Speaker: Zhiwen Zhang

Title:Small noise expansions and fast numerical methods for elliptic equation on domains with random boundary

Abstract: We consider the solution to an elliptic partial differential equation on the domain whose geometric boundary is subject to certain small perturbations. As the perturbation amplitude $\eps$ goes to zero, we propose an asymptotic approach to construct, up to any order in theory, the approximate series of the true solution on the random domain as the sum of the solutions on the unperturbed regular domain with random boundary conditions. Our asymptotic series have the extraordinary advantage in computation when the perturbation is random. The $m$-th order approximate solution in the asymptotic series can be computed by solving $N^m$ deterministic equations on regular domain in parallel, when the random boundary is described by $N$ terms of Karhunen-Loeve expansion or polynomial chaos.