Alternating Energy Minimization Methods for Multi-term Matrix Equations

TL;DR

This paper introduces an iterative low-rank approximation method for solving multi-term matrix equations, especially those from stochastic PDE discretizations, using an energy minimization framework with rank adaptivity.

Contribution

It develops a novel rank-adaptive alternating energy minimization approach for efficiently solving multi-term matrix equations with improved accuracy techniques.

Findings

Effective in approximating solutions of multi-term matrix equations.

Demonstrated success on stochastic Galerkin finite element discretizations.

Provides computational procedures for low-rank solution enhancement.

Abstract

We develop computational methods for approximating the solution of a linear multi-term matrix equation in low rank. We follow an alternating minimization framework, where the solution is represented as a product of two matrices, and approximations to each matrix are sought by solving certain minimization problems repeatedly. The solution methods we present are based on a rank-adaptive variant of alternating energy minimization methods that builds an approximation iteratively by successively computing a rank-one solution component at each step. We also develop efficient procedures to improve the accuracy of the low-rank approximate solutions computed using these successive rank-one update techniques. We explore the use of the methods with linear multi-term matrix equations that arise from stochastic Galerkin finite element discretizations of parameterized linear elliptic PDEs, and demonstrate their effectiveness with numerical studies.