Preconditioning Stochastic Finite Element Matrices
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Thu, 28/01/2010 14:00 |
Dr. Catherine Powell (University of Manchester) |
Computational Mathematics and Applications  |
3WS SR |
| In the last few years, there has been renewed interest in stochastic finite element methods (SFEMs), which facilitate the approximation of statistics of solutions to PDEs with random data. SFEMs based on sampling, such as stochastic collocation schemes, lead to decoupled problems requiring only deterministic solvers. SFEMs based on Galerkin approximation satisfy an optimality condition but require the solution of a single linear system of equations that couples deterministic and stochastic degrees of freedom. This is regarded as a serious bottleneck in computations and the difficulty is even more pronounced when we attempt to solve systems of PDEs with random data via stochastic mixed FEMs. In this talk, we give an overview of solution strategies for the saddle-point systems that arise when the mixed form of the Darcy flow problem, with correlated random coefficients, is discretised via stochastic Galerkin and stochastic collocation techniques. For the stochastic Galerkin approach, the systems are orders of magnitude larger than those arising for deterministic problems. We report on fast solvers and preconditioners based on multigrid, which have proved successful for deterministic problems. In particular, we examine their robustness with respect to the random diffusion coefficients, which can be either a linear or non-linear function of a finite set of random parameters with a prescribed probability distribution. | |||