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2009.88: Preconditioning stochastic Galerkin saddle point systems

2009.88: Catherine E. Powell and Elisabeth Ullmann (2009) Preconditioning stochastic Galerkin saddle point systems.

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Mixed finite element discretizations of deterministic second-order elliptic partial differential equations (PDEs) lead to saddle point systems for which the study of iterative solvers and preconditioners is mature. Galerkin approximation of solutions of stochastic second-order elliptic PDEs, which couple standard mixed finite element discretizations in physical space with global polynomial approximation on a probability space, also give rise to linear systems with familiar saddle point structure. For stochastically nonlinear problems, the solution of such systems presents a serious computational challenge. The blocks are sums of Kronecker products of pairs of matrices associated with two distinct discretizations and the systems are large, reflecting the curse of dimensionality inherent in most stochastic approximation schemes. Moreover, for the problems considered herein, the leading blocks of the saddle point matrices are block-dense and the cost of a matrix vector product is non-trivial.

We implement a stochastic Galerkin discretization for the steady-state diffusion problem written as a mixed first-order system. The diffusion coefficient is assumed to be a lognormal random field, approximated via a nonlinear function of a finite number of unbounded random parameters. We study the resulting saddle point systems and investigate the efficiency of block-diagonal preconditioners of Schur complement and augmented type, for use with MINRES. By introducing so-called Kronecker product preconditioners we improve the robustness of cheap, mean-based preconditioners with respect to the statistical properties of the stochastically nonlinear diffusion coefficients.

Item Type:MIMS Preprint
Uncontrolled Keywords:saddle point matrices, preconditioning, Kronecker product, multigrid, stochastic Galerkin finite element method, lognormal random field, H(div) approximation
Subjects:MSC 2000 > 35 Partial differential equations
MSC 2000 > 65 Numerical analysis
MIMS number:2009.88
Deposited By:Dr C.E. Powell
Deposited On:19 November 2009

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