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An Adaptive ANOVA-Based Data-Driven Stochastic Method for Elliptic PDEs with Random Coefficient

Published online by Cambridge University Press:  03 June 2015

Zhiwen Zhang ,
Xin Hu ,
Thomas Y. Hou ,
Guang Lin  and
Mike Yan
Zhiwen Zhang
Affiliation:
Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Xin Hu
Affiliation:
Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Thomas Y. Hou*
Affiliation:
Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA
Guang Lin*
Affiliation:
Pacific Northwest National Laboratory, Richland, WA 99352, USA
Mike Yan
Affiliation:
Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA
*
Corresponding author.Email:[email protected]
Corresponding author.Email:[email protected]
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Abstract

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.

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 highdimensional 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.

Keywords

60H3565C3035R6062H99Analysis of variancestochastic partial differential equationsdata-driven methodsKarhunen-Loeve expansionuncertainty quantificationmodel reduction.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 571 - 598
Copyright
Copyright © Global Science Press Limited 2014

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