Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems
Efficient Iterative Solvers for Stochastic Galerkin Discretizations of Log-Transformed Random Diffusion Problems
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Date
2011-06-22
Authors
Ullmann, Elisabeth
Elman, Howard C.
Ernst, Oliver G.
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Abstract
We consider the numerical solution of a steady-state diffusion problem
where the diffusion coefficient is the exponent of a random field. The
standard stochastic Galerkin formulation of this problem is
computationally demanding because of the nonlinear structure of the
uncertain component of it. We consider a reformulated version of this
problem as a stochastic convection-diffusion problem with random
convective velocity that depends linearly on a fixed number of
independent truncated Gaussian random variables. The associated Galerkin
matrix is nonsymmetric but sparse and allows for fast matrix-vector
multiplications with optimal complexity. We construct and analyze two
block-diagonal preconditioners for this Galerkin matrix for use with
Krylov subspace methods such as the generalized minimal residual method.
We test the efficiency of the proposed preconditioning approaches and
compare the iterative solver performance for a model problem posed in
both diffusion and convection-diffusion formulations.