Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

22/2008

Analysis and implementation issues for the numerical approximation of parabolic equations with random coefficients

Thursday 23rd October 2008

Nobile, Fabio; Tempone, Raul

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-Lo eve 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