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Xiu D. Numerical Methods for Stochastic Computations. A Spectral Method Approach

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Xiu D. Numerical Methods for Stochastic Computations. A Spectral Method Approach
Princeton University Press, 2010. — 144 p. — ISBN 978-0-691-14212-8.
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
Basic Concepts of Probability Theory
Survey of Orthogonal Polynomials and Approximation Theory
Formulation of Stochastic Systems
Generalized Polynomial Chaos
Stochastic Galerkin Method
Stochastic Collocation Method
Miscellaneous Topics and Applications
Some Important Orthogonal Polynomials in the Askey Scheme
The Truncated Gaussian Model G(α, β)
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