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View CartTaylor Approximations for Stochastic Partial Differential Equations
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- 1 Princeton University, Princeton, New Jersey
- 2 Goethe University, Frankfurt am Main, Germany
Description
This book presents a systematic theory of Taylor expansions of evolutionary-type stochastic partial differential equations (SPDEs). The authors show how Taylor expansions can be used to derive higher order numerical methods for SPDEs, with a focus on pathwise and strong convergence.
In the case of multiplicative noise, the driving noise process is assumed to be a cylindrical Wiener process, while in the case of additive noise the SPDE is assumed to be driven by an arbitrary stochastic process with Hölder continuous sample paths. Recent developments on numerical methods for random and stochastic ordinary differential equations are also included since these are relevant for solving spatially discretised SPDEs as well as of interest in their own right.
The authors include the proof of an existence and uniqueness theorem under general assumptions on the coefficients as well as regularity estimates in an appendix.
Keywords: stochastic partial differential equation, Taylor expansions, Taylor approximations, mild solutions, numerical methods
Table of Contents
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Front Matter
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1. Introduction
- I Random and Stochastic Ordinary Differential Equations
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2. Taylor Expansions and Numerical Schemes for RODEs
- II Stochastic Partial Differential Equations
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5. Stochastic Partial Differential Equations
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6. Numerical Methods for SPDEs
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Back Matter
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Excerpt
The numerical approximation of stochastic partial differential equations (SPDEs), specifically, stochastic evolution equations of the parabolic or hyperbolic type, encounters all of the difficulties that arise in the numerical solution of both deterministic PDEs and finite dimensional stochastic ordinary differential equations (SODEs) as well as many more due to the infinite dimensional nature of the driving noise processes. The state of development of numerical schemes for SPDEs compares with that for SODEs in the early 1970s. Most of the numerical schemes that have been proposed to date have a low order of convergence, especially in terms of an overall computational effort, and only recently has it been shown how to construct higher order schemes.
©2011 SIAM
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