RT Journal Article
ID 62860d63447fe689
A1 Sousedik, Bedrich
A1 Ghanem, Roger
T1 TRUNCATED HIERARCHICAL PRECONDITIONING FOR THE STOCHASTIC GALERKIN FEM
JF International Journal for Uncertainty Quantification
JO IJUQ
YR 2014
FD 2014-07-28
VO 4
IS 4
SP 333
OP 348
K1 stochastic Galerkin finite element methods
K1 iterative methods
K1 Schur complement method
K1 Gauss-Seidel method
K1 hierarchical and multilevel preconditioning
AB Stochastic Galerkin finite element discretizations of partial differential equations with coefficients characterized by arbitrary distributions lead, in general, to fully block dense linear systems.We propose two novel strategies for constructing preconditioners for these systems to be used with Krylov subspace iterative solvers. In particular, we present a variation of the hierarchical Schur complement preconditioner, developed recently by the authors, and an adaptation of the symmetric block Gauss-Seidel method. Both preconditioners take advantage of the hierarchical structure of global stochastic Galerkin matrices, and also, when applicable, of the decay of the norms of the stiffness matrices obtained from the polynomial chaos expansion of the coefficients. This decay allows to truncate the matrix-vector multiplications
in the action of the preconditioners. Also, throughout the global matrix hierarchy, we approximate solves with certain
submatrices by the associated diagonal block solves. The preconditioners thus require only a limited number of stiffness matrices obtained from the polynomial chaos expansion of the coefficients, and a preconditioner for the diagonal blocks of the global matrix. The performance is illustrated by numerical experiments.
PB Begell House
LK http://dl.begellhouse.com/journals/52034eb04b657aea,670f36d96da30eed,62860d63447fe689.html