Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods
This work addresses the computational bottleneck of solving stochastic PDEs with Galerkin methods, offering a scalable preconditioning strategy for practitioners in uncertainty quantification.
The paper proposes a hierarchical Schur complement preconditioner for solving large linear systems arising from stochastic Galerkin finite element methods, achieving efficient iterative solution by exploiting the recursive block structure of the matrices. Numerical experiments demonstrate improved convergence, with condition number bounds provided for a model elliptic problem.
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 one 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.