Generalizing Reduced Rank Extrapolation to Low-Rank Matrix Sequences

TL;DR

This paper extends Reduced Rank Extrapolation (RRE) to efficiently accelerate the iterative solution of large-scale low-rank matrix equations with changing mapping functions.

Contribution

It introduces two novel generalizations of RRE for low-rank matrix sequences and variable fixed-point mappings, enabling broader application.

Findings

Effective computation of RRE for low-rank matrix sequences

New RRE formulation for changing fixed-point mappings

Successful numerical examples on Lyapunov and Riccati equations

Abstract

Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.