Preconditioning techniques for generalized Sylvester matrix equations

TL;DR

This paper introduces algebraic, parameter-free preconditioning methods for efficiently solving generalized multiterm Sylvester matrix equations, leveraging low Kronecker rank approximations to improve computational performance.

Contribution

It proposes novel preconditioning techniques that use low Kronecker rank approximations, compatible with existing iterative solvers for generalized Sylvester equations.

Findings

Low Kronecker rank inverse approximations enable faster solutions.

The methods are compatible with sparse approximate inverse techniques.

They require only matrix-matrix multiplications, suitable for modern architectures.

Abstract

Sylvester matrix equations are ubiquitous in scientific computing. However, few solution techniques exist for their generalized multiterm version, as they now arise in an increasingly large number of applications. In this work, we consider algebraic parameter-free preconditioning techniques for the iterative solution of generalized multiterm Sylvester equations. They consist in constructing low Kronecker rank approximations of either the operator itself or its inverse. While the former requires solving standard Sylvester equations in each iteration, the latter only requires matrix-matrix multiplications, which are highly optimized on modern computer architectures. Moreover, low Kronecker rank approximate inverses can be easily combined with sparse approximate inverse techniques, thereby enhancing their performance with little or no damage to their effectiveness.