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Zusatztext "[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 Mathé, Mathematical Reviews Informationen zum Autor Dongbin Xiu is associate professor of mathematics at Purdue University. Klappentext "Short and comprehensive, this book is appropriate for novices of polynomial chaos. Many diverse fields are adopting this method, and this book can be used for first-year graduate studies as well as senior undergraduate courses. The book includes important new developments, such as non-Gaussian processes and stochastic collocation methods." --George Karniadakis, Brown University Zusammenfassung 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 Inhaltsverzeichnis Preface xi Chapter 1: Introduction 1 1.1 Stochastic Modeling and Uncertainty Quantification 1 1.1.1 Burgers' Equation: An Illustrative Example 1 1.1.2 Overview of Techniques 3 1.1.3 Burgers' Equation Revisited 4 1.2 Scope and Audience 5 1.3 A Short Review of the Literature 6 Chapter 2: Basic Concepts of Probability Theory 9 2.1 Random Variables 9 2.2 Probability and Distribution 10 2.2.1 Discrete Distribution 11 2.2.2 Continuous Distribution 12 2.2.3 Expectations and Moments 13 2.2.4 Moment-Generating Function 14 2.2.5 Random Number Generation 15 2.3 Random Vectors 16 2.4 Dependence and Conditional Expectation 18 2.5 Stochastic Processes 20 2.6 Modes of Convergence 22 2.7 Central Limit Theorem 23 Chapter 3: Survey of Orthogonal Polynomials and Approximation Theory 25 3.1 Orthogonal Polynomials 25 3.1.1 Orthogonality Relations 25 3.1.2 Three-Term Recurrence Relation 26 3.1.3 Hypergeometric Series and the Askey Scheme 27 3.1.4 Examples of Orthogonal Polynomials 28 3.2 Fundamental Results of Polynomial Approximation 30 3.3 Polynomial Projection 31 3.3.1 Orthogonal Projection 31 3.3.2 Spectral Convergence 33 3.3.3 Gibbs Phenomenon 35 3.4 Polynomial Interpolation 36 3.4.1 Existence 37 3.4.2 Interpolation Error 38 3.5 Zeros of Orthogonal Polynomials and Quadrature 39 3.6 Discrete Projection 41 Chapter 4: Formulation of Stochastic Systems 44 4.1 Input Parameterization: Random Parameters 44 4.1.1 Gaussian Parameters 45 4.1.2 Non-G...
