From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation

TL;DR

This paper introduces an efficient rational Krylov subspace method with adaptive shifts for solving Lyapunov equations with rank-1 right-hand sides, leveraging connections to linear ODEs and low-rank approximation techniques.

Contribution

It presents a novel adaptive shift selection strategy for rational Krylov subspace methods tailored to Lyapunov equations with rank-1 right-hand sides.

Findings

Numerical experiments demonstrate the method's effectiveness.

The approach improves computational efficiency over existing methods.

Adaptive shifts enhance convergence and accuracy.

Abstract

We propose a new method for the approximate solution of the Lyapunov equation with rank-$1$ right-hand side, which is based on extended rational Krylov subspace approximation with adaptively computed shifts. The shift selection is obtained from the connection between the Lyapunov equation, solution of systems of linear ODEs and alternating least squares method for low-rank approximation. The numerical experiments confirm the effectiveness of our approach.