A low-rank algorithm for solving Lyapunov operator $\varphi$-functions within the matrix-valued exponential integrators

TL;DR

This paper introduces a low-rank algorithm for efficiently computing Lyapunov operator $\

Contribution

It presents a novel low-rank recursive method with error analysis for large-scale Lyapunov $\

Findings

Effective in reducing computational cost for large matrices.

Validated through numerical experiments on differential Lyapunov and Riccati equations.

Suitable for matrix-valued exponential integrators.

Abstract

In this work we present a low-rank algorithm for computing low-rank approximations of large-scale Lyapunov operator $\varphi$-functions. These computations play a crucial role in implementing of matrix-valued exponential integrators for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank.The proposed method employs a scaling and recursive procedure, complemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost is primarily determined by the multiplication of sparse matrices with block vectors. Numerical experiments validate the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in solving differential Lyapunov equations and Riccati equations.