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Numerical Methods for Stochastic Computations : A Spectral Method Approach - Dongbin Xiu

Numerical Methods for Stochastic Computations

A Spectral Method Approach

Hardcover Published: 21st July 2010
ISBN: 9780691142128
Number Of Pages: 144

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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

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"[A]s a newbie to this field, by reading this lively written text I was able to gain insight into this really interesting and challenging matter."--Peter Mathe, Mathematical Reviews

Prefacep. xi
Introductionp. 1
Stochastic Modeling and Uncertainty Quantificationp. 1
Burgers' Equation: An Illustrative Examplep. 1
Overview of Techniquesp. 3
Burgers' Equation Revisitedp. 4
Scope and Audiencep. 5
A Short Review of the Literaturep. 6
Basic Concepts of Probability Theoryp. 9
Random Variablesp. 9
Probability and Distributionp. 10
Discrete Distributionp. 11
Continuous Distributionp. 12
Expectations and Momentsp. 13
Moment-Generating Functionp. 14
Random Number Generationp. 15
Random Vectorsp. 16
Dependence and Conditional Expectationp. 18
Stochastic Processesp. 20
Modes of Convergencep. 22
Central Limit Theoremp. 23
Survey of Orthogonal Polynomials and Approximation Theoryp. 25
Orthogonal Polynomialsp. 25
Orthogonality Relationsp. 25
Three-Term Recurrence Relationp. 26
Hypergeometric Series and the Askey Schemep. 27
Examples of Orthogonal Polynomialsp. 28
Fundamental Results of Polynomial Approximationp. 30
Polynomial Projectionp. 31
Orthogonal Projectionp. 31
Spectral Convergencep. 33
Gibbs Phenomenonp. 35
Polynomial Interpolationp. 36
Existencep. 37
Interpolation Errorp. 38
Zeros of Orthogonal Polynomials and Quadraturep. 39
Discrete Projectionp. 41
Formulation of Stochastic Systemsp. 44
Input Parameterization: Random Parametersp. 44
Gaussian Parametersp. 45
Non-Gaussian Parametersp. 46
Input Parameterization: Random Processes and Dimension Reductionp. 47
Karhunen-Loeve Expansionp. 47
Gaussian Processesp. 50
Non-Gaussian Processesp. 50
Formulation of Stochastic Systemsp. 51
Traditional Numerical Methodsp. 52
Monte Carlo Samplingp. 53
Moment Equation Approachp. 54
Perturbation Methodp. 55
Generalized Polynomial Chaosp. 57
Definition in Single Random Variablesp. 57
Strong Approximationp. 58
Weak Approximationp. 60
Definition in Multiple Random Variablesp. 64
Statisticsp. 67
Stochastic Galerkin Methodp. 68
General Procedurep. 68
Ordinary Differential Equationsp. 69
Hyperbolic Equationsp. 71
Diffusion Equationsp. 74
Nonlinear Problemsp. 76
Stochastic Collocation Methodp. 78
Definition and General Procedurep. 78
Interpolation Approachp. 79
Tensor Product Collocationp. 81
Sparse Grid Collocationp. 82
Discrete Projection: Pseudospectral Approachp. 83
Structured Nodes: Tensor and Sparse Tensor Constructionsp. 85
Nonstructured Nodes: Cubaturep. 86
Discussion: Galerkin versus Collocationp. 87
Miscellaneous Topics and Applicationsp. 89
Random Domain Problemp. 89
Bayesian Inverse Approach for Parameter Estimationp. 95
Data Assimilation by the Ensemble Kalman Filterp. 99
The Kalman Filter and the Ensemble Kalman Filterp. 100
Error Bound of the EnKFp. 101
Improved EnKF via gPC Methodsp. 102
Some Important Orthogonal Polynomials in the Askey Schemep. 105
Continuous Polynomialsp. 106
Hermite Polynomial Hn (x) and Gaussian Distributionp. 106
Laguerre Polynomial Ln() (x) and Gamma Distributionp. 106
Jacobi Polynomial Pn(, ß) (x) and Beta Distributionp. 107
Discrete Polynomialsp. 108
Charlier Polynomial Cn(x; a) and Poisson Distributionp. 108
Krawtchouk Polynomial Kn (x; p, N) and Binomial Distributionp. 108
Meixner Polynomial Mn (x; ß, c) and Negative Binomial Distributionp. 109
Hahn Polynomial Qn (x; , ß, N) and Hypergeometric Distributionp. 110
The Truncated Gaussian Model G(, ß)p. 113
Referencesp. 117
Indexp. 127
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691142128
ISBN-10: 0691142122
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 144
Published: 21st July 2010
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.27
Weight (kg): 0.34

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