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Numerical Methods for Stochastic Partial Differential Equations with White Noise (Hardcover, 1st ed. 2017)
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Numerical Methods for Stochastic Partial Differential Equations with White Noise (Hardcover, 1st ed. 2017)
Series: Applied Mathematical Sciences, 196
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This book covers numerical methods for stochastic partial
differential equations with white noise using the framework of
Wong-Zakai approximation. The book begins with some motivational
and background material in the introductory chapters and is divided
into three parts. Part I covers numerical stochastic ordinary
differential equations. Here the authors start with numerical
methods for SDEs with delay using the Wong-Zakai approximation and
finite difference in time. Part II covers temporal white noise.
Here the authors consider SPDEs as PDEs driven by white noise,
where discretization of white noise (Brownian motion) leads to PDEs
with smooth noise, which can then be treated by numerical methods
for PDEs. In this part, recursive algorithms based on Wiener chaos
expansion and stochastic collocation methods are presented for
linear stochastic advection-diffusion-reaction equations. In
addition, stochastic Euler equations are exploited as an
application of stochastic collocation methods, where a numerical
comparison with other integration methods in random space is made.
Part III covers spatial white noise. Here the authors discuss
numerical methods for nonlinear elliptic equations as well as other
equations with additive noise. Numerical methods for SPDEs with
multiplicative noise are also discussed using the Wiener chaos
expansion method. In addition, some SPDEs driven by non-Gaussian
white noise are discussed and some model reduction methods (based
on Wick-Malliavin calculus) are presented for generalized
polynomial chaos expansion methods. Powerful techniques are
provided for solving stochastic partial differential equations.
This book can be considered as self-contained. Necessary background
knowledge is presented in the appendices. Basic knowledge of
probability theory and stochastic calculus is presented in Appendix
A. In Appendix B some semi-analytical methods for SPDEs are
presented. In Appendix C an introduction to Gauss quadrature is
provided. In Appendix D, all the conclusions which are needed for
proofs are presented, and in Appendix E a method to compute the
convergence rate empirically is included. In addition, the authors
provide a thorough review of the topics, both theoretical and
computational exercises in the book with practical discussion of
the effectiveness of the methods. Supporting Matlab files are made
available to help illustrate some of the concepts further.
Bibliographic notes are included at the end of each chapter. This
book serves as a reference for graduate students and researchers in
the mathematical sciences who would like to understand
state-of-the-art numerical methods for stochastic partial
differential equations with white noise.
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