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White Noise Analysis for Stochastic Partial Differential Equations

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Numerical and Symbolic Scientific Computing

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

Stochastic partial differential equations arise when modelling uncertain phenomena. Here the emphasis is on uncertain systems where the randomness is spatial. In contrast to traditional slow computational approaches like Monte Carlo simulation, the methods described here can be orders of magnitude more efficients. These more recent methods are based on some kind stochastic Galerkin approximations, approximating the unknown quantities as functions of independent random variables, hence the name “white noise analysis”. We outline the steps leading to the fully discrete equations, commenting on one possible numerical solution method. Key to many of the developments is tensor product structure of the solution, which must be exploited both theoretically and numerically. For two examples with polynomial nonlinearities the computations are shown to be quite explicit and can be performed largely analytically.

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Authors and Affiliations

  1. Institute of Scientific Computing, Technische Universität Braunschweig, 38092, Braunschweig, Germany

    Hermann G. Matthies

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  1. Hermann G. Matthies
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Correspondence to Hermann G. Matthies .

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  1. Inst. Numerische Mathematik, Universität Linz, Altenbergerstr. 69, Linz, 4040, Austria

    Ulrich Langer

  2. , Research Institute for Symbolic Computat, Johannes Kepler University, Altenbergerstr. 69, Linz, 4040, Austria

    Peter Paule

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Matthies, H.G. (2012). White Noise Analysis for Stochastic Partial Differential Equations. In: Langer, U., Paule, P. (eds) Numerical and Symbolic Scientific Computing. Texts & Monographs in Symbolic Computation. Springer, Vienna. https://doi.org/10.1007/978-3-7091-0794-2_8

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  • DOI: https://doi.org/10.1007/978-3-7091-0794-2_8

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