Solving Stochastic Collocation Systems with Algebraic Multigrid

Gordon, Andrew D. and Powell, Catherine E. (2010) Solving Stochastic Collocation Systems with Algebraic Multigrid. [MIMS Preprint]

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Stochastic collocation methods facilitate the numerical solution of partial differential equations (PDEs) with random data and give rise to long sequences of similar discrete linear systems. When elliptic PDEs with random diffusion coefficients are discretized with standard finite element methods in the physical domain, the resulting collocation systems are symmetric and positive definite. Such systems can be solved iteratively with the conjugate gradient (cg) method and algebraic multigrid (amg) is a highly robust preconditioner. When mixed finite element methods are applied, amg is also a key tool for solving the resulting sequence of saddle point systems via the preconditioned minimal residual (minres) method. In both cases, the stochastic collocation systems are trivial to solve when considered individually. The challenge lies in exploiting the systems' similarities to recycle information and minimize the cost of solving the entire sequence. We apply full tensor and sparse grid stochastic collocation schemes to a model stochastic elliptic problem and discretize in physical space using standard piecewise linear finite elements and Raviart-Thomas mixed finite elements. We propose efficient solvers for the resulting sequences of linear systems, that are more robust than other solution strategies in the literature. In particular, we show that it is feasible to use finely-tuned amg preconditioning for each system if key set-up information is reused. Crucially, the preconditioners are robust with respect to variations in the discretization and statistical parameters for both stochastically linear and nonlinear diffusion coefficients.

Item Type: MIMS Preprint
Uncontrolled Keywords: random data, stochastic collocation, sparse grids, finite elements, mixed finite elements, preconditioning, multigrid
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 60 Probability theory and stochastic processes
MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr C.E. Powell
Date Deposited: 10 Feb 2010
Last Modified: 08 Nov 2017 18:18

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