Krylov-subspace recycling via the POD-augmented conjugate-gradient method
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.
Cite
@article{arxiv.1512.05820,
title = {Krylov-subspace recycling via the POD-augmented conjugate-gradient method},
author = {Kevin Carlberg and Virginia Forstall and Ray Tuminaro},
journal= {arXiv preprint arXiv:1512.05820},
year = {2016}
}