Solving rank structured Sylvester and Lyapunov equations

TL;DR

This paper introduces efficient hierarchical matrix-based methods for solving large-scale Sylvester and Lyapunov equations with rank-structured data, demonstrating their effectiveness through numerical experiments and a MATLAB toolbox.

Contribution

It presents novel estimates of numerical rank and solution schemes leveraging hierarchical matrices for rank-structured Sylvester and Lyapunov equations.

Findings

Methods are efficient for large-scale problems with banded data.

Numerical experiments confirm the effectiveness of the proposed approaches.

A MATLAB toolbox facilitates easy application and testing of the methods.

Abstract

We consider the problem of efficiently solving Sylvester and Lyapunov equations of medium and large scale, in case of rank-structured data, i.e., when the coefficient matrices and the right-hand side have low-rank off-diagonal blocks. This comprises problems with banded data, recently studied by Haber and Verhaegen in "Sparse solution of the Lyapunov equation for large-scale interconnected systems", Automatica, 2016, and by Palitta and Simoncini in "Numerical methods for large-scale Lyapunov equations with symmetric banded data", SISC, 2018, which often arise in the discretization of elliptic PDEs. We show that, under suitable assumptions, the quasiseparable structure is guaranteed to be numerically present in the solution, and explicit novel estimates of the numerical rank of the off-diagonal blocks are provided. Efficient solution schemes that rely on the technology of hierarchical matrices are described, and several numerical experiments confirm the applicability and efficiency of the approaches. We develop a MATLAB toolbox that allows easy replication of the experiments and a ready-to-use interface for the solvers. The performances of the different approaches are compared, and we show that the new methods described are efficient on several classes of relevant problems.