TY - JOUR
T1 - An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient
AU - Zhiwen Zhang, Xin Hu, Thomas Y. Hou, Guang Lin & Mike Yan
JO - Communications in Computational Physics
VL - 3
SP - 571
EP - 598
PY - 2014
DA - 2014/12
SN - 16
DO - http://doi.org/10.4208/cicp.270913.020414a
UR - https://global-sci.org/intro/article_detail/cicp/7054.html
KW - 
AB - <p style="text-align: justify;">In this paper, we present an adaptive, analysis of variance (ANOVA)-based
data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven
stochastic method. To handle high-dimensional stochastic problems, we investigate
the use of adaptive ANOVA decomposition in the stochastic space as an effective
dimension-reduction technique. To improve the slow convergence of the generalized
polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the
data-driven stochastic method (DSM) for speed up. An essential ingredient of the
DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary
conditions.<br/>Our ANOVA-DSM consists of offline and online stages. In the offline stage, the
original high-dimensional stochastic problem is decomposed into a series of low-dimensional stochastic subproblems, according to the ANOVA decomposition technique. Then, for each subproblem, a data-driven stochastic basis is computed using
the Karhunen-Loève expansion (KLE) and a two-level preconditioning optimization
approach. Multiple trial functions are used to enrich the stochastic basis and improve
the accuracy. In the online stage, we solve each stochastic subproblem for any given
forcing function by projecting the stochastic solution into the data-driven stochastic
basis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposed
stochastic subproblems using the DSM. An adaptive ANOVA strategy is also provided
to further reduce the number of the stochastic subproblems and speed up our method.
To demonstrate the accuracy and efficiency of our method, numerical examples are
presented for one- and two-dimensional elliptic PDEs with random coefficients.</p>