An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient (Communications in Computational Physics, 2014)
In this paper, we present an adaptive, analysis of variance (ANOVA)-baseddata-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-drivenstochastic method. To handle high-dimensional stochastic problems, we investigatethe use of adaptive ANOVA decomposition in the stochastic space as an effectivedimension-reduction technique. To improve the slow convergence of the generalizedpolynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt thedata-driven stochastic method (DSM) for speed up. An essential ingredient of theDSM 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 boundaryconditions.Our ANOVA-DSM consists of offline and online stages. In the offline stage, theoriginal 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 usingthe Karhunen-Loève expansion (KLE) and a two-level preconditioning optimizationapproach. Multiple trial functions are used to enrich the stochastic basis and improvethe accuracy. In the online stage, we solve each stochastic subproblem for any givenforcing function by projecting the stochastic solution into the data-driven stochasticbasis constructed offline. In our ANOVA-DSM framework, solving the original high-dimensional stochastic problem is reduced to solving a series of ANOVA-decomposedstochastic subproblems using the DSM. An adaptive ANOVA strategy is also providedto further reduce the number of the stochastic subproblems and speed up our method.To demonstrate the accuracy and efficiency of our method, numerical examples arepresented for one- and two-dimensional elliptic PDEs with random coefficients.