Block-Diagonal Preconditioning for Spectral Stochastic Finite Element Systems
Block-Diagonal Preconditioning for Spectral Stochastic Finite Element Systems
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Date
2007-06
Authors
Powell, Catherine E.
Elman, Howard C.
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Abstract
Deterministic models of fluid flow and the transport of chemicals in flows
in heterogeneous porous media incorporate PDEs whose material parameters
are assumed to be known exactly. To tackle more realistic stochastic flow
problems, it is fitting to represent the permeability coefficients as
random fields with prescribed statistics. Traditionally, large numbers of
deterministic problems are solved in a Monte Carlo framework and the
solutions averaged to obtain statistical properties of the solution
variables. Alternatively, so-called stochastic finite element methods
(SFEMs) discretise the probabilistic dimension of the PDE directly leading
to a single structured linear system. The latter approach is becoming
extremely popular but its computational cost is still perceived to be
problematic as this system is orders of magnitude larger than for the
corresponding deterministic problem. A simple block-diagonal
preconditioning strategy, incorporating only the mean component of the
random field coefficient and based on incomplete factorisations has been
employed in the literature, and observed to be robust, for problems of
moderate variance, but without theoretical analysis. We solve the
stochastic Darcy flow problem in primal formulation via the spectral SFEM
and focus on its efficient iterative solution. To achieve optimal
computational complexity, we base our block-diagonal preconditioner on
algebraic multigrid. In addition, we provide new theoretical eigenvalue
bounds for the preconditioned system matrix and highlight the dependence
of the iteration counts on all the SFEM parameters.