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Title: Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method

This paper 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 the benefits of the proposed method over existing approaches for Krylov-subspace recycling.
 [1] ;  [2] ;  [1]
  1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
  2. Univ. of Maryland, College Park, MD (United States)
Publication Date:
Report Number(s):
Journal ID: ISSN 0895-4798; 618975
Grant/Contract Number:
Accepted Manuscript
Journal Name:
SIAM Journal on Matrix Analysis and Applications
Additional Journal Information:
Journal Volume: 37; Journal Issue: 3; Journal ID: ISSN 0895-4798
Research Org:
Sandia National Lab. (SNL-CA), Livermore, CA (United States)
Sponsoring Org:
USDOE National Nuclear Security Administration (NNSA)
Country of Publication:
United States
97 MATHEMATICS AND COMPUTING; Krylov-subspace recycling; proper orthogonal decomposition; augmented Krylov methods; model reduction; conjugate-gradient method
OSTI Identifier: