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Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
Author(s) -
Nobile F.,
Tempone Raul
Publication year - 2009
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.2656
Subject(s) - mathematics , random field , collocation (remote sensing) , stochastic partial differential equation , partial differential equation , polynomial chaos , parabolic partial differential equation , parametric statistics , random variable , monte carlo method , tensor product , polynomial , mathematical analysis , pure mathematics , statistics , remote sensing , geology
Abstract We consider the problem of numerically approximating statistical moments of the solution of a time‐dependent linear parabolic partial differential equation (PDE), whose coefficients and/or forcing terms are spatially correlated random fields. The stochastic coefficients of the PDE are approximated by truncated Karhunen–Loève expansions driven by a finite number of uncorrelated random variables. After approximating the stochastic coefficients, the original stochastic PDE turns into a new deterministic parametric PDE of the same type, the dimension of the parameter set being equal to the number of random variables introduced. After proving that the solution of the parametric PDE problem is analytic with respect to the parameters, we consider global polynomial approximations based on tensor product, total degree or sparse polynomial spaces and constructed by either a Stochastic Galerkin or a Stochastic Collocation approach. We derive convergence rates for the different cases and present numerical results that show how these approaches are a valid alternative to the more traditional Monte Carlo Method for this class of problems. Copyright © 2009 John Wiley & Sons, Ltd.