Efficient tensor-based approach to solving linear systems involving Kronecker sum of matrices

TL;DR

This paper introduces a new tensor-based method for efficiently solving linear systems involving Kronecker sums, which are common in high-dimensional PDE discretizations, demonstrating improved performance on classical equations.

Contribution

The paper presents a novel tensor formula for solving Kronecker sum linear systems, extending Sylvester equation solutions to high-dimensional problems.

Findings

Successfully solved 2D and 3D Poisson equations

Efficiently handled 2D convection-diffusion equations

Method is suitable for high-dimensional PDEs

Abstract

A novel tensor-based formula for solving the linear systems involving Kronecker sum is proposed. Such systems are directly related to the matrix and tensor forms of Sylvester equation. The new tensor-based formula demonstrates the well-known fact that a Sylvester tensor equation has a unique solution if the sum of spectra of the matrices does not contain zero. We have showcased the effectiveness of the method by efficiently solving the 2D and 3D discretized Poisson equations, as well as the 2D steady-state convection-diffusion equation, on a rectangular domain with Dirichlet boundary conditions. The results suggest that this approach is well-suited for high-dimensional problems.