An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDE with Random Coefficients

Generalized polynomial chaos (gPC) methods have been successfully applied to various stochastic problems in many physical and engineering fields. However, realistic representation of stochastic inputs associated with various sources of uncertainty often leads to high dimensional representations that are computationally prohibitive for classic gPC methods. Additionally in the classic gPC methods, the gPC bases are determined based on the probabilistic distribution of stochastic inputs. However, the stochastic outputs may not share the same probabilistic distribution as the stochastic inputs. Hence, the gPC bases may not be the optimal bases for such systems, which causes the slow convergence of gPC methods for such stochastic problems. Here we present a general framework that integrates the adaptive ANOVA decomposition technique and the data-driven stochastic method to alleviate both of the two limitations. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique for high-dimensional stochastic problems. Three different ANOVA adaptive criteria are discussed. To improve the slow convergence of gPC methods, we use the data-driven stochastic method (DDSM) which was developed by Cheng-Hou-Yan in [5]. This method has an offline computation and an online computation. In the offline computation, optimal gPC bases are obtained by Karhunen-Loéve (K-L) expansion of the covariance matrix of stochastic outputs obtained by ANOVA-based sparsegrid PCM. In the online computation, a Galerkin-projection based gPC method with the optimal bases developed in the offline computation is employed, which greatly speeds up the convergence. Numerical examples are presented for one, two-dimensional elliptic PDE with random coefficients, and a two-dimensional Corresponding author. Email address: [email protected] (Guang Lin)