Krylov-subspace recycling via the POD-augmented conjugate-gradient method

TL;DR

This paper introduces a novel Krylov-subspace recycling method using goal-oriented POD for efficient solutions of sequences of symmetric-positive-definite linear systems, outperforming existing approaches.

Contribution

It proposes a POD-inspired truncation strategy for Krylov subspace recycling, tailored for symmetric-positive-definite matrices, with a hybrid solution method and goal-oriented basis selection.

Findings

Numerical experiments show improved efficiency over existing methods.

The method effectively handles sequences with varying right-hand sides.

Low-dimensional subspaces retain solution accuracy for inexact tolerances.

Abstract

This work presents a new Krylov-subspace-recycling method for efficiently solving sequences of linear systems of equations characterized by varying right-hand sides and symmetric-positive-definite matrices. As opposed to typical truncation strategies used in recycling such as deflation, we propose a truncation method inspired by goal-oriented proper orthogonal decomposition (POD) from model reduction. This idea is based on the observation that model reduction aims to compute a low-dimensional subspace that contains an accurate solution; as such, we expect the proposed method to generate a low-dimensional subspace that is well suited for computing solutions that can satisfy inexact tolerances. In particular, we propose specific goal-oriented POD `ingredients' that align the optimality properties of POD with the objective of Krylov-subspace recycling. To compute solutions in the resulting `augmented' POD subspace, we propose a hybrid direct/iterative three-stage method that leverages 1) the optimal ordering of POD basis vectors, and 2) well-conditioned reduced matrices. Numerical experiments performed on solid-mechanics problems highlight te benefits of the proposed method over existing approaches for Krylov-subspace recycling.