Numerical Methods for Stochastic Computations: A Spectral Method Approach

✍ Scribed by Dongbin Xiu

• Publisher: Princeton University Press
• Year: 2010
• Tongue: English
• Leaves: 142
• Category: Library

✦ Synopsis

The first graduate-level textbook to focus on fundamental aspects of numerical methods for stochastic computations, this book describes the class of numerical methods based on generalized polynomial chaos (gPC). These fast, efficient, and accurate methods are an extension of the classical spectral methods of high-dimensional random spaces. Designed to simulate complex systems subject to random inputs, these methods are widely used in many areas of computer science and engineering. The book introduces polynomial approximation theory and probability theory; describes the basic theory of gPC methods through numerical examples and rigorous development; details the procedure for converting stochastic equations into deterministic ones; using both the Galerkin and collocation approaches; and discusses the distinct differences and challenges arising from high-dimensional problems. The last section is devoted to the application of gPC methods to critical areas such as inverse problems and data assimilation. Ideal for use by graduate students and researchers both in the classroom and for self-study, Numerical Methods for Stochastic Computations provides the required tools for in-depth research related to stochastic computations.The first graduate-level textbook to focus on the fundamentals of numerical methods for stochastic computations Ideal introduction for graduate courses or self-study Fast, efficient, and accurate numerical methods Polynomial approximation theory and probability theory included Basic gPC methods illustrated through examples

✦ Table of Contents

Numerical Methods for Stochastic Computations: A Spectral Method Approach......Page 4
Contents......Page 8
Preface......Page 12
1.1.1 Burgers’ Equation: An Illustrative Example......Page 16
1.1.2 Overview of Techniques......Page 18
1.1.3 Burgers’ Equation Revisited......Page 19
1.2 Scope and Audience......Page 20
1.3 A Short Review of the Literature......Page 21
2.1 Random Variables......Page 24
2.2 Probability and Distribution......Page 25
2.2.1 Discrete Distribution......Page 26
2.2.2 Continuous Distribution......Page 27
2.2.3 Expectations and Moments......Page 28
2.2.4 Moment-Generating Function......Page 29
2.2.5 Random Number Generation......Page 30
2.3 Random Vectors......Page 31
2.4 Dependence and Conditional Expectation......Page 33
2.5 Stochastic Processes......Page 35
2.6 Modes of Convergence......Page 37
2.7 Central Limit Theorem......Page 38
3.1.1 Orthogonality Relations......Page 40
3.1.2 Three-Term Recurrence Relation......Page 41
3.1.3 Hypergeometric Series and the Askey Scheme......Page 42
3.1.4 Examples of Orthogonal Polynomials......Page 43
3.2 Fundamental Results of Polynomial Approximation......Page 45
3.3.1 Orthogonal Projection......Page 46
3.3.2 Spectral Convergence......Page 48
3.3.3 Gibbs Phenomenon......Page 50
3.4 Polynomial Interpolation......Page 51
3.4.1 Existence......Page 52
3.4.2 Interpolation Error......Page 53
3.5 Zeros of Orthogonal Polynomials and Quadrature......Page 54
3.6 Discrete Projection......Page 56
4.1 Input Parameterization: Random Parameters......Page 59
4.1.1 Gaussian Parameters......Page 60
4.1.2 Non-Gaussian Parameters......Page 61
4.2.1 Karhunen-Loeve Expansion......Page 62
4.2.3 Non-Gaussian Processes......Page 65
4.3 Formulation of Stochastic Systems......Page 66
4.4 Traditional Numerical Methods......Page 67
4.4.1 Monte Carlo Sampling......Page 68
4.4.2 Moment Equation Approach......Page 69
4.4.3 Perturbation Method......Page 70
5.1 Definition in Single Random Variables......Page 72
5.1.1 Strong Approximation......Page 73
5.1.2 Weak Approximation......Page 75
5.2 Definition in Multiple Random Variables......Page 79
5.3 Statistics......Page 82
6.1 General Procedure......Page 83
6.2 Ordinary Differential Equations......Page 84
6.3 Hyperbolic Equations......Page 86
6.4 Diffusion Equations......Page 89
6.5 Nonlinear Problems......Page 91
7.1 Definition and General Procedure......Page 93
7.2 Interpolation Approach......Page 94
7.2.1 Tensor Product Collocation......Page 96
7.2.2 Sparse Grid Collocation......Page 97
7.3 Discrete Projection: Pseudospectral Approach......Page 98
7.3.1 Structured Nodes: Tensor and Sparse Tensor Constructions......Page 100
7.3.2 Nonstructured Nodes: Cubature......Page 101
7.4 Discussion: Galerkin versus Collocation......Page 102
8.1 Random Domain Problem......Page 104
8.2 Bayesian Inverse Approach for Parameter Estimation......Page 110
8.3 Data Assimilation by the Ensemble Kalman Filter......Page 114
8.3.1 The Kalman Filter and the Ensemble Kalman Filter......Page 115
8.3.2 Error Bound of the EnKF......Page 116
8.3.3 Improved EnKF via gPC Methods......Page 117
Appendix A - Some Important Orthogonal Polynomials in the Askey Scheme......Page 120
A.1.2 Laguerre Polynomial L(α)n(x) and Gamma Distribution......Page 121
A.1.3 Jacobi Polynomial P(α,β)n(x) and Beta Distribution......Page 122
A.2.2 Krawtchouk Polynomial Kn(x; p,N) and Binomial Distribution......Page 123
A.2.3 Meixner Polynomial Mn(x; β, c) and Negative Binomial Distribution......Page 124
A.2.4 Hahn Polynomial Qn(x; α, β,N) and Hypergeometric Distribution......Page 125
Appendix B - The Truncated Gaussian Model G(α, β)......Page 128
References......Page 132
Index......Page 142