[Paper Review] Hierarchical Schur complement preconditioner for the stochastic Galerkin finite element methods Dedicated to Professor Ivo Marek on the occasion of his 80th birthday.

Bedřich Sousedík, Roger Ghanem+1 more|arXiv (Cornell University)|Jan 1, 2014

Probabilistic and Robust Engineering Design37 references21 citations

TL;DR

This paper proposes a hierarchical Schur complement preconditioner for stochastic Galerkin finite element methods, exploiting the recursive two-by-two block structure of global matrices to accelerate Krylov subspace iterations. By approximating inverse actions via inner Krylov loops on mean-value problem blocks and avoiding explicit matrix assembly, it achieves robust convergence with a bounded condition number for elliptic problems, validated numerically.

ABSTRACT

SUMMARY 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-Lo` eve 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. Submitted as preprint to ArXiv.

Motivation & Objective

• Address the challenge of solving large, structured linear systems arising from stochastic Galerkin discretization of elliptic PDEs with random coefficients.
• Overcome the high computational cost of iterative solvers for tensor-product discretization spaces in spatial and stochastic dimensions.
• Develop a preconditioner that leverages the recursive hierarchical structure of global matrices without explicit matrix assembly.
• Ensure scalability and efficiency by approximating inverse actions through inner Krylov iterations on mean-value problem blocks.
• Provide theoretical condition number bounds and demonstrate robust performance through numerical experiments.

Proposed method

• Exploit the recursive two-by-two block structure of the global matrix, where one submatrix is block diagonal with blocks corresponding to the deterministic mean-value problem.
• Apply the Schur complement technique recursively at each level of the hierarchical matrix structure to decouple and precondition the system.
• Approximate the inverse of the diagonal blocks using inner Krylov subspace iterations, reusing a preconditioner for the mean-value problem.
• Perform matrix-vector multiplications only for off-diagonal blocks, avoiding full matrix assembly.
• Use stiffness matrices from the truncated Karhunen-Loève expansion as input, without forming the global matrix or preconditioner explicitly.
• Combine outer Krylov iterations with multiple independent inner Krylov loops to efficiently approximate the preconditioning action.

Experimental results

Research questions

• RQ1 How can the hierarchical block structure of stochastic Galerkin matrices be exploited to design an efficient, matrix-free preconditioner?
• RQ2 What is the theoretical condition number bound of the preconditioned system for a model elliptic problem?
• RQ3 Can inner Krylov iterations on mean-value problem blocks effectively approximate the action of the inverse without explicit matrix assembly?
• RQ4 How does the proposed method compare in convergence behavior and computational cost to standard preconditioning approaches?
• RQ5 To what extent does the method scale with increasing stochastic dimension and mesh refinement?

Key findings

• The hierarchical Schur complement preconditioner achieves a condition number bound that is independent of the stochastic dimension for a model elliptic problem.
• The method avoids explicit formation of the global matrix and preconditioner, relying only on stiffness matrices and a preconditioner for the mean-value problem.
• Numerical experiments confirm robust convergence behavior across various stochastic discretization levels and mesh refinements.
• The use of inner Krylov iterations on mean-value blocks provides an effective and efficient approximation of the inverse action.
• The preconditioner maintains efficiency even for high-dimensional stochastic problems due to its recursive, matrix-free structure.
• The approach demonstrates scalability and stability, with convergence rates largely unaffected by increasing stochastic dimension.