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

2013 ◽  
Vol 1 (1) ◽  
pp. 452-493 ◽  
Author(s):  
Mulin Cheng ◽  
Thomas Y. Hou ◽  
Mike Yan ◽  
Zhiwen Zhang
Keyword(s):  
Random Coefficients ◽  
Stochastic Method ◽  
Data Driven ◽  
Elliptic Pdes

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

2015 ◽  
Vol 13 (1) ◽  
pp. 173-204 ◽  
Author(s):  
Zhiwen Zhang ◽  
Maolin Ci ◽  
Thomas Y. Hou
Keyword(s):  
Random Coefficients ◽  
Stochastic Method ◽  
Data Driven ◽  
Elliptic Pdes

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

2014 ◽  
Vol 16 (2) ◽  
pp. 571-598 ◽  
Author(s):  
Zhiwen Zhang ◽  
Xin Hu ◽  
Thomas Y. Hou ◽  
Guang Lin ◽  
Mike Yan
Keyword(s):  
Stochastic Method ◽  
Data Driven ◽  
High Dimensional ◽  
Compact Representation ◽  
Elliptic Pdes ◽  
Decomposition Technique ◽  
Stochastic Problem ◽  
Forcing Function ◽  
Speed Up ◽  
Stochastic Basis

AbstractIn 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.

scholarly journals Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients

2013 ◽  
Vol 125 (3) ◽  
pp. 569-600 ◽  
Author(s):  
A. L. Teckentrup ◽  
R. Scheichl ◽  
M. B. Giles ◽  
E. Ullmann
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Random Coefficients ◽  
Elliptic Pdes ◽  
Multilevel Monte Carlo

scholarly journals Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications

2011 ◽  
Vol 230 (10) ◽  
pp. 3668-3694 ◽  
Author(s):  
I.G. Graham ◽  
F.Y. Kuo ◽  
D. Nuyens ◽  
R. Scheichl ◽  
I.H. Sloan
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Methods ◽  
Random Coefficients ◽  
Elliptic Pdes ◽  
Quasi Monte Carlo
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