Adaptive Stochastic Methods: In Computational Mathematics and Mechanics / Edition 1

by Dmitry G. Arseniev, Vladimir M. Ivanov, Maxim L. Korenevsky

Overview

This monograph develops adaptive stochastic methods in computational mathematics. The authors discuss the basic ideas of the algorithms and ways to analyze their properties and efficiency. Methods of evaluation of multidimensional integrals and solutions of integral equations are illustrated by multiple examples from mechanics, theory of elasticity, heat conduction and fluid dynamics.

Contents

Part I: Evaluation of Integrals
Fundamentals of the Monte Carlo Method to Evaluate Definite Integrals
Sequential Monte Carlo Method and Adaptive Integration
Methods of Adaptive Integration Based on Piecewise Approximation
Methods of Adaptive Integration Based on Global Approximation
Numerical Experiments
Adaptive Importance Sampling Method Based on Piecewise Constant Approximation

Part II: Solution of Integral Equations
Semi-Statistical Method of Solving Integral Equations Numerically
Problem of Vibration Conductivity
Problem on Ideal-Fluid Flow Around an Airfoil
First Basic Problem of Elasticity Theory
Second Basic Problem of Elasticity Theory
Projectional and Statistical Method of Solving Integral Equations Numerically

Product Details

ISBN-13:	9783110553642
Publisher:	De Gruyter
Publication date:	01/09/2018
Pages:	290
Product dimensions:	6.69(w) x 9.45(h) x (d)
Age Range:	18 Years

About the Author

Dmitry Arsenyev, Vladimir Ivanov and Maxim Korenevskii, St. Petersburg Polytechnical University, Russia.

Table of Contents

Preface V

Introduction: Statistical Computing Algorithms as a Subject of Adaptive Control 1

Part I Evaluation of Integrals

1 Fundamentals of the Monte Carlo Method to Evaluate Definite Integrals 9

1.1 Problem setup 9

1.2 Essence of the Monte Carlo method 10

1.3 Sampling of a scalar random variable 11

1.3.1 The inverse function method 11

1.3.2 The superposition method 14

1.3.3 The rejection method 15

1.4 Sampling of a vector random variable 16

1.5 Elementary Monte Carlo method and its properties 18

1.6 Methods of variance reduction 20

1.6.1 Importance sampling 20

1.6.2 Control variate sampling 21

1.6.3 Advantages and relations between the methods of importance sampling and control variate sampling 21

1.6.4 Symmetrisation of the integrand 22

1.6.5 Group sampling 23

1.6.6 Estimating with a faster rate of convergence 24

1.7 Conclusion 25

2 Sequential Monte Carlo Method and Adaptive Integration 27

2.1 Sequential Monte Carlo method 27

2.1.1 Basic relations 27

2.1.2 Mean square convergence 29

2.1.3 Almost sure convergence 36

2.1.4 Error estimation 39

2.2 Adaptive methods of integration 41

2.2.1 Elementary adaptive method of one-dimensional integration 42

2.2.2 Adaptive method of importance sampling 44

2.2.3 Adaptive method of control variate sampling 46

2.2.4 Generalised adaptive methods of importance sampling and control variate sampling 47

2.2.5 On time and memory consumption 47

2.2.6 Regression-based adaptive methods 49

2.2.7 Note on notation 56

2.3 Conclusion 57

3 Methods of Adaptive Integration Based on Piecewise Approximation 59

3.1 Piecewise approximations over subdomains 59

3.1.1 Piecewise approximations and their orders 59

3.1.2 Approximations for particular classes of functions 60

3.1.3 Partition moments and estimates for the variances D_k 62

3.1.4 Generalised adaptive methods 63

3.2 Elementary one-dimensional method 65

3.2.1 Control variate sampling 66

3.2.2 Importance sampling 67

3.2.3 Conclusions and remarks 69

3.3 Sequential bisection 70

3.3.1 Description of the bisection technique 70

3.3.2 Control variate sampling 72

3.3.3 Importance sampling 76

3.3.4 Time consumption of the bisection method 78

3.4 Sequential method of stratified sampling 79

3.5 Deterministic construction of partitions 80

3.6 Conclusion 82

4 Methods of Adaptive Integration Based on Global Approximation 83

4.1 Global approximations 83

4.1.1 Approximations by orthonormalised functions: Basic relations 84

4.1.2 Conditions for algorithm convergence 86

4.2 Adaptive integration over the class S_p 92

4.2.1 Haar system of functions and univariate classes of functions S_p 92

4.2.2 Adaptive integration over the class S_p: One-dimensional case 93

4.2.3 Expansion into parts of differing dimensionalities: Multidimensional classes S_p 95

4.2.4 Adaptive integration over the class S_p: Multidimensional case 97

4.3 Adaptive integration over the class E^α_s 103

4.3.1 The classes of functions E^α_s 104

4.3.2 Adaptive integration with the use of trigonometric approximations 104

4.4 Conclusion 108

5 Numerical Experiments 111

5.1 Test problems setup 111

5.1.1 The first problem 111

5.1.2 The second problem 112

5.2 Results of experiments 113

5.2.1 The first test problem 113

5.2.2 The second test problem 121

6 Adaptive Importance Sampling Method Based on Piecewise Constant Approximation 123

6.1 Introduction 123

6.2 Investigation of efficiency of the adaptive importance sampling method 123

6.2.1 Adaptive and sequential importance sampling schemes 123

6.2.2 Comparison of adaptive and sequential schemes 126

6.2.3 Numerical experiments 128

6.2.4 Conclusion 131

6.3 Adaptive importance sampling method in the case where the number of bisection steps is limited 132

6.3.1 The adaptive scheme for one-dimensional improper integrals 132

6.3.2 The adaptive scheme for the case where the number of bisection steps is limited 134

6.3.3 Peculiarities and capabilities of the adaptive importance sampling scheme in the case where the number of bisection steps is fixed 135

6.3.4 Numerical experiments 136

6.3.5 Conclusion 140

6.4 Solution of a problem of navigation by distances to pin-point targets with the use of the adaptive importance sampling method 141

6.4.1 Problem setup 141

6.4.2 Application of the adaptive importance sampling method to calculating the optimal estimator of the object position 143

6.4.3 A numerical experiment 145

6.4.4 Conclusion 148

Part II Solution of Integral Equations

7 Semi-Statistical Method of Solving Integral Equations Numerically 151

7.1 Introduction 151

7.2 Basic relations 152

7.3 Recurrent inversion formulas 154

7.4 Non-degeneracy of the matrix of the semi-statistical method 155

7.5 Convergence of the method 161

7.6 Adaptive capabilities of the algorithm 163

7.7 Qualitative considerations on the relation between the semi-statistical method and the variational ones 165

7.8 Application of the method to integral equations with a singularity 165

7.8.1 Description of the method and peculiarities of its application 165

7.8.2 Recurrent inversion formulas 168

7.8.3 Error analysis 168

7.8.4 Adaptive capabilities of the algorithm 171

8 Problem of Vibration Conductivity 173

8.1 Boundary value problem of vibration conductivity 173

8.2 Integral equations of vibration conductivity 174

8.3 Regutarisation of the equations 180

8.4 An integral equation with enhanced asymptotic properties at small β 184

8.5 Numerical solution of vibration conductivity problems 187

8.5.1 Solution of the test problem 187

8.5.2 Analysis of the influence of the sphere distortion and the external stress character on the results of the numerical solution 190

9 Problem on Ideal-Fluid Flow Around an Airfoil 193

9.1 Introduction 193

9.2 Setup of the problem on flow around an airfoil 193

9.3 Analytic description of the airfoil contour 195

9.4 Computational algorithm and optimisation 198

9.5 Results of numerical computation 199

9.5.1 Computation of the velocity around an airfoil 199

9.5.2 Analysis of the density adaptation efficiency 201

9.5.3 Computations on test cascades 205

9.6 Conclusions 207

9.7 A modified semi-statistical method 208

9.7.1 Computational scheme 209

9.7.2 Ways to estimate the variance in the computing process 210

9.7.3 Recommendations and remarks to the scheme of the modified semi-statistical method 211

9.7.4 Numerical experiment for a prolate airfoil 211

10 First Basic Problem of Elasticity Theory 215

10.1 Potentials and integral equations of the first basic problem of elasticity theory 215

10.1.1 The force and pseudo-force tensors 215

10.1.2 Integral equations of the first basic problem 217

10.2 Solution of some spatial problems of elasticity theory using the method of potentials 218

10.2.1 Solution of the first basic problem for a series of centrally symmetric spatial regions 219

10.2.2 Solution of the first basic problem for a sphere 220

10.2.3 Solution of the first basic problem for an unbounded medium with a spherical cavity 220

10.2.4 Solution of the first basic problem for a hollow sphere 221

10.3 Solution of integral equations of elasticity theory using the semi-statistical method 223

10.4 Formulas for the optimal density 225

10.5 Results of numerical experiments 227

11 Second Basic Problem of Elasticity Theory 231

11.1 Fundamental solutions of the first and second kind 231

11.2 Boussinesq potentials 234

11.3 Weyl tensor 235

11.4 Weyl force tensors 237

11.5 Arbitrary Lyapunov surface 238

12 Projectional and Statistical Method of Solving Integral Equations Numerically 241

12.1 Basic relations 241

12.2 Recurrent inversion formulas 244

12.3 Non-degeneracy of the matrix of the method 246

12.4 Convergence of the method 250

12.5 Advantages of the method and its adaptive capabilities 253

12.6 Peculiarities of the numerical implementation 255

12.7 Another computing technique: Averaging of approximate solutions 257

12.8 Numerical experiments 259

12.8.1 The test problem 259

12.8.2 The problem on steady-state forced small transverse vibration of a pinned string caused by a harmonic force 264

Afterword 271

Bibliography 273

Index 277