Scientific Computing Group Seminars - Detail View

Speaker: E. Ullmann

Affiliation: Technische Universität Bergakademie Freiberg

Talk Title: Preconditioning Stochastic Galerkin Saddle Point Systems

Invited by: George Karniadakis

Time: Oct. 16 2009 11 a.m.

Location: 182 George Street, Room 110

Abstract:

We study efficient iterative solvers for Galerkin equations associated with mixed finite element discretizations of second-order elliptic partial differential equations (PDEs) with random coefficient functions. Such systems arise, for example, from discretized Darcy flow problems with random permeability coefficients. The Galerkin matrix has a familiar saddle point structure, however, due to the coupling of (standard) mixed finite element discretizations in the physical space and global polynomial chaos approximations on a probability space, the number of unknowns is huge. Moreover, the leading blocks of the saddle point matrix are sums of Kronecker products of pairs of matrices associated with the physical and stochastic discretization, respectively. Depending on the models employed for the random coefficient function, this block can be block-dense, and the cost of a matrix-vector product is non-trivial. We analyze block-diagonal preconditioners of Schur complement and augmented type for use with MINRES. First, we consider an inexpensive mean-based preconditioner based on fast solvers for scalar diffusion problems. For stochastically linear random coefficient functions, which arise, for example, from a truncated Karhunen-Loève expansion, with moderate fluctuations in the data relative to the mean value, we demonstrate that this gives an efficient way for solving the large coupled Galerkin system. If, on the other hand, the random diffusion coefficient is a lognormal random field approximated via a nonlinear function of random parameters, mean-based preconditioners are not effective. For this case, we combine so-called Kronecker product preconditioners with the Schur complement and augmented preconditioning approach, respectively. We study the spectral properties of the preconditioned saddle point matrix and demonstrate numerically the improved robustness of the Kronecker product preconditioners compared to the mean-based approach with respect to key statistical parameters of the random diffusion coefficient. This is joint work with Catherine Powell, David Silvester (Manchester, UK) and Oliver Ernst (Freiberg, Germany).