Probability Seminar: Chia Lee (The University of North Carolina at Chapel Hill)

An application of the Wiener chaos expansion and Malliavin calculus to the numerical error estimates for a stochastic finite element method.

We consider the numerical solution of a class of elliptic and parabolic SPDEs driven by a Gaussian white noise using a stochastic finite element method. To derive a priori error estimates and quantify the optimal rate of convergence of the numerical method, we tap on ideas from the theory for deterministic finite element, where the extension of numerical error estimates from elliptic PDE to parabolic PDE is a standard technique.  By expressing the stochastic perturbation in terms of the Malliavin divergence operator, we are able to use tools from the Malliavin calculus to derive the error estimates for the parabolic SPDE, in close mimicry with the techniques from the deterministic finite element theory. In particular, the analysis employs a formal stochastic adjoint problem arising from the adjoint relationship between the Malliavin derivative and the Malliavin divergence operator, to obtain the optimal order of spatial convergence. Some numerical simulations will be shown.

Refreshments will be served at 3:45pm in the 3rd floor lounge of Hanes Hall