Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods

Abstract

Use of the stochastic Galerkin finite element methods leads to large systems of linear equations obtained by the discretization of tensor product solution spaces along their spatial and stochastic dimensions. These systems are typically solved iteratively by a Krylov subspace method. We propose a preconditioner, which takes an advantage of the recursive hierarchy in the structure of the global matrices. In particular, the matrices posses a recursive hierarchical two-by-two structure, with one of the submatrices block diagonal. Each of the diagonal blocks in this submatrix is closely related to the deterministic mean-value problem, and the action of its inverse is in the implementation approximated by inner loops of Krylov iterations. Thus, our hierarchical Schur complement preconditioner combines, on each level in the approximation of the hierarchical structure of the global matrix, the idea of Schur complement with loops for a number of mutually independent inner Krylov iterations, and several matrix-vector multiplications for the off-diagonal blocks. Neither the global matrix nor the matrix of the preconditioner need to be formed explicitly. The ingredients include only the number of stiffness matrices from the truncated Karhunen-Loève expansion and a good preconditioned for the mean-value deterministic problem. We provide a condition number bound for a model elliptic problem, and the performance of the method is illustrated by numerical experiments.HIERARCHICAL SCHUR COMPLEMENT PRECONDITIONER 137 Both methods are defined using tensor product spaces for the spatial and stochastic discretizations. Collocation methods sample the stochastic PDE at a set of collocation points, which yields a set of mutually independent deterministic problems. Because one can use existing software to solve this set of problems, collocation methods are often referred to as non-intrusive. However, the number of collocation points can be quite prohibitive when high accuracy is required or when the stochastic problem is described by a large number of random variables.On the other hand, the stochastic Galerkin method is intrusive. It uses the spectral finite element approach to transform a stochastic PDE into a coupled set of deterministic PDEs, and because of this coupling, specialized solvers are required. The design of iterative solvers for systems of linear algebraic equations obtained from discretizations by stochastic Galerkin finite element methods has received significant attention recently. It is well-known that suitable preconditioning can significantly improve convergence of Krylov subspace iterative methods. Among the most simple, yet quite powerful methods, belongs the mean-based preconditioner by Powell and Elman [6], see also [7]. Further improvements include, for example, the Kronecker product preconditioner by Ullmann [8]. We refer to Rosseel and Vandewalle [9] for a more complete overview and comparison of various iterative methods and preconditioners, including matrix splitting and multigrid techniques. Also, an intere...

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