Computational Krylov-based methods for large-scale differential Sylvester matrix problems

TL;DR

This paper introduces Krylov-based numerical methods for efficiently solving large-scale differential Sylvester matrix equations with low-rank constant terms, combining integral and projection techniques.

Contribution

It presents two novel Krylov-based approaches for differential Sylvester equations, including theoretical error bounds and residual norm expressions.

Findings

The methods effectively solve large-scale problems.

Numerical experiments compare the two approaches.

Theoretical results provide error bounds and residual expressions.

Abstract

In the present paper, we propose Krylov-based methods for solving large-scale differential Sylvester matrix equations having a low rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low-dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as BDF or Rosenbrock method and the obtained solution is used to build the low rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.