A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data

We investigate the convergence rate of approximations by finite sums of rank-1 tensors of solutions of multiparametric elliptic PDEs. Such PDEs arise, for example, in the parametric, deterministic reformulation of elliptic PDEs with random field inputs, based, for example, on the M-term truncated Karhunen–Lo`eve expansion. Our approach could be regarded as either a class of compressed approximations of these solutions or as a new class of iterative elliptic problem solvers for high-dimensional, parametric, elliptic PDEs providing linear scaling complexity in the dimension M of the parameter space. It is based on rank-reduced, tensor-formatted separable approximations of the high-dimensional tensors and matrices involved in the iterative process, combined with the use of spectrally equivalent low-rank tensor-structured preconditioners to the parametric matrices resulting from a finite element discretization of the high-dimensional parametric, deterministic problems. Numerical illustrations for the M-dimensional parametric elliptic PDEs resulting from sPDEs on parameter spaces of dimensions M ≤ 100 indicate the advantages of employing low-rank tensor-structured matrix formats in the numerical solution of such problems.


Published in:
Siam Review, 52, 2, 317–355
Year:
2010
Publisher:
Society for Industrial and Applied Mathematics
ISSN:
0036-1445
Keywords:
Laboratories:


Note: The status of this file is: EPFL only


 Record created 2012-07-26, last modified 2018-03-13

n/a:
Download fulltext
PDF

Rate this document:

Rate this document:
1
2
3
 
(Not yet reviewed)