Applied and Computational Math Seminar: Numerical Approximation of Parabolic SPDE’s

Numerical Approximation of Parabolic SPDE’s

Noel J. Walkington

Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA, USA

Abstract

Convergence theory for numerical schemes to approximate solutions of stochastic parabolic equations of the form form

du + A(u) dt = f dt + g dW, u(0) = u0,

will be reviewed. Here u is a random variable taking values in a function space U, A : U → U′ is partial differential operator, W = {Wt}t≥0 a Wiener process, and f, g, and u0 are data. This talk will illustrate how techniques from stochastic analysis and numerical partial differential equations can be combined to obtain a realization of the Lax–Richtmeyer meta–theorem

A numerical scheme converges if (and only if) it is stable and consistent.

Structural properties of the partial differential operator(s) and probabilistic methods will be developed to establish stability and a version of Donsker’s theorem for discrete

′

processes in the dual space U .
This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).