On solving stochastic collocation systems with algebraic multigrid

Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physica...

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Bibliographic Details
Published in:IMA journal of numerical analysis 2012-07, Vol.32 (3), p.1051-1070
Main Authors: Gordon, A. D., Powell, C. E.
Format: Article
Language:English
Subjects:
Algebra
Collocation
Discretization
Linear systems
Mathematical models
Numerical analysis
Partial differential equations
Stochasticity
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Summary:Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar linear systems. When elliptic PDEs with random diffusion coefficients are discretized with mixed finite element methods in the physical domain we obtain saddle point systems. These are trivial to solve when considered individually; the challenge lies in exploiting their similarities to recycle information and minimize the cost of solving the entire sequence. We apply stochastic collocation to a model stochastic elliptic problem and discretize in physical space using Raviart-Thomas elements. We propose an efficient solution strategy for the resulting linear systems that is more robust than any other in the literature. In particular, we show that it is feasible to use finely-tuned algebraic multigrid preconditioning if key set-up information is reused. The proposed solver is robust with respect to variations in the discretization and statistical parameters for stochastically linear and nonlinear data.
ISSN:0272-4979
1464-3642
DOI:10.1093/imanum/drr034