Preconditioned Recycling Krylov subspace methods for self-adjoint problems

TL;DR

This paper introduces a recycling Krylov subspace method that leverages Ritz vectors from previous solves to efficiently address sequences of self-adjoint linear systems, significantly reducing computation time.

Contribution

It presents a novel recycling Krylov subspace approach that automatically extracts Ritz vectors for deflation in self-adjoint problems, compatible with arbitrary inner products and preconditioners.

Findings

Substantial decrease in computation time with recycling

Effective for nonlinear Schrödinger equations

Compatible with various inner products and preconditioners

Abstract

The authors propose a recycling Krylov subspace method for the solution of a sequence of self-adjoint linear systems. Such problems appear, for example, in the Newton process for solving nonlinear equations. Ritz vectors are automatically extracted from one MINRES run and then used for self-adjoint deflation in the next. The method is designed to work with arbitrary inner products and arbitrary self-adjoint positive-definite preconditioners whose inverse can be computed with high accuracy. Numerical experiments with nonlinear Schr\"odinger equations indicate a substantial decrease in computation time when recycling is used.