebra, implies that the number h of distinct possible arrangements of dimensionless groups is equal to the number of combinations of m linearly independent columns of the dimensional matrix (see [2]). These unique 74

75

solutions form the S-matrix. Although the solutions are all algebraically equivalent, they give different estimators for a dependent variable, by regression to the same data. As a matter of fact most researchers rely on an arbitrarily selected solution for parameter estimation, possibly using only the subjective criterion of recognizing some groups in this solution, as widely used in technical literature. In the present work, we have designed/developed/implemented an algorithmic procedure for (a) finding out all independent feasible solutions and (b) selecting the best solution by means of quantitative criteria and some kind of filtering performed by an Ontological Knowledge Base (Om) which is continually enriched either in the course of running the software on a variety of dimensional problems or exogenously via an intelligent agent [3]. It is worthwhile noting that the methodology presented herein can be easily extended to cover fractal VPCs, as they are defined in [4], provided that corresponding data are available. 2.

Methodology

The algorithmic procedure, especially designed for (a) the DA of a process, (b) the choice of optimal combination of independent dimensionless groups, and (c) the creation/maintenance of the OKB, includes the following stages. Figure 1 illustrates the interconnection of stages within an R & D programme, represented by the corresponding number or letter, in the case of activity or decision node. resvectivelv. I . Collection ofall VPds (totalled n), relating to the full-scale process, which is the objective of the R & D programme. 2. Design and testing of the laboratory scale simulator of the process. 3. Ranking of all VPCs in order of decreasing importance. 4. Measurements of all VPCs by means of the simulator. 5. Selection of the q VPCs ranked first (out of total n) to start the iterative part of the algorithm. Initially, k = q. 6. Computer aided Dimensional Analysis to form the h,x(k-m) S,-matrix, where hi is the number of independent solutions at step j . 7. Filtering of the S,-matrix to obtain a partial Pj-matrix by fulfilling the criterion of minimum number of dimensionless groups ‘recognized’ by the O m . A reasonable choice for this filtering criterion is (k-m). 8. Estimation of all parameter values appearing in the solutions, represented by the rows of the P,-matrix, by non-linear regression. 9. Ranking of the solutions, in order of increasing SEE. Let SEE, be the SEE of the solution ranked first.

76 10. Scale up of the process under development to pilot plant by taking into consideration the relation corresponding to last stored SEE in stage 1 1.

Executive line

..................Information line Figure 1. Flow chart of the algorithmic procedure designeddeveloped for dimensional analysis, scale up of the processes and creatiodenrichment of the OKB.

11. 12. 13. Q. R.

Let a = SEE, (initially a = M, where M is a very big number). Creation/Enrichment/Usage of an OKB. Addition of the next VPC ranked next in stage 3. Is the SEE, of the solution ranked first in stage 9 less than a? Is the equalityj = (n - q - I ) valid?

3.

Implementation

We implemented the above algorithmic procedure in the case of mass transfer under the form of liquid drops moving in immiscible liquids, a process which plays important role in liquid-liquid extractors, in separators used in distillation columns, and in packed towers when the packing is not wetted by the disperse phase. The first nine VPCs collected are the following ranked in order of decreasing importance (see stages 1-3 in Figure 1): U = terminal velocity of drop E [L T-'I, Ap = difference in densities of drop liquid and water E [M L"], D = equivalent spherical drop diameter E[L], g = local acceleration due to gravity E [L T-'], CT = interfacial tension between drop liquid and water E [M T' 2 1, p = density of water E [M L-3], I.( = viscosity of water E [M L-' T-'I, = viscosity of drop liquid E [M L-' T-'I, and d = diameter of container E [L]. The first seven were selected to start the iterative part of the algorithm (stage 5 ) , i.e. herein n=9, m=3, q=7. Instead of performing measurements by means of the simulator (stage 4), numerical data from technical literature [5] were used. The nrcdimensionless groups (r = I, ..., ho, c = 1, 2, ..., q-m) constitute the hox (km ) or 18x4 So-matrix (stage 6). For filtering the So-matrix, we set the criterion

77

of hlfilling the maximum number (i.e. k-m = 4) of dimensionless groups recognized by the OKB in order to obtain the partial Po-matrix (stage 7), which consists of only two rows (lines No.8 and 11 of Table 1). The solution No.8 was ranked first by performing stages 8, 9, and its SEE0 value was stored according to stage 11, as j # (n - q - 1). the stage 13 is activated and the 8" dimensional parameter po enters in stage 6. This iteration gives a 3 1x5 S1-matrixfrom which a 2x5 PI-matrix is obtained by setting the same criterion of fulfilling the maximum number (i.e. k-m = 5) of dimensionless groups recognized by the OKB. The No.12 solution {IIIz,~ = plm P-'I3 - ~g U, n12.2=p2I3P-23 g 113 D, n12.3 = p1I3p-Il3 g-'I3 0,1712,4 = p-' Ap, E11z,5 = p po}, with SEEl = 12.2624, is passing the filter and as SEEl < a this solution dominates. This means that the contribution of po to the modeling of the process is significant even when a strict filter is applied. As j # (n - q - I ) , the stage 13 will again be activated to examine the contribution of parameter d but there is also a provision in our software design for terminating the program in case that the user considers the last SEE-value as adequately small (a termination clause that might be also set a priori). It is worthwhile noting that all information gained during this procedure has been stored in the OKB (stage 12). Table 1. The hj x (k-rn) = 18 x (7 - 3) Sj-matrix (forj=O), with SEE of non-linear regression for each solution; groups in gray are recognizable by the OKB in the stage of filtering.

SEE 12.89856 12.85352 12.79775 12.79961 12.78040 12.79682 12,79450 12.77123 12.81673 12.83649 15.85902 12.79193 12.87053 12.78061 12.81102 12.79239 21.62942 21.14292

78 4.

Discussion

The predominant solution in the above case example is found when j=l as there is no solution passing the filter for j=2; it happens also that SEE1 < a. A problem might arise if SEE1 is slightly higher that SEE,, but the inclusion of within the equation of state increases the explanatory ability of the independent variables. To provide against such a contradiction, we have inserted a man-machine interaction node in the computer program so that the user can judge on whether the marginal utility added (by increasing j ) counterbalances possible slight increase of accepted SEE; so, if he decides to keep the additional VPC, he knows exactly the price paid for the sake of more explicit scientific reasoning. A similar but simpler question arises as regards the selection of the q initial VPCs, which constitute the minimal set of VPCs capable to describe the process under consideration at j=O. E.g. in the above case example, the software user might be an expert in process engineering who might consider the wall effect not negligible enough to omit the container diameter d; consequently, he would start the algorithmic procedure with 8 (including 6) instead of 7 VPCs even if the corresponding information extracted from the OKJ3 is in favour of omitting this parameter. Inversely, the OKB is enrichedupdated by means of such human intervention; the neuro-fuzzy rules within it learn from experience resulting to eventual change of suggested proposals (i.e. change of dominance hierarchy) in next runningslsessions. 5.

Conclusions

We have designed an algorithmic procedure for (a) finding all feasible solutions of independent dimensionless groups that describe a process, and (b) selecting the best solution by means of quantitative criteria and some kind of filtering performed by an Ontological Knowledge Base (OKB). By applying this procedure we avoid the arbitrary choice of a, possibly dependent, combination of dimensionless groups, which is either the result of an empirical procedure (lacking theoretical foundation), or the subjective choice of a person who usually makes decisions on the basis of which dimensionless groups seem familiar to him and/or are cited in other studies, regardless of semantics. The aforementioned procedure was implemented in a user-friendly computer application developed using MS .Net Architecture; moreover, the OKB was implemented using widely accepted Internet standards (XML-W3C specifica-tions) to allow interaction with other similar applications or agents, possibly across the Internet [3]. The software was successfully applied in the

79 case of mass transfer under the form of liquid drops moving in immiscible liquids, a process which plays important role in liquid-liquid extractors, in separators used in distillation columns, and in packed towers when the packing in not wetted by the disperse phase. By checking (a) the procedure and the software functionality (by letting a common user -not an expert- to perform all possible operations) and (b) the results qualitatively (by means of technical literature survey) and quantitatively (by using linear algebra) we found no problematic response of the system. Nevertheless we have installed a manmachine interface to permit human interaction when the user has a reason to believe that another (acceptable but sub-optimal) solution might be better fitted under certain circumstances; in such a case, the OW3 is enriched by human reasoning and new predominant rules may emerge eventually.

6. Acknowledgments The authors kindly acknowledge financial support provided by the Research Centre of the University of Piraeus.

References R. E. Johnstone and M. W. Thring, Pilot Plants, Models, and Scale-up Methods in Chemical Engineering, McGraw-Hill, New York (1957). 2. W. D. Curtis, J. D. Logan, and W. A. Parker, Linear Algebra and its Applications, 47, 1 17 (1982). 3. F. A. Batzias, and E. C. Marcoulaki, Computer-Aided Chem. Engineering 10,829 (2002). 4. M. Rybaczuk, and W. Zielinski, Chaos, Solitons and Fractals, 12, 2517 (2001). 5. P. M. Krishna, D. Venkateswarlu, and G. S. R. Narasimhamurty, Journal of Chemical and Engineering Data, 4,336, 340 (1959). 1.

80 Appendix Table A.I. Sample of the Variables, Parameters, and Dimensional Constants (VPCs) included in the OKB. Symbol

U Ap D g a P u Mo d N DA V

D! APra

kLa Us

DP P MT

e

Name Terminal velocity of drop Difference in densities of drop liquid and water Equivalent spherical drop diameter Local acceleration due to gravity Interfacial tension between drop liquid and water Density of water Viscosity of water (or liquid) Viscosity of drop liquid Diameter of container Impeller/mixing/stirrer speed Molecular Diffusivity Kinematic Viscosity of liquid Impeller/stirrer diameter Pressure difference Volumetric Liquid-side mass transfer coefficient Slip velocity Particle diameter Power Total Mass of particle Power input per unit mass of fluid

Dimension [LT1] [ML0]

[L]

[LT2] [MT2] [ML-3] [M L-' T1] [M L'1 T1] [L]. [T1]

[L2T'] [L'T1] [L] [M L'1 T2] [T1] [LT1] [L] [M L2 T'3] [M] [L2T3]

Table A.2. Sample of the recognizable Dimensionless Groups included in the OKB; groups in gray have been used for filtering in the implementation presented herein. Symbol |

Name Reynolds Number Weber Group : Drag Coefficient of Drops ; Froude Number : B group Gravity Group G Group (G) P Group : Sd Group ; Terminal velocity Group Schmidt number (Sc) Power Number (Po) Solid Concentration or Quantity Group (U) (Sh) Sherwood number Euler number (Eu) (NReP) Particle Reynolds Number Turbulent Reynolds Number (Re) Impeller Reynolds Number (Re)

vimm^ :;§fl:

. :-"?W : ; ;ilp. -«iep:. - (wt)

; :XP) ^•::|l!iA

I

Variables D p a"1 U D p U2 a 1 4/3 D U"2 g Ap p'1 DU' 2 g !/4D 2 0 -'gAp (4/3pA P n- 2 g) 1 / 3 D gD 4 U 4 P 3 °- 3 u"4 c3 g.! p2 Ap"1

y^W'V^cAp-1'3

U 3/41/3 p2'3 u-"3 g"3 Ap-"3 U D A -'v P N'3 DI ~5 p-' MT p 1 D3

MO^DA*2

APp^D,- ^ 2 D p u s v-'

v D 4/3 £ l/3

D,2 N p a'1

81

Figure A. I . The Graphical User Interface of the software application that implements the herein presented algorithmic procedure.

Figure A.2. The 8-variable dimensional problem, discussed in the implementation section, is being defined by the non-expert user through a user-friendly dialog form.

82

Table A.3. The hi x (k-m) = 31 x (8 - 3) Sj-matrix (forj=l), with SEE of non-linear regression for each solution; groups in gray are recognizable by the O D in the stage of filtering. Dimensionless Groups

I

SEE 12.806 12.626 12.598 12.464 12.557 12.271 12.449 12.459 12.271 12.278 12.271 12.262 12.459 12.460 15.214 12.872 12.346 12.304 12.280 12.288 12.882 12.354 12.313 12.379 13.394 12.313 12.263 23.173 20.364 23.387 15.177

NEW DEVELOPMENTS ON EQUIVALENT THERMAL IN HYDROTHERMAL OPTIMIZATION. AN ALGORITHM OF APPROXIMATION

L. BAYON, J. M. GRAU, M. M. RUIZ AND P. SUAREZ University of Ouiedo. Department of Mathematics. E. U.I. T.I. C./ Manuel Llaneza 75,Gijbn, 33208, Asturias, Spain. E-mail: [email protected]. es

1. Introduction

This work is embedded in the line of research entitled “Optimization of hydrothermal systems”. In a previous paper [l]we considered the possibility of substituting a problem with m thermal plants and n hydroplants (HnT,) by an equivalent problem (H,-Tl) with a single thermal power station: the equivalent thermal plant. In said paper we calculated the equivalent minimizer in the case where the cost functions are second-order polynomials. We proved that the equivalent minimizer is a second-order polynomial with piece-wise constant coefficients; moreover, it belongs to the class C1. In this paper we shall present two fundamental contributions: first, new theoretical results relative to the equivalent thermal plant and, second, an algorithm for the approximate calculus for a general model. We assume throughout the paper the following definitions. Let Fi : Di 5 R R (i = 1,.. . ,m) be the cost functions of the thermal power stations. We assume that

-

V< E

D

= D1

+ ... + Dm G R,3(<1,.. . , < m )E

n m

Di

i=l

the unique minimum of

-

m

m

i= 1

i=l

CFi(zi)with the condition Czi = <.

Definition 1.1. Let us call the i-th distribution function, the function Qi

: D1

+ . + D, 1 .

Di defined by

Qi(t)= & , V i = 1,.. . , m 83

84

-

Definition 1.2. We will denote as the equivalent minimizer of {Fi}?, the function Q : D1+ . . . Dm R defined by

+

m

i=l

with the constraint

m C x i i=l

= J.

2. New Theoretical Developments

In this paper we continue the theoretical studies of the equivalent thermal plant. First we prove, under certain assumptions, the existence and uniqueness of the equivalent minimizer Q. Theorem 2.1. Let {Fi}El c C1[O,0o) be a set o f f u n c t i o n s such that F,! i s strictly increasing (i = l,...,m), with Fl(0)I F[+l(0), and let the m

function F {(XI,.

. . ,x,)

:

[0,0

0 ) ~4

R be F ( z 1 , . . . ,

~ m := )

:=

a= 1

m

E Rm

CFi(zi). Let C a

I xi 2 0 A X X =~a}. i= 1

{Qi}zl

Then, there exists a unique set such that: (1) ( Q l ( u ) ,. . . ,*,(a)) is the minimum of F over C,, 'da 2 0. (2) I t holds that (Q1(u),..

. ,Q m ( a ) )E

6, *

m

a > ( C Fl-l a=

1

0

FA)(O)

-1

i=l

being

(3) ( ~ l ( a .). ,. , Q ~ ( U ) )$

6, * f o r certain i E {I, ...,m - 1)

Qi(u) = Qi+l(U) = ... = Qm(u)= 0 In the previous theorem we also obtain the distribution functions Qk. Now we define the equivalent thermal plant piece-wisely, taking into account the restriction of power positivity. Theorem 2.2. Let { F i } E l , F, and C, be defined as in Theorem 2.1. Then there exists { 6 k } F s 1 c (with Sm+l = 00) and {Qk};ZZ=l c

85

C[O,oo) such that f o r every a > 0, the minimum of F over C, attains at ( Q l ( a ) , . .. ,Qm(a)),being

Also, we shall prove that, for a general model, the equivalent thermal plant belongs to the class C1.

Theorem 2.3. Let {Fi}En=, c C1[O,oo) be a set of functions defined as in Theorem 2.1. T h e n the function m

belongs t o the class C' and Q'(0) = F i ( 0 ) . To conclude this section, we analyze the situation that arises when the thermal plants are constrained to restrictions of the type m

C,

:= { ( X I , .

. . , xm>E I W ~ I P5 xi , ~5~P,,,

A

Exi = a> i= 1

3. An Algorithm of Approximation

We have developed a new algorithm for the approximate calculus of the thermal equivalent of m thermal power plants whose cost functional is general (non-quadratic). The outline is the following: i) We linearly approximate the derivative of the cost function of each thermal plant, F:(x), i = 1,...,m in the power generation interval of each plant. This approximation may be done as finely as one wishes by simply increasing the number of splines in said interval. The integration of these functions leads us to the piece-wise defined functions Q,(x), i = 1,...,m that approximate the cost function of each thermal plant considered

-

+ P z k X + r a k z 2 if 6& + &x + r z l ~if2

5 X < &+I; x2

k

= 1,..., 1 - 1

ha1

86

ii) We next demonstrate that each function $ i ( x ) can be considered as the minimizing equivalent of 1 fictitious thermal plants, whose cost functions, denoted by { Fil(x), F i 2 (x), . . . ,F i l (x)} , are second-order polynomials

&(x) = (Yik

+ &x + ^ / i k x 2 ; k = 1, ..., 1

The aforementioned coefficients, deduced from those obtained in [l],are given by

-

oak

= 27ikbik

+Pik

-

where ( b i k , b i k + l ) is the domain of Q i ( x ) . iii) Finally, we construct the equivalent minimizer of all the functions obtained

{ Fij}i= 1,...,m j=l,...,1

We finally show, using an example, that the developed algorithm offers very good approximate results in comparison with prior methods, such as for instance [2]. References 1. L. B a y h , J. M. Grau and P. SuBrez, A new formulation of the equivalent thermal in optimization of hydrothermal systems, Math. Probl. Eng. (2002). 2. L. Bay&, J . M. Grau and P. SuBrez, A New Algorithm for the Optimization of a Simple Hydrothermal Problem, Proceedings CMMSE 2002, Vol. I pp. 6170(2002).

UNIQUE VIRTUES OF THE PADE APPROXIMANT FOR HIGH-RESOLUTION SIGNAL PROCESSING

DZEVAD BELKIC Department of Medical Radiation Physics, Karolinska Institute, P. 0. Box 260, S-171 76 Stockholm, Sweden

This lecture reviews quantum-mechanical signal processing based upon the Pad6 approximant (PA). We presently link the PA and the Lanczos algorithm to design the Pad6-Lanczos approximant (PLA). The PLA is operationalized with the recursive algorithm called the fast Pad6 transform (FPT) for both parametric and nonparametric estimations of spectra. The FPT for any given power series is defined by the unique quotient of two polynomials. This processor provides a meaningful result even when the original expansion diverges. It can significantly accelerate slowly converging sequences/series. As opposed to a single polynomial, e.g. the fast Fourier transform (FFT), the FPT can analytically continue general functions outside their definition domains. Moreover, we show that the FPT is an efficient solver of generalized eigen-problems e.g. the quantummechanical evolution/relaxation matrix U comprised of auto-correlation functions. These generic functions can be either computed theoretically or measured experimentally. Such a concept, put forward as a computational tool, surpasses its initial purpose. Indeed auto-correlation functions represent a veritable alternative formulation of quantum mechanics. This is not just because all the major observables e.g. complete energy spectra, local density of states, quanta1 rate constants, etc, are expressible through the auto-correlation functions. It is also because these and other observables could be given completely in terms of some appropriate, relatively small informational parts that can be singled out and analyzed separately from the unwantedlredundant remainder of the full data set of auto-correlation functions. The needed dimensionality reduction of original large problems treated by the FPT can be achieved by e.g. windowing using the bandlimited decimation. Alternatively, as abundantly done in this work, the Lanczos tridiagonalization can be employed yielding sparse Jacobi matri87

88

ces in terms of the Lanczos coupling parameters {an,&}that have their very important physical interpretations. The FPT is naturally ingrained in the Schrodinger picture of quantum mechanics and in the total timeindependent Green function for the studied system. This yields a versatile framework for a unified treatment of spectroscopy and collision within signal processing and quantum mechanics. In the quantum-mechanical method of nearest neighbors or tight bindings, we use experimentally measured time signals as the only input data to derive the exact analytical expressions for the FPT, the Lanczos polynomials {Pn(w),Q,(w)} , the coupling parameters {an, Pn} , the Lanczos state vectors qbn , the total wave function T(w) at any frequency w and the ‘Hamiltonian’ operator 6. An extension of the FPT to multi-dimensional analysis is also given for usage in very diverse fields. A joint potential power of all parametric methods is their ability to extract the peak parameters from a given time signal as the result of data processing. By contrast, the Fourier-based spectral analysis must resort to post-processing fits to perform quantification of resonances in a studied spectrum. However, a common weakness of all parametric estimators is the inevitable production of spurious or extraneous resonances. Spurious peaks are due to the noise corruption of the time signal and to the socalled overdetermination problem. If the signal of the length N happens to have less than N / 2 peaks, the problem becomes algebraically overdetermined, since there are more equations than unknowns. This unavoidably leads to singular values associated with false peaks that represent spurious resonances. In fact, the adequacy and utility of all the existing parametric methods ultimately depend upon their ability to unequivocally identify and regularize spurious resonances. In quantum chemistry within Nuclear Magnetic Resonance (NMR), more than two decades ago, the method called Linear Predictor (LP) showed an initial success for parametric processing of experimentally measured time signals. Long computer processing times and unavailability of automated software of the LP were largely tolerated, but the critical lack of an objective procedure to adequately solve the problem with spurious poles was the main reason for the noticeable absence of this method among modern parametric estimators. We address this key stability issue within the Pad6 approximant by modeling a complex spectrum through a function determined uniquely by a rational polynomial i.e. a polynomial quotient for the given power series of the resolvent or the Green function with time signal points as the expansion coefficients. Extraneous roots are present in both the numerator

89

and denominator polynomial of the PA. Most harmful are the denominator spurious roots of the PA as they lead to unphysical spikes in the Pad6 magnitude or power spectrum. At first glance, the numerator spurious roots of the PA might seem innocuous, but they must also be viewed as unwelcome since they fill out the valleys with unphysical anti-resonances and this destroys the phase minimum, as well as the uniqueness feature of the PA. We examine this important problem within the inherently unstable PA by using the so-called constrained root reflection which is an analytical procedure for regularizing spurious roots. First we unequivocally separate the genuine from spurious resonances in the Pad6 power spectrum which is itself the PadQChebyshev approximant (PCA). Second, the unstable PCA containing diverging and converging exponentials is properly stabilized. This is done by a special root reflection which reverses the sign of diverging exponentials so that they are relocated on the side of genuine resonances. Such a procedure is accomplished under the constraint that the parameter and the shape spectra of the PA are the same. The shape spectrum is the Pad6 polynomial quotient evaluated at selected frequencies. The pai rameter spectrum is the sum of partial fractions built from the found peak parameters (position, width, height and phase). The ensuing constrained root reflection has its strong physical significance in full preservation of the total energy of the signal. The resulting method is called the PadQSchur approximant (PSA) which, as a stable estimator, possesses only converging exponentials. The PSA describes the Pad6 power spectrum as the unique ratio of two Schur polynomials whose computed roots are all adequately regularized and, as such, located on the side of physical resonances. In this way, rather than attempting to eliminate or reduce the noise content from the measured data, as repeatedly done previously in the literature on signal processing but with the risk of losing weak genuine spectral features, the PSA processes noise as it comes together with the physical signal. At the CMMSE 2003 Conference we shall present a number of illustrations establishing the unique virtues of the Pad6 approximant by assessing its utility and versatility for signal as well as image processing with a special emphasis on accuracy, stability, efficiency, robustness and simplicity in user-friendly automatized applications across interdisciplinary research.

A FEEDBACK LINEARIZATION TECHNIQUE BY USING NEURAL NETWORKS: APPLICATION TO BIOPROCESS CONTROL Y.S. BOUTALIS' AND 0.1.KOSMIDOU Automatic Control Systems Laboratory Department of Electrical & Computer Engineering Democritus University of Thrace GR-67100 XANTHI. Greece

Most physical systems operations are nonlinear in nature and hence they should be described by means of nonlinear mathematical models. Since nonlinear models are not convenient for control purposes, due to both theoretical and computational reasons, they are often linearized by using appropriate exact or approximate linearization techniques [ 5 ] , [4]. Among them, feedback linearization [ 5 ] is the most theoretically rigorous method; it consists in finding a feedback control law and a state variable transformation (diffeomorphism) such that the closed loop system model becomes linear, in the new coordinate variables. However, feedback linearization requires some strong constraints to be satisfied by the original nonlinear system, and thus its applicability is quite restricted. If, in addition, the original system is characterized by uncertain parameters, external disturbances and unmeasured state variables, as is the case of bioprocess control systems, the linearization problem becomes particularly complex and almost inextricable. In the present paper a method for linearizing a nonlinear system is proposed by using neural networks. More precisely, a nonlinear n-th order system is considered, in state space description; it is assumed that the system is characterized by uncertain time-varying parameters and that some of the state variables can not be accurately measured. The feedback linearizing controller applied to the system, yields a linear model in state space canonical form, of order equal to the relative degree r of the original nonlinear system. In this model, the r-th state equation is of the form

Xr =f

( x ) + g(x)u + d

x : the state vector u : the control input d : external disturbance

'

Corresponding author. Tel. +30 25410 79504, FAX +30 25410 26473,,E-muil:

vhoirr (rj;w .duth. P r

90

91 which is expressed in terms of the controller components, and, in some cases, in terms of the effect of the system’s zero dynamics. Therefore, it implies an additional first order nonlinear differential equation depending on the system’s non-linearity and uncertainty. In the proposed method, this equation is approximated by an artificial neural network (ANN). Both cases of f(x) and/or g(x) being uncertain and not completely known are considered. In all cases NN approximators are used to emulate the performance of the unknown non-linear functions. A problem often arising in control problems using NNs with conventional training algorithms (e.g Backpropagation) is that these algorithms do not guarantee that the performance of NNs will not deteriorate the stability of the whole process. This is overcome by describing the control objective as a tracking stability problem and the weights of the NNs are obtained on-line so that the tracking stability is ensured. This way, we ensure that the error tracking system is not going to become unstable due to the performance of the NNs. The above technique is applied on fermentation processes, such as the bioprocess of the production of Saccharomyces Cerevisiae (baker’s yeast), or the bioprocess related to waste water treatment. It is well known that fermentation processes are characterized by the growth of bacteria whose behavior is highly nonlinear and time varying. Moreover, many of the process parameters are uncertain or naturally varying during the process operation. In addition, external disturbances are often present at the system input. Under these conditions it is very important to ensure the process “stability” as well as a desired level of productivity [2]. Appropriate state space non-linear models for these processes have been proposed in [7], [6]. These models are linearized by the proposed method. Simulation results are given to show the ability of multilayer ANNs to approximate the nonlinear behavior and to act as reliable emulators of the nonlinear terms of equation (1).

REFERENCES 1. G. Bastin and G. Dochain, “On line Estimation and Adaptiva Control of Bioreactors”, Elsevier, 1990. 2. G. Bastin and J.F. Van Impre, “Nonlinear and Adaptive Control in Biotechnology: A Tutorial”, European Journal of Control, 1, 1995, pp. 37-53. 3. Y.S. Boutalis and 0.1. Kosmidou, “Design of a Neural Network Observer to Control a Feed-Batch Bioprocess”, European Control Conference ECC 99, Karlsruhe, Germany, 1999.

92 4. G.O. Guardabassi and S.M. Savaresi, “Approximate Linearization via Feedback - an Overview”, Automatica, 37, pp. 1-15,2001. 5. A.Isidori, “Nonlinear Control Systems”, Springer-Verlag, 1995. 6. M. Pengov, “Applications des observateurs non lintaires i la commande des bioproctdts”, Thkse de Docteur en Sciences, Universitk de Metz, France, 1998. 7. A. Rajab, J.-M. Engasser, P. Germain and A. Miclo, “A Physiological Model of Yeast Aerobic Growth”, 3rdEuropean Congress on Biotechnology, Miinchen, Germany, 1984. 8. D.E. Rumelhart, G.E. Hinton and R.J. Williams, “Learning Internal Representations by Error Propagation”, in Parallel Distributed Processing, D.E. Rumelhart and J.L. McLelland, Eds., Cambridge, MA: MIT Press, 1986.

SOME NUMERICAL METHODS FOR STIFF PROBLEMS

J. C. BUTCHER In the modelling of many physical systems, stiff differential equations need t o be solved numerically. Because stiff problems require implicit methods, implementation costs are an important consideration in the assessment of contending algorithms. We will consider a number of alternatives to the popular backward difference methods. These standard algorithms are incapable of being both A-stable and of order greater than 2 and we therefore focus on multistage methods with special structures t o guarantee low implementation costs. High order RungeKutta methods suffer the natural disadvantage of high-dimensionality in the algebraic systems required for stage evaluation. This is partly overcome by the use of singlyimplicit methods and we will consider how far this approach can be taken. Further developments involve general linear methods and the key t o finding good methods within this large family seems to be t o restrict attention to methods with inherent Rung-Kutta stability.

1. Introduction

Stiff differential equation systems arise frequently in the modelling of physical and engineering problems. They are characterised by the existence of rapidly decaying components in the solution; these rapidly decaying components cannot be faithfully approximated by traditional explicit methods unless abnormal restrictions are placed on the stepsize. In contrast to this unsatisfactory situation, implicit methods sometimes possess a property known as A-stability which guarantees stable behaviour for many stiff problems. The simplest examples of an (unsatisfactory) explicit method and of a (satisfactory) implicit method are, respectively, the Euler method and the implicit Euler method. The most straightforward motivation for these is to write the change of solution over a single step as a simple approximation to the integral of the derivative of the solution. Consider a standard initial-value problem

X ( t 0 ) = xo.

X’ = F ( t ,X ) ,

If n - 1 steps, each of size h, have been completed to give an approximation

x,-1

M

X(t,-,)

= X(to 93

+ ( n - 1)h).

(1)

94

the next approximation can be found from the formula

X ( t 0 + nh) = X ( t o + ( n- 1)h) +

tn

Ll

F ( t ,X ( t ) ) d t .

The two simplest ways of approximating the right-hand side are to replace the first term by its approximation given by (l),and the integral term by the width of the interval h, multiplied by the integrand computed at either the start of the step or the end of the step. This gives the two numerical methods

and

X" = X " _ l + hF(t",X").

(3)

These are known as the Euler method and the implicit Euler method respectively. To see why (2) is unsatisfactory when applied to a stiff problem, consider the linear differential equation system

X' = M X ,

X ( 0 ) = xo,

where M is supposed to have moderately sized eigenvalues (that is, eigenvalues close to zero) and to have, in addition, further eigenvalues with very negative real parts. In practice we will want to solve non-linear problems, and it will not be possible to de-couple the problem into sub-problems in which the two eigenvalue scales can be treated separately, but for the purpose of analysis, we will take as a prototypical special case a matrix M of the form

where q is negative with large magnitude. The exact solution is

[

X ( t ) = ex:(t)

ex:qt)

3 xo.

Because of the rapidly decaying nature of the solution to the second component, a numerical scheme should have its stepsize selected to model the behaviour of the first component in a faithful way, in accordance with whatever accuracy is required. In the case of the Euler method (2), the numerical approximation is

95

To approximate the first component, a stepsize h M 0.1 may be a suitable choice. However, it is the second component that will cause trouble. Rather than decaying rapidly, as for the exact solution, this component of the approximation will have a magnitude which actually increases, unless lhql 5 2. This is usually an unreasonable requirement. On the other hand, for the implicit form of the Euler method given by (3), the numerical approximation becomes

[

xn=

(1 - h)-n 0 0 (l-hq)-n]Xo

and the second component decays in magnitude for any negative value of hQ. Considerations of the behaviour of this simple type of linear problem motivates the following definition. Definition 1. A numerical method is A-stable if the sequence of numerical approximations produced by the method, when applied to the differential equation X‘ = q X , is bounded whenever hq is a (possibly complex) number with negative real part. The search for good methods for solving stiff problems is typically centred on the consideration of numerical schemes which satisfy this definition and have a reasonably high order of accuracy but, at the same time, do not impose severe computational demands. It is known that linear multistep methods cannot be A-stable and have order greater than 2. Hence we will look instead for suitable Runge-Kutta methods and at other multistage methods within the large general linear family. In Section 2 we will discuss implicit RungeKutta methods as possible contenders and in Section 3 we introduce a promising new class of general linear methods. 2. A-stable implicit Runge-Kutta

methods

RungeKutta methods are usually represented by a coefficient tableau such as

96

In taking a step from t,-l to t, we need to compute approximate solutions at t,-l+ h q , where c1 = - $ and c2 = 3 This explains the numbers in the first column of the tableau. The approximations at these "off-step" points are given by

+ 9.

4

X(t,-i

+ hci) = x(t,-l) + hmX'(t,-l + hcl) + hailtX'(t,-l + hcz),

where

A=

a22]

[& a a 1

all a12 =

and X',at the points where it is required, are found from the differential equation being solved. This explains more of the numbers in the tableau for this method. The final row in the tableau is related to the formula for the solution at the end of the step, which is given by the approximation

X(t,)

x

X ( L - 1 ) + hblX'(t,-1

+ hci) + hb2X'(tn-i + hcz),

with bl = b2 = 51. This particular implicit RungeKutta method has two stages because it uses approximations at this many points in order to complete the step. The cost of implementing the method goes up rapidly with s, the number of stages. The cost is also related to the structure of A. We present two further methods for comparison. These are respectively

and

It is clear that the method with tableau (5) has definite advantages over (4) because the two stages can be evaluated in sequence, rather as components of a larger coupled system. On the other hand, (4) has order of accuracy 4, rather than just 2 in the case of (5). The coefficient matrix A for the fully coupled method given by (6) has the same eigenvalues as for the decoupled method (5). By carrying out the computation in the most efficient way, this method can be implemented with a similar cost to (5).

97

Moreover ( 6 ) has computational advantages related to the fact that each of the stages can be computed with a similar accuracy as for the final output result. A-stable methods such as (6) exist for quite high orders but they suffer their own disadvantages for increasingly high orders. However, some of these difficulties can be partly overcome using a generalization known as “effective order”.

3. A-stable general linear methods w i t h the IRKS property General linear methods constitute an enormous family of computational schemes. Not only do they have a multiplicity of stages but they also have a multiplicity of values passed from one step to the next. Instead of representing the method in terms of two principal arrays, A and bT, as in a RungeKutta method, we can now use four matrices which we denote by A , U,B and V. It is convenient to arrange these as a partitioned ( s T - ) x (s T - ) matrix, where s is the number of stages and T is the number of values computed in a step and passed on to the next step. Thus we use a tableau of the form

+

+

[t ]; to represent one of these methods. In the interests of efficient computation, the s x s submatrix A should be chosen to be lower triangular; furthermore we would like the diagonal elements to be equal. In addition to these requirements, we would like the method to be A-stable and of high order. High order alone is often not enough for stiff problems; the stages also need to be highly accurate as approximations to the corresponding offstep values. It has been discovered recently that all of these conditions can be met by imposing a restriction known as “inherent Runge-Kutta stability”. This guarantees, amongst other benefits, that the stability of the method can be looked at in just the same way as for Runge-Kutta methods. It is possible to derive A-stable methods of quite high orders within this family. What is not known is how to select from the wide range of methods that exist with these properties, a particular choice in the most propitious way. It is also not known how to overcome some of the technical difficulties and complications that are thrown up in implementing these new methods. In spite of these complications, reasonable solutions are already known to many of these questions. But improvements are undoubtedly possible and a rich set of opportunities exists for further investigation.

PERIOD TWO TRICHOTOMY ON a Y X N - l + 6XN-2 XN+1

=

+

XN

+

XN-2

E. CAMOUZIS Department of Mathematics, University of the Aegean, Karlowasi, samos, Greece

R. DEVAULT Northwestern State University, Natchitoches, LA 71497, USA G. PAPASCHINOPOULOS Democritus University of Thrace, Department of Electrical and Computer Engineering, GR-67100 Xanthi, Greece

Our aim in this paper is to investigate the boundedness, global stability and periodic character of solutions of the difference equation Xn+1 =

QI

+ yxn-1+ bxn-2 , n = 0,1, ..., B x n + Dxn-2

where the parameters a,y, 6,B , D and the initial conditions are nonegative real numbers and

Bxn

+ Dx,-2

for all n.

98

>0

LEAST SQUARES FITS OF POTENTIALS BY USING ENERGY AND GRADIENT DATA : VIBRATIONAL ANHARMONIC SPECTRA FOR H2CO FROM DFT CALCULATIONS

P. CARBONNIERE,D. BEGUE, A. DARGELOS AND C. POUCHAN Laboratoire de Chimie Structurale. UMR 5624 I.F.R. Rue Jules F e r v , 64000 PAU, FRANCE E-mail: [email protected]

We present a least-squares fitting procedure to obtain a quartic force field by using energy and gradient data arising from B3LYP/cc-pVTZ calculations on a << simplex-sum )) of Box and Behnken grid of points. We illustrate and we test for H2C0 the quartic force field and the resulting vibrational anharmonic spectra performed from 44 simplex-sum configurations and we compare our results to those obtained by using the classical 168 energy calculations.

1. Introduction The construction of an accurate complete quartic force field is an important stake in the theoretical studies of vibrational anharmonic spectra. Generally, the conventional approach consists to accumulate a great number of energy data resulting from ab initio calculations carried out in a grid of points representing the geometrical variations and to deduce from it the analytical potential h c t i o n by using a standard least squares method [l]. When increase the size of the molecule the number of data point required in this latter approach increases dramatically limiting his use to four or five atoms systems. This limitation has recently stimulated new development to intend obtaining accurate quartic force field for larger molecules. Among these methods two approaches using energy, gradient and Hessian data have been done recently in the literature [2,3] including an extended least squares procedure consisting jointly in fitting all the data obtained in the grid. In this work we present the procedure implemented in our code REGRESS EGH [4] for determining the analytical form of the potential with a reduced number of points suitably chosen in the grid following a simplex-sum design known for its efficiency and accuracy. Illustration and tests presented here concern the formaldehyde molecule.

99

100

2.

Method

The extended least squares method taking into account the first and second derivatives presented elsewhere [5,6] leads to an extended merit function defined by :

in which

V(s,,s2,..., snV) correspond to the model of potential function which can be

written as a Taylor expansion in terms of displacement coordinates :

where (anv/qasi) are the coefficients to be determined. Vi (sI,s2,..., snv) and vib (sl ,s2,..., snV) are respectively the first and second derivatives of the model with respect to S, and sb displacement coordinates, Nv the number of vibrations.

(g)m( and

Em

'

a2E

are respectively the value of energy, gradient and

Hessian terms analytically obtained by a b initio calculations in each point of the grid.

3. Example of (E-G) method :application and tests for H2C0 In this case the expression of the merit h c t i o n is truncated as R = R, + R, . We have shown [6] that the simplex-sum grid of Box and Behnken [7] truncated to the third sum seems well suited to the resolution of this problem by fulfilling the criteria of efficiency and accuracy. This grid leads to the following number of computations reported in Table 1 with regard to the number of variables si for the (E-G) method.

101 Table 1. Number of calculations with the (E-G)method with a simplex sum design truncated to the third sum. Comparison with an usual (E) linear regression procedure. (results obtained without symmetry consideration). Nterm Npoint 35 70 6 210 420 9 715 1430 12 1820 3640 15 3876 7752 18 7315 14630 21 12650 25300 24 20475 40950 -

(E) Redundancy 2 2 2 2 2 2 2 2

Simplex-sum design (E-G) Npoint Redundancy Gain 15 1.7 4.7 64 2.1 6.6 176 2.5 8.1 378 2.7 9.6 697 2.9 11.1 1160 3.0 12.6 1794 3.1 14.1 2626 3.2 15.6

The use of the (E-G) method results in significant gains that increase with the size of the system studied, since the number of calculations is divided by a factor of about 5 for a triatomic molecule and close to 16 for a molecule containing 10 atoms. Nevertheless, the computational cost of the analytical gradient is added to that for energy at each calculated point of the potential grid. Consequently, the gain in term of time appears less important. The computational costs of the H2C,0 series using the B3LYP/6-31 lG* method estimated by the two ways show that in spite of time limitations due to calculation of the gradient, the gain remains satisfying since it rises by a factor of 3 to 13 when n increases from 1 to 7 illustrating the degree of effectiveness of the (E-G) method. For HZCO a comparison of the quartic fit based on 168 energy data with those based on 44 (energy + gradient) data shows a higher accuracy for the former system. The RMS error are respectively 0.05 cm-' and 0.16 cm-' for the determination of the 84 non null parameters expressed with respect to dimensionless normal coordmates. It is worth noticing however that the mean dispersion between the cubic and the quartic force constants obtained from the two ways are found to be 0.1 and 0.9 cm-' respectively. We have reported in table 2 the wavenumbers calculated for H2CO fiom the two quartic force field using a perturbative approach. The agreement is remarkably good for the findamental bands (mean difference = 0.8 cm-') and for single combinations (mean difference = 1.5 cm-I). This mean difference don't exceed 3 cm'' for all low-lying overtones. To illustrate this good agreement we show in figure 1 the difference obtained for all bands calculated from the two ways until

102 12 000 cm"1. It should be noted that the mean difference is only 0.85 cm"1 in the medium IR region with a maximum dispersion of 4 cm"1. This mean difference increases reasonably in the near IR region justifying the effectiveness of the (EG) method to calculate an accurate quartic force field. Table 2. Wavenumbers for H2CO calculated by 2nd order Perturbation Theory from B3LYP/cc-pVTZ quartic force field. Comparison between (E-G) method and (E) method.

(E) method (E-G) method 1197 1197 V] 1256 1256 V2 1521 1521 V3 1798 1798 V4 2734 2738 V5 2772 2773 V6 mean difference for V;: 0.8 cm" label

2v, 2v2 2v3 2v4 2v5 2v6

2411 2514

2411 2514

3044 3044 3575 3575 5412 5399 5469 5468 mean difference for 2vj : 2.5 cm"1 3641 3v, 3640 3774 3v2 3776 4571 3v3 4570 5333 3v4 5331 8003 3v5 7997 8093 3v6 8087 mean difference for 3vj: 2.8 cm"1

(E) method (E-G) method 2445 2445 V]+V 2 2709 2709 V]+V 3 2775 2774 V 2 +V 3 2988 2987 Vi+V 4 3047 3047 V 2 +V 4 3309 3309 V 3 +V 4 Vl+Vs 3926 3929 3955 3956 Vi+V 6 3985 3988 V 2 +V 5 4022 4021 V 2 +V 6 4247 4251 V 3 +V 5 4284 4283 V 3 +V 6 4537 4542 V 4 +V 5 4578 4577 V 4 +V 6 5370 5376 V 5 +V 6 mean difference for v,+Vj: 1.5 cm"1 label

103

5t

---------;

I

0

8000

4000

12000

Wavenumber (em")

Figure 1. Difference between vibrational bands calculated fiom the (E-G) method and (E) method until 12000 cm-'.

References 1. G. C. Schatz in Reaction and Molecular Dynamics (Proceedings of the European School on Computational Chemistry, Perugia, Italy, July 1999), edited by A. Lagana and A. Riganelli (Springer, New York, 2000). 2. C. S. Ewig, R. Berry, U. Dinur, J. R. Hill, M. J. Hwang, H. Li, C. Liang, J. Mapple, Z. Peng, T. P. Stockfisch, T. S. Thacher, L. Yan, X. Ni, A. T. Hagler, J. Comput. Chem. 22, 1782 (2000). 3. T. Xie, J. M. Bowman, J. Chem. Phys. 117, 10487 (2002) 4. REGRESS EGH, Ph. Carbonnibre, D. Btgut, A. Dargelos, C. Pouchan, Laboratoire de Chimie Thtorique et Physico-Chimie Moltculaire, UMR CNRS 5624,2001. 5. J. R. Maple, M. J. Hwang, T. P. Stockfisch, U. Dinur, M. Waldman, C. S. Ewig, A. T. Hagler, J. Comput. Chem. 15, 162 (1994). 6. Ph. Carbondre, D. BtguB, A. Dargelos, C. Pouchan, J. Chem. Phys. (to be published). 7. G. E. P. Box, D. W. Behnken, Ann. Mat. Stat. 31,838 (1960).

DERIVING PREDICTION INTERVALS FOR NEUROFUZZY NETWORKS G. CASTELLANO, A.M. FANELLI, AM) C. MENCAR CILAB - Computational Intelligence LABoratory Department of Computer Science, University of Bari v. E. Orabona, 4 - 70126 - Bari, ITALY E-mail: {castellano, fanelli, mencar}@di. uniba.it In this paper, we describe a method to calculate prediction intervals for neurofuzzy networks used as predictive systems. The method also allows defining prediction intervals for the fuzzy rules that constitute the rule base of the neuro-fuzzy network, resulting in a more readable knowledge base. Moreover, the method does not depend on a specific architecture and can be applied to a variety of neuro-fuzzy models. An illustrative example is reported to show the validity of the proposed approach.

1. Introduction A neuro-fuzzy network is a fuzzy system that uses a learning algorithm derived from or inspired by neural network theory to determine its parameters (fuzzy sets and fuzzy rules) by processing data samples [I]. Several neuro-fuzzy networks exist in literature, and most of them acquire an input/output mapping from data in form of fuzzy rules that can be used for prediction or classification. However, in many real-world problems, data are corrupted by noise or follow complex relationships that are hardly discovered by simple models; in such cases, a model that provides a prediction interval as output rather than a single numerical value is more appreciable. Many methods have been proposed for estimating confidence intervals and prediction intervals for neural networks [2], but many of such techniques are dependent on the specific neural architecture. As a consequence, such techniques are not applicable for different models, like neuro-fuzzy networks. In this paper, we propose an approach to derive prediction intervals for neuro-fuzzy networks so as the system provides an estimate of the uncertainty associated with predicted output values. The proposed method does not depend on the specific architecture, so as to be applied on a variety of models. In addition, it can be applied on each rule of the neuro-fuzzy network, resulting in a more readable knowledge base. 104

105

2.

A neuro-fuzzy network

The neurofuzzy network used as pilot model for prediction interval calculation is the Adaptive Network-based Fuzzy Inference System (ANFIS) of order 0 [3], as depicted in Figure 1. Li

L:

L

ir

Figure I . The Olh order ANFIS network (one output only)

The rule set that is expressed by such model follows the schema:

Ri : IF xis Ai THEN y1 = a,,, ...,y,,,= aim

(1)

where x is an input variable defined over R", Ai are n-dimensional fuzzy sets, yj are output variables and aYmare constants in R . For simplicity, we assume that m = 1, that is, the system has only one output. For m > 1 the extension of the proposed method is straightforward. When an input vector is presented to the network, the system output is defined by the following inference rule:

r=l

where the rule strength pA, coincides with the membership value of input x to the fuzzy set A,, for each of the R rules that define the rule set. The ANFIS network can be trained in several ways like through backpropagation or second order methods [3]. At the end of the training process, the parameters of the network - namely the parameters of the fuzzy membership functions and the consequent constants - are adjusted so as to minimize a given loss function, usually the Mean Squared Error.

106 3. Prediction interval derivation

Here we describe the proposed method to derive prediction intervals for the neurofuzzy system outputs. Suppose that there exist a supervisor that provides, for each input x , the exact value g of an underlying unknown function g = ?(XI . We can assume that the supervisor is memory-less, that is, the returned value is independent on the previously calculated values. A neuro-fuzzy system, like the previously described model, is trained so as to provide an approximated value for the same input x . The following error function can be defined as: e(x) = G(x)- Y(x)

(3)

Suppose that a finite training set T = {(x, ,gP) E Rn+’p = 1,2,..., N } is available for training. The examples of the training set are assumed to be drawn independently from a single source, so as to be identically distributed. After training, a finite number of errors e,, e,, ...,eN are available. Such errors can be considered as independent and identically distributed random variables; hence their mean value

e

N

=+Eel

(4)

i=l

is a random variable that approaches the Normal distribution for increasing N. For a normally distributed random variable, prediction intervals can be calculated. A prediction interval [La,U,] of confidence level a represents an interval that will include the error enew of a newly drawn example xnew, with probability greater than 1- a! [4]. Formally: p (en,, E [L,,u,]) L 1- a

(5)

The last relation is equivalent to the following:

P (g E [G - u, ,5 - La1) L 1- a Relation (6) defines a statistical method to estimate the true value of the underlying function approximated by the neuro-fuzzy network, with a desirable confidence level. Formally, a prediction interval is defined by the following relations:

107

where t%,,N-llis the value of the Student distribution with N - 1 degrees of freedom corresponding to the critical value a/2, and s is the sampled standard deviation:

The width of the prediction interval is directly related on the model accuracy. As a consequence, the less accurate is the model, or the smaller is the training set cardinality, the wider is the prediction interval. In the proposed approach, the prediction intervals are derived on the basis of the approximated outputs inferred by the neuro-fuzzy network. However, for representation pursuits, prediction intervals can be calculated for each rule consequent, resulting in the following representation: I F x i s A, THENy E [a, - Ua,a, - L a ] (1 - a )

(10)

The calculation of prediction intervals for the fuzzy rules leads to a more explanatory knowledge base, since it helps users to perceive an entire range of validity of each rule, instead of a single numerical value. It should be noted, however, that such intervals are used only for rules representation, while the derivation of the prediction interval for the network output must follow relation (6). 4. 'Illustrativeexample In order to validate the proposed method, we trained a Oth order ANFIS based on 2-dimensional data. The available dataset was split into a training set of 25 examples and a test set of 26 examples. The training dataset and the inputloutput mapping defined by the trained ANFIS are depicted in Figure 2. Then,

108 prediction intervals are calculated for a = 10%,1%,0.5%,0.1% and are illustrated in Figure 3. 0.8,

Figure 2: The training dataset (circles) anf the input!output mapping defined by the neuro-fuzzy system (line)

Figure 3: Prediction intervals of the input/output mapping for four confidence levels (lo%, 1%, 0.5%, 0.1%)

For each value of the confidence level, the test set has been used to check whether the desired outputs fall within the prediction interval. The results are reported in

Table 1. As it can be seen, the number of desired outputs that fall outside the prediction interval is coherent with the confidence level. In the design phase, the choice of the appropriate value of the confidence level should be a tradeoff between precision of the mapping (narrow intervals) and good predictive estimation (large intervals).

109

Table 1: Prediction errors with different confidence levels ~~

Confidenc e level 10% 1% 0.05% 0.01%

5.

examples outside the prediction interval 2 1 0 0

(7.69%) (3.85%)

Conclusions

In this paper we have proposed a method to assign prediction intervals to neurofuzzy networks used as predictive systems. The method is quite general and can be applied to a variety of neuro-fuzzy models. Moreover, the method can be used to define fuzzy rules with interval-valued consequents, which result in more readable knowledge base. Future research is in the direction of defining prediction intervals that are functionally related to the input, so as to make interval width related to model accuracy. References 1. D. Nauck, F. Klawonn, and R. Kruse, Foundations of Neuro-Fuzzy Systems, Wiley, Chichester, (1997) 2. R. Dybowski, S. Roberts, ConJdence intervals and prediction intervals for feed-forward neural networks, In Dybowski R, Gant V. (eds.) Clinical Applications of Artificial Neural Networks. Cambridge: Cambridge University Press, pp 298-326 (2001) 3. J. S. R. Jang, ANFIS: Adaptive--network-- based fuzzy inference systems, IEEE Trans. on System, Man and Cybernetics, Vol. 23, No. 3, pp. 665-685 (1993) 4. J. Neter, W. Wasserman, and M. H. Kutner, Applied linear statistical models: Regression, analysis of variance, and experimental designs. Homewood, IL: Irwin (1985)

MARANGONI EFFECTS IN A HORIZONTAL SOLIDIFICATION PROCESS IN MICROGRAVITY *

M. M. CERIMELE, D . MANSUTTI AND F. PISTELLA Istituto per le Applicazioni del Calcolo ”M. Picone” (C.N.R.) Viale del Policlinico, 137 00161 R o m a - Italy E-mail: pistellaaiac. cnr.it

The present work was stimulated by the need to plan solidification experiments on a space platform and the general aim is to provide data from numerical simulation to be used to set properly the experimental apparatus. The material typically used in this kind of experiments is succinonitrile (SCN) for it assembles the characteristics of semi-conductor materials and metals although being transparent and allowing easy observation of the flow structures, phase front and deformations. We shall present the numerical simulation of the horizontal Bridgman solidification process where, at the open top of the parallelepipedic crucible, the shear stress induced by the surface tension gradients at the air/melt interface (Marangoni effect) is taken into account. In order to better understand the mechanism of interaction between the buoyancy and the thermocapillary forces we shall compare the numerical results obtained for SCN with those from the simulation of the solidification of silicon, a material chacterized by negligible surface tension. The mathematical model here adopted describes the flow of the liquid phase - considered as an incompressible newtonian fluid -, the heat transport phenomena within the whole sample and the evolution of the phase front. The stream-function/vorticity formulation for the flow of the liquid phase and the front-fixing treatment of the moving phase front are used. The numerical approximation is based upon a second order E N 0 scheme combined with a second order time scheme. The validation of the mathematical *This work has been developed within the project ”EnvironmentalProcesses” at Istituto per le Applicazioni del Calcolo ”M. Picone” (C.N.R.). 110

111

and numerical models versus physical direct observation was discussed in for a melting experiment and in for the present experimental setting in a full gravity environment. Here we shall compare streamlines and isothermal lines at thermodynamical equilibrium in microgravity and g-jittering (oscillatory microgravity) fields versus the case in full gravity field. As referencing experimental setup we adopt the one described in but for the top of the crucible that in our case is assumed to be open. In the horizontal Bridgman technique for artificial crystal growth the molten material is placed in a (idealized) shallow parallelepipedic crucible; the crystal grows as a device moves from one extremum to the other on the crucible, by taking the heat out and carrying along the solidification front. During this process a convective flow starts within the melt due to the horizontal temperature gradient. In the following we detail the equations governing the evolution of the material. The unsteady Navier-Stokes equations for incompressible fluids are adopted to represent the melt dynamics:

PL

av (at + (V . 0 )V)

= -Vp

+ pV2v - pL[1-

cr (TL- Tp)]g

(1)

Here v is the velocity field, p is the pressure, TL is the temperature of the melt, g is the acceleration due to gravity, p is the dynamic viscosity, a is the volumetric thermal expansion coefficient, CL is the specific heat and k L is the thermal conductivity of the melt. The spatial and temporal derivatives are meant with respect to the quadruple (z,y, z, t ) , where (z,y, z ) varies within the domain 0: c R3 occupied by the melt and t 2 t o . The above equations are built on the basis of the Boussinesq approximation: p~ is the density at the melting point and within the buoyancy term the density p is supposed to vary according to the linear law p = p ~ [ 1 ~ - ( T-LT p ) ]where , (I: and Tp are respectively the thermal expansion coefficient and the melting temperature. In the solid phase only heat diffusion occurs, which is governed by the equation

112

where Ts, ps, cs and ks are respectively the temperature, the density, the specific heat and the thermal conductivity of the solid. The above equation holds on Di,that is the spatial domain occupied by the solid (Dic R3) for t 2 to. On the phase front, that is the surface separating the domain of the melt DZ and the domain of the solid Di,the equation

holds, where 1 is the latent heat characteristic of the material, the derivatives are meant to be with respect to the outer normal to the front in Di and n is the corresponding normal unit vector. Equation (5) is the Stefan condition which expresses the balance of the energy exchanged between the two phases whose difference has to be equal to the latent heat. The above equations are completed by the following boundary conditions: a condition on the velocity field of the melt is required for the momentum equation (1);we impose

v=o

(no- slip)

at the walls of the crucible (impermeability) and also at the phase front. At the open top of the crucible (within the restriction of flat (horizontal) free-surface), we impose

au -- -aaaTL _ ay at ax v=o where x and y are the coordinates respectively along the horizontal and the vertical directions, u and v are respectively the horizontal and vertical components of the velocity field of the melt and is the temperature gradient of surface tension accounting for the thermocapillary effects.

3

113

0

the energy equations for the melt (3) and the solid (4) require conditions on the temperature fields or their normal derivative on the whole boundary. As we suppose that the material is pure, the phase-change temperature is a sharp value that we call Tp. Then we impose

TL = TH

(TH2 Tp, superheating)

at the vertical wall of the crucible in 0;;

TL = Ts = Tp

(phase - change temperature)

at the phase boundary;

Ts = Tc

(Tc < Tp,subcooling)

at the vertical wall of the crucible in 0;;

-man = Lo ,

m S =o

(adiabatic wall)

an at the horizontal crucible walls (open top case included). The initial conditions in our simulations will describe a portion of material in solid phase at temperature (Ts(z, y, z , t o ) < T p )and the remaining melt, restly, at temperature (TL(z, y, z , t o ) > T p ) .

Acknowledgments Beside the institutional funding from C.N.R., the development of this work was made possible by the finantial support of A.S.I. within the project "Simulazione numerica per lo studio della convezione naturale in processi di solidificazione", 2002-2003. References 1. M. M. Cerimele, D. Mansutti, F. Pistella Computers and Fluids 31,437 (2002). 2. M. M. Cerimele, D. Mansutti, F. Pistella Mathematics in Industry I, 197 (2002). 3. G. H. Yeoh, G. de Vahl Davis, E. Leonardi, H.C. de Groh I11 and M. A. Yao J. of Crystal Growth 173, 492 (1997).

SIMULATION OF INCOMPRESSIBLE FLOWS ACROSS MOVING BODIES USING MESHLESS FINITE DIFFRERENCING C. S. CHEW, K. S. YE0 AND C. SHU Department of Mechanical Engineering, National University of Singapore 10 Kent Ridge Crescent Singapore 119260 Republic of Singapore E-mail: engp031I @nus.edu.sg, [email protected]

Computational fluid dynamics (CFD) is increasingly becoming the tool of choice for researchers and engineers facing complex fluid flow problems. One area of CFD that has attracted significant attention in recent years is the development of meshless or mesh-free methods'. These methods have so far been used in many applications like fracture mechanics, astrophysics and fluid flow. The key advantage of meshless methods over mesh-based methods like finite volume (FV) and finite element (FE) methods is its non-requirement of pre-specified connectivity between nodes in order to derive approximation and interpolation. This leads to the ability of finite difference (FD) methods to treat irregular domains and boundaries. A direct result of this is another advantage of mesh-free methods having the adaptability for problems involving large movement or wholesale motion of boundaries or embedded bodies. In this area, it has the edge over composite or overset grid methods in being able to use only one coordinate frame. A convecting meshless scheme for boundary driven fluid flow is described. This scheme can be applied to self-propulsion problems, or, in general, flows involving motion of rigid or deformable bodies embedded within the flow. The generalized finite difference method' (GFD) was chosen this scheme. A meshless cloud of nodes is embedded around the body surface along with the structured Cartesian background nodes for the entire flow field. By incorporating a subdomain for each node and its neighbours, the use of Taylor series expansion enables the derivatives to be found. Two forms of incompressible momentum equations are used: the standard Eulerian form in primitive variables for the structured nodes and the Arbitrary LagrangianEulerian (ALE) form for the meshless cloud of nodes. GFD with moving least squares approximation is used for spatial discretization of the convecting meshless nodes. The projection method with Crank-Nicolson discretization is used to achieve second order accurate time discretization. Following prescribed body motion or fluid structure interaction, the meshless cloud acquire new locations in the next time step and the momentum equations in ALE form can then be solved.

114

115 Test cases were performed to verify code usability and method accuracy and convergence. The simulation of the decaying vortex has been a popular benchmark for the gauging of accuracy3. Basically the exact solutions of the time-dependent function values are known and both residues and absolute errors were calculated. Solving using different grid sizes proved the present method to be of second order spatial accuracy. The meshless method was then tried on the driven cavity flow at Reynolds number 1000. The results were well-compared with popular benchmark results4. In considering the effects of implementing the convecting meshless nodes across a normal flow field, the decaying vortex was again simulated, with a moving patch of nodes within the Cartesian field. The residues were well controlled and the errors were not significantly increased. Finally, a couple of external flows cases were simulated, basically on a flow past a stationary bluff body and a flapping ellipse inside an enclosed body of fluid. Future works in the pipeline include writing a parallel code for larger scale simulation and simulating oscillating cylinders and flapping of elliptic aerofoil(s). TOPICS + KEYWORDS: Computational fluid mechanics, meshless, generalized finite difference, moving boundary, Arbitrary Lagrangian-Eulerian, incompressible, Navier Stokes equations, bluff body, fractional step. PREFERENCE: Oral References T. Belytschko, Y. Krongauz, D. Organ, M. Fleming, P. Krysl, Comput. Methods Appl. Engrg. 139,3 (1996) 2. N. Perrone, R. Kaos, Computers and Structures 5,45 (1975) 3. D. Kim and H. Choi, J. Comput. Phys. 162,411 (2000) 4. U . Ghia, K. N. Ghia and C. T. Shin, J. Comput. Phys. 48,387 (1982) 1.

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V velocity conponert dflow a c m elipse

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External flow across ellipse, u velocity contour and v velocity contour.

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Vorticity plot and streamlines plot of flapping ellipse within confined space.

AUTOMATIC GENERATION OF SOFTWARE COMPONENTS FOR REAL OPTIONS MODELLING

A. CHORTARAS, Y. GUO, M. M. GHANEM, F. 0. BUNNIN Department of Computing, Imperial College London E-mail: (ac901, yg, mmg, f o b l } @doc.ic.ac.uk

We describe the design and implementation of a software system that, from high level specification documents, generates source code for the numerical valuation of real options. The documents allow the description of both the flexibility present in investment projects and the dynamics of the underlying stochastic variables, independently of any valuation methodspecific details. By applying symbolic transformations to the specifications the system generates efficient and reusable software components, which are combined with software components that implement numerical methods in order for the final real option valuation code to be generated. 1. Introduction The complexity of the interactions among the real options that real-life projects include and the use of complex stochastic models to describe the underlying uncertainty sources often make numerical techniques the only tractable way for valuating real options. By exploiting the analogy between financial and real options [4], real options may be valuated by extending the financial option pricing methodology [ 5 ] ,[3]. However, developing code for the valuation of varying project structures in combination with different stochastic models is an expensive, error prone and time-consuming process. A system automating this process can help the efficient and rapid design of the management’s investment plans by shortening development time, reducing implementation costs and offering increased flexibility in project analysis. The system that we present extends earlier work in [l]and [2] 118

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in the real option valuation domain by providing support for incomplete markets and interacting real option collections modelling, while maintaining the flexibility of handling complex stochastic models. 2. System Design

The design of our system allows it to produce real option valuation code in a systematic and transparent way, by interacting at a high level with the end-user. It deploys the technologies of automatic code generation and software components. Its kernel is implemented in Java and Mathematica while the product of the automatic code generation are software components implemented as C++ classes. The use of the software components technology facilitates the mapping of financial and mathematical concepts to software entities at a high level of abstraction and offers reusability of the generated code. The specifications of the real option valuation problems are provided to the system in the form of XML documents. The architecture of the system is illustrated in Fig. 1 and consists of three parts, each working at a different level of abstraction and modelling a different aspect of a given valuation problem. Real Option X M L SpecificationDocuments

Stochastic Model X M L Specification Documents

Real Option Components Generation

Stochastic Mode ComponentsGeneration

Ciasses Library Components Library

Components Generation

New C++ Valuation Classes

Figure 1. System architecture.

Real Option Specification The real option specification part provides a high level way of specifying individual real options and real option collections independently of any stochastic model or valuation method-specific

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details. The XML specification documents define the properties of individual real options (underlying variables, payoff, time to expiration and exercise style) and of collections of real options (ordering in time and type of interaction). Single and compound options are supported. Compound options may be contingent either on single or on sets of mutually exclusive options, thus sequences of interacting options can easily be modelled. From the specification documents two types of software components are generated: individual real option and real option collection components. The generated components are based on abstract class hierarchies that define the functions that concrete classes, corresponding to specific real o p tions, have to implement. The individual real option components abstract the structural and semantic similarities and differences among real options in an effective way that permits the development of computationally efficient valuation components. The collection components model the general project structure and are made up by individual real option components. Stochastic Model Specification The modelling of the dynamics of the options underlying variables is done at the stochastic model specification part. The underlying variables are assumed to follow continuous diffusion processes. The generated components capture all the mathematical details of the model in a systematic way that facilitates the development of the valuation components. The components are generated from XML documents defining the details of a stochastic model (drift, diffusion, dividend structure, uncertainty sources and their correlation). Given that there is no best numerical method for all valuation problems the system is designed without opting for a particular valuation method. The stochastic model components comply with this requirement by providing to the numerical components all the mathematical quantities the latter need and can be directly derived from the mathematical model. Given the different nature of the several numerical methods the stochastic model components must accommodate different needs. For example, finite difference methods require the calculation of the coefficients of a partial differential equation, while Monte Carlo methods the calculation of the risk neutral dynamics. The application of symbolic transformations (using Mathematica) on the original specifications and the use of abstract class hierarchies that model the needs of the different numerical methods is how this objective is achieved. Valuation Components The valuation components, generated by the valuation code generation part, are the final product of the system and

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those that perform the actual real option valuation. They are generated by combining a real option collection and a stochastic model component and hence they inherit the complete specifications of a given problem. They have the form of classes with a well-defined external interface so that project valuation applications can easily be developed. Their most significant part is the numerical component they are based on. The numerical components implement numerical methods suitable for the valuation of the options that can be specified in the specification parts of the system. The applicability of a numerical method depends on the characteristics of each particular problem. The numerical components form a numerical library, which is extensible so that new components may be added. The current implementation includes finite differences and lattice method components for options contingent on one or two stochastic variables. The software components generated by each part are semantically and operationally independent and reusable. This allows the generation of several valuation components by combining different real option collection components with different stochastic model components. 3. Conclusions

Our system provides a problem solving environment and applies the software synthesis technology to the real options domain. It facilitates the rapid development of transparent and theoretically sound investment project valuation programs and provides management with the opportunity to formulate projects as real option collections at a natural level and perform flexible analysis for varying model assumptions and project structures. This flexibility increases understanding of the uncertainty inherent in operations and results in faster and better-informed decision making. References 1. F.O. Bunnin, Automatic Generation of Software Components for Financial Modeling, PhD Thesis, Imperial College, London, 2001. 2. F.O. Bunnin, Y. Guo, Y. Ren, and J. Darlington, Design of High Performance Financial Modeling Environment, Parallel Computing, vol 26, 2000. 3. A. Gamba, An Extension of Least Square Monte Car10 Simulation for Multioption Problems, 6th Annual Real Options Conference, Cyprus, 2002. 4. L. Trigeorgis, Real Options: Managing Flexibility and Strategy in Resource Allocation, MIT Press, 1996. 5 . L. Trigeorgis, The Nature of Option Interactions and the Valuation of Investments with Multiple Real Options, Journal of Financial and Quantitative Analysis, vol. 28, 1993.

MODELING THE STATE AND BEHAVIOR OF AN ENZYME USING UML - AN OBJECT ORIENTED APPROACH

v. N. CHRISTOFILAKIS~' Institute of Informatics and Telecommunications National Centre of Scientific Research, "DEMOKRITOS" Ag.Paraskevi, Athens 15310, GREECE E-mail: [email protected]

CH. ALEXOPOULOS Laboratory of Peptides Chemistry Department, University of Ioannina Panepistimioupoli, Ioannina 451 10, GREECE E-mail: me00051 @cc.uoi.gr In this paper an object oriented approach is used to visualize a family of enzymes that are widespread in nature and can be found in many animals and plant species. To achieve this assay, Unified Modeling Language (UML) is utilized to describe accurately and in detail the chemical information. In this survey we intend to provide multiple dimensional access to the inherent information concerning a biochemical process, and as an example the acid phosphatase is chosen.

1. Extended Abstract

It is very common to represent real word systems using models. Especially when the system is too complicated, models help you to visualize a system as it is or as you want it to be. Complementally, modeling is important because: It allows you to specify the structure and behavior of a system Gives you a template that guides you in constructing a system Documents the decisions you have made Gives the possibility of better and faster comprehension of significances UML [l], which was adopted by the Object Management Group OMG [2] on November 1997 as a standard formal language, is used for visualizing,

. ..

Electronics - Telecommunicationsand Applications Laboratory, Physics Department, University of Ioannina, Panepistimioupoli, Ioannina 45 110, Greece. * Corresponding Author

122

123 specifying, constructing and documenting the artifacts of systems the range of which is very broad. By applying object-oriented approach through the diagrams of UML it is possible to obtain reusable, change-tolerant and stable models of any system. UML encompasses nine types of diagrams: use-case, class, object, state, sequence, collaboration, activity, deployment and component. These could be categorized according to their features in structural, behavioral and implementation diagrams. (Figure 1)". Depending on the system that is modeled, it is possible to use one or more from the diagrams that are illustrated in Figure 1 to represent different aspects of the system. In this approach it is shown how the functionality is designed inside a set of chemical data, originating from an enzyme catalyzed system. The enzyme that used is Acid Phosphatase (AP) [3]. AP is a general term, associated with nonspecific phosphomonoesterases with optimum activity in the pH range of 4-6.

C

BEHAVIORAL

SEQUENCE

COLLABORATION

Figure 1: The classification of nine diagrams of UML.

You can obtain figures 1,2, 3 from Web server at URL: httD://www.telecomlab.ar/extendedabstractACCMSEI a

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Figure 2: Phosphate ester hydrolysis catalyzed by acid phosphatase (E.C.3.1.3.2) on pnitrophenylphosphate(pNPP) as substrate.

AP belongs to the family of hydrolases (E.C.3.), by acting on ester bonds (E.C.3.1.) and more precisely, it constitutes a phosphoric monoester hydrolase (E.C.3.1.3). We are focused on acid phosphatase, characterized by E.C.3.1.3.2, which could also be appeared with other name, such as: acid phosphomonoesterase or phosphomonoesterase or glycerophosphatase. By using the enzyme databases available in the World Wide Web, (ExPaSy, KEGG, WIT, BRENDA) we noticed that there are 24 PDB entries in enzyme class E.C.3.1.3.2. It is possible to use artificial phosphate esters since the enzyme is rather non-specific and will catalyze phosphate ester hydrolysis on many different substrates. Phosphate ester of p-nitrophenol is a good substrate to use, since the product formed after ester hydrolysis, p-ninitrophenol, can be easily detected and measured. Utilizing the Rational Unified Process framework [4] of Rational Suite Development Studio [5] a detailed representation of the enzyme catalyzed system is given through structural and behavioral diagrams of UML. In Figure 3 the class diagram [footnote], at a higher level of perspective, of the enzyme catalyzed system is illustrated. The classes of enzymes, phosphate esters, chemical reactions, products, their instances and the relationships among them are shown in this structural diagram. Furthermore, taking advantage of the interoperability of UML with extensible Markup Language (XML), [6] an efficient, general, powerful and extensible mechanism for handling both the “capture” and the publication of chemical information is offered [7]. To accomplish this task the XML document is developed by modeling it as a class diagram in Rational Rose XML Document Type Definition (DTD) [8].

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14/ EC.3.1 .3.2

Oh4.W. +concentration 6pecificity

lpNpp4

I

I

I

I

aerperature

\ xoncentration mame

I Chemical RcacCons

G-

-

Color Reacnon QPti

Xolor aAbsorbance

Figure 3: Class diagram of the enzyme catalyzed system

It is the first time to model through UML an enzyme catalyzed system. This representation offers the possibility to create a new data-bank in the Web, accessible to everybody and an innovative, powerful tool in chemical education - computational Chemistry. References 1. OMG UniJed Modeling Language Specification, Version 1.4 2. Object Management Group OMG, http://www.omg.org 3. H. Bull, PG. Murray, D. Thomas, AM. Fraser and PN. Nelson, Mol. Pathol., 2002,55 (2), 65-72. 4. P. Kruchten , The Rational Unified Process An Introduction, AddisonWesley, 2"dEdition. 5 . Rational Software, http://www.rational.com 6. Extensible Markup Language (XML), htt~://www.w3.org/XML/ 7. P. Murray-Rust, H. S . Rzepa, Markup Languages-How to Structure Chemistry-Related Documents, Chem. International, Vo1.24 No.4, (2002) 8. Rational Solutions, Online Documentation, v2002.05.00

ATOMISTIC MONTE CARL0 SIMULATION STUDIES OF POLYMER MELTS GRAFTED ON SOLID SUBSTRATES K. DAOULAS AND V.G. MAVRANTZAS Institute of Chemical Engineering and High-Temperature Chemical Processes (FOR TH-ICE/HT) Stadiou Street, Platani,GR 26500,Patras, GREECE E-mail: daoulas@physics. upatraxgr. [email protected]

We discuss the results of a recent atomistic Monte Carlo simulation study of polyethylene (PE) melts grafted by one of their ends on a solid substrate. The simulations have been executed with a rather detailed forcefield which describes interactions at the level of individual atoms. The shortest length scale in the simulation is the Carbon-Carbon bond length I (=1.54A).Results are presented for the main thermodynamic and conformational properties of these grafted layers in the vicinity of two types of surfaces: a non-interacting hard wall and a graphite basal plane.

1. Introduction

Polymers adsorbed on solid surfaces are used extensively in many industries to manufacture surfaces with controlled friction and wear characteristics, manipulate wettability by aqueous and organic liquids, stabilize suspensions against flocculation and improve the compatibility of immiscible interfaces. In the latest years, particular emphasis is given to polymers adsorbed on solid surfaces by one of their two ends. The layers formed in this case are highly ordered and this property has been exploited in the fabrication of nanoscale structures with immediate applications in areas such as molecular electronics, corrosion inhibition, molecular recognition and design of biosensors. In an effort to understand molecular packing and orientation in these systems, we present here the first results of a recent atomistic simulation study of PE melts grafted on a solid substrate. Such an analysis has immediate applications to the class of materials known as self-assembled monolayers (SAM’S), which are formed, for example, by the adsorption of alkanethiols on a Au( 11 1) surface and by alkyl chains on a Si( 1 1 1) surface.

I

2. Molecular model The system examined is a thin film of a PE melt exposed to a solid surface (a semi-infinite graphite basal plane or a wall) at the lower face and to a vacuum or air at the upper face. The simulations are conducted in an orthorhombic box of dimensions LY,L! and L, in the directions x, y and z, respectively, which is filled with segments from several polymer chains; z is the direction perpendicular to

126

127 the interface. Some or all of these chains are permanently grafted on the substrate. In the simulations, the united-atom representation has been adopted, with each methyl and methylene group along the chain backbone regarded as a single interacting site. The molecular model is described in detail in Refs 2,3 and comprises: (a) intramolecular and intermolecular interactions described by a Lennard-Jones (LJ) potential, (b) a bending potential governing the distribution of C-C-C bond angles, (c) a torsional potential governing the distribution of torsional angles and (d) the interaction of all PE carbon atoms with the substrate. Bond lengths are considered to be frozen to their equilibrium value, 1=1.54A. Two types of substrates have been considered: (a) graphite and (b) a hard, noninteracting wall. The functional form of the interaction with the graphite is described in Refs. 4,5 and is shown schematically in Fig. 1.

-

-n 80

.

-0.85

u; -n go c

f- -0.95

0

0

Figure 1. Polymedgraphite potential plotted over a graphite lattice unit cell at a distance z = 3 0 from the first layer of graphite atoms (gray circles). Positions o f local energy minima, where adsorption is more likely to take place, are shown as void circles.

In the course of the simulations, all grafted chains are assumed to have their first carbon atom permanently anchored at a distance from the graphite surface equal to the value of the C-C bond length (i.e., equal to 1.54A). As regards the upper face of the simulation box, the strategy followed in the present work depends on the system studied: For the systems consisting exclusively of grafted chains, the upper face of the system is assumed to be exposed to vacuum. On the other hand, for the systems consisting of mixtures of grafted and free chains, the upper face of the simulation box is assumed to be exposed to air.

3. Simulation method

128 The very powerful end-bridging MC method (EBMC) was chosen to cany out the simulations in a semigrand canonical ensemble where the following parameters are held fixed: the pressure P, the temperature T, the total number of chains NL/?,the total number of mers n, and the spectrum of relative chemical potentials p* of all chain species in the melt except two, which are taken as reference species. Initial configurations at the desired grafting density with which the subsequent MC simulation is carried out were obtained after modifying the amorphous cell method of Theodorou-Suter, properly modified to take into account the presence of the solid substrate and the grafting on the solid. For systems composed of both grafted and free chains, with the upper face of the film exposed to air, the MC simulations are executed in the semigrand N m P T p* ensemble, where the dimension Lz of the simulation box in the z direction is allowed to vary, responding to the value of the pressure P applied. To prevent chains from exiting from the top face of the simulation box, through 7 the upper box surface, a soft, ramp-like potential was assumed there . In the case of a fully grafted melt, the soft, ramp-like potential was disregarded and chains were allowed to freely dangle; the simulations in this case are also executed in the NchnTPp* ensemble but with P=O atm. In all simulations, T=450 K. 4. Results

Fig. 2a,b presents typical snapshots of a system containing: (a) only grafted chains (chain length=C7x, 0=2.62nm-') and (b) a mixture of grafted and free chains C7x chains grafted on a hard surface at 0=1.31nm-*. The highly ordered structure of both systems is striking.

129

Figure 2a,b. Atomistic views of a) a Cx PE mclt dl chains of which are grafted on a hard surface at o=2.62nm2 and b) a PE mclt consisting of fee (gray) and graftcd chains @lack) at 0=1.31nm?

The strong orientational order of the grafted melts can be quantified by means of the components of the conformation tensor C. This is the tensor of the secondmoment of the chain end-to-end vector properly normalized so that at equilibrium it reduces to the unit tensor. Fig. 3 presents the time evolution of the CZzcomponent of C perpendicular to the substrate, as function of U, for the C7x melt. It is seen that with increasing U, the chains assume a conformation which is highly extended along the z direction indicative of the strong anisotropy that grafting induces to these macromolecular systems. Additional information about the conformational properties of the grafted layers can be obtained by studying the average chain conformational path. This is defined as the average height <<(i)> of backbone atom i above the grafting plane. This is shown in Fig. 4 as a function of the scaled bond coordinate s=i/N, where N is the chain length. Also shown in the Figure is the curve proposed by the analytical brush theories, 8,9?10,1I which suggest a universal dependence of <<(s)> on s.

130

a = 2.62 nm"

-a=

40

0.0

0.1

03

1.75nm'

. - .

03

a4

5

CPU time (10~s) Figure 3. Evolution of the C,, component of the chain conformation tensor C with CPU time for tethered C78 PE melts, as a function of grafting density, at o=1.75nni2 (filled (open circles) and 0=2.62nm+~ (thick solid line). circles), 0=2.18nm-~

A

1'2

n

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n A W rA

f

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' -w-

rA

s

t

u = 2.62nni2 -a-u=2.lW2 -oa = 1.75nm.' Theay (Refs. 8,9)

-.-

s=i/N Figure 4. f i e normalized mean height [(s) shown as a function of the normalized carbon atom coordinate s along chain contour, of a tethered C n PE mdt for three grafting densities 0. f i e prediction of the Milner et al.8*9theoryisalso shown as a dashed line.

Fig. 5 presents the variation with distance from the substrate of the density p of the C78 PE melt simulated, all chains of which are grafted: (a) on a hard substrate and (b) on graphite, at grafting density ~ 2 . 1 nm 8 -2 . In the latter case, the strong dependence of p on z is evident, particularly at the first layers where p

131 attains values considerably higher than in the bulk due to polymer adsorption on graphite.

*r

E

n N v

-hard surface

1.0

--O-graphite

0.8 0.6

a

0.4 0.2 0.0 0

15

45

30

z(

60

5

1

Figure 5 . Local density p plotted as a h c t i o n of distance fiom the graffing plane for a C s PE melt tethered at o=2.18nni2,above a graphite (open circles) or B hard substrate (solid line). The numbers 1 through 3, for the case of graphite, denote the positions ofthe first three local maxima in the density due to polymer adsorption.

By properly analyzing the mean orientational characteristics of the grafted layers at the level of individual segments, one can develop a methodology for calculating their deuterium ( H) NMR spectrum. Such calculations have been presented in a recent article3 and the resulting spectrum for the C78 PE melt (0=2.62 nm-l) are shown in Fig. 6. The spectrum exhibits its highest peak at frequencies characterized by a doublet splitting on the order of 0.4kHz. This value is in qualitative agreement with available experimental data reported in the literature for long PDMS chains grafted on silica characterized by similar chain overlaps as those of the shorter systems studied here.

'

132

--

I

~

C,, 0=2.62nm-~

n V

3

5

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Figure 6. 2H-NMR rpcetnun of t h e c78 PE systcm graficd at 0=2.62nm'~ealeulatcd from the atomistic simulation data

References

2.

A.V. Shevade, J. Zhou, M.T. Zin and S. Jiang, Langmuir, 17, 7566 (2001): L. Zhang, K. Wesley and S. Jiang, Langmuir, 17,6275 (2001). K.Ch. Daoulas, A.F. Terzis and V.G. Mavrantzas, J. Chem. Phys., 116,

3.

K. Ch. Daoulas, V.G. Mavrantzas and D.J. Photinos, J. Chem. Phys., 118,

4. 5. 6. 7. 8.

W.A. Steele, Surface Sci., 36, 3 17 (1973). K.F. Mansfield and D.N. Theodorou, Macromolecules, 24,4295 (1991). D.N. Theodorou and U.W. Suter, Macromolecules, 18, 1467 (1985). K.F. Mansfield and D.N. Theodorou, Macromolecules, 23,4430 (1990). S.T. Milner, T.A. Witten and M.E. Cates, Macromolecules, 21,2610

1.

11028 (2002). 1521 (2003).

(!988). 9.

S.T. Milner, T.A. Witten and M.E. Cates, Macromolecules, 22,853 (1 989).

10. T.M. Birshtein, Yu.V. Liatskaya and E.B. Zhulina, Polymer, 31, 2 185( 1990). 11. E.B. Zhulina, O.V. Borisov, V.A. Pryamitsyn and T.M. Birshtein, Macromolecules, 24, 140 (1991).

MARKOV'S PROPERTY AND GENERLIZED PADE -TYPE APPROXIMANTS NICHOLAS J. DARAS Jean Moreas 19, 15232 Chalandri, Athens, Greece

Pad6 approximants are rational functions whose expansion in ascending powers of the variable coincides with the Taylor power series expansion of analytic functions into a disk as far as possible, that is up to the sum of the degree of the numerator and denominator. The numerator and denominator of a Pad6 approximant are completely determined by this condition and no freedom is left. On the contrary, PadC-type approximants are rational functions with an arbitrary denominator, whose numerator is determined by the condition that the expansion of the PadB-type approximant matches the Taylor expansion of analytic functions up to the degree of the numerator. The great advantage of PadC-type approximants over Pad6 approximants lies in the free choice of the poles which may lead to a better approximation ([2-61). One would like to adapt the proofs of the one variable to the several variables case, however some major obstacles present themselves. It is therefore reasonable to suspect that the outlet lies with the consideration of another type of series representation for functions. Thus, in [8], we investigated PadC-type (and Padt) approximants to the Fourier series expansion of a real-valued function harmonic in the unit disk. Moreover, in [ 113, we introduced PadC-type (and Padd) approximation to the Fourier representation of a complex-valued harmonic function. This coordinate procedure is defined to be a composed approximation. Any classical Padt-type approximant to an analytic function coincides with a composed PadC-type approximant to this function. With this background, in [9] we used the solution of the Dirichlet problem in order to discuss the possibility of the numerical evaluation of a 2n-periodic

Lp function on [-n,n] (or on the unit circle) by means of composed PadCtype approximants to its Fourier series representation. Further, in [12] we considered integral representation formulas for all the above Padd-type approximants to Fourier series and in this connection we defined and studied the corresponding Padt-type operators. Finally, the definition and effectiveness of a PadC-type approximation to the Fourier series representation of a 2n -periodic finite Baire measure on [-n, n](or on the unit circle) are investigated in [ 101. Following the above direction, in [ 131, by using interpolating generalized polynomials for the Bergman Kernel function into an open bounded subset of we defined the so-called generalized Padt-type approximants to any f

a

en,

133

134 which are of class L2. HL2(R) of all analytic functions in The terminology is due to H . Van Rossum who in [23] was first introduced the in the space

notion of generalized Pad6 approximants. The characteristic property of such an approximant is that its Fourier series representation with