Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients
KAUST DepartmentApplied Mathematics and Computational Science Program
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Stochastic Numerics Research Group
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AbstractWe 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 approxi- mating 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. © 2009 John Wiley & Sons, Ltd.
SponsorsThe first and second authors were partially supported by the University of Austin Subcontract (Project Number 024550. Center for Predictive Computational Science) The first author acknowledges the Italian fund PRIN 2007 Numerical modeling, For scientific computing and advanced applications'. The Second acknowledges his Dahlquist fellowship at the Royal Institute of Technology in Stockholm. Sweden and his Start up funds at SC. Florida State University. He also would like to acknowledge the support of UdelaR in Uruguay.