A Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficients

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--Loève (KL) expansion. A two-level preconditioning optimization approach and a randomized SVD algorithm are used to reduce the offline computational cost. 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. Compared with a generalized polynomial chaos method (gPC), the ratio of the computational complexities between DSM (online stage) and gPC is of order $O((m/N_p)^2)$. Here $m$ and $N_p$ are the numbers of elements in the basis used in DSM and gPC, respectively. Typically we expect $m\ll N_p$ when the effective dimension of the stochastic solution is small. A timing model, which takes into account the offline computational cost of DSM, is constructed to demonstrate the efficiency of DSM. Applications of DSM to stochastic elliptic problems show considerable computational savings over traditional methods even with a small number of queries. We also provide a method for an a posteriori error estimate and error correction.