A Class of Data-driven Methods for Stochastic Partial Differential Equations

Abstract

We propose a data-driven stochastic method (DSM) to study stochastic partial differential equations (SPDEs) in the multiquery setting. An essential ingredient of the proposed method is to construct a data-driven stochastic basis under which the stochastic solutions to the SPDEs enjoy a compact representation for a broad range of forcing functions and/or boundary conditions. Our method consists of offline and online stages. A data-driven stochastic basis is computed in the offline stage using the Karhunen-Loeve (KL) expansion. In the online stage, we solve a relatively small number of coupled deterministic PDEs by projecting the stochastic solution into the data-driven stochastic basis constructed offline. Applications of DSM to stochastic elliptic problems show considerable computational savings over traditional methods even with a small number of queries.