# Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method

## Abstract

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.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)
- Univ. of Maryland, College Park, MD (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1251146

- Report Number(s):
- SAND-2016-0828J

Journal ID: ISSN 0895-4798; 618975

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Journal Article: 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

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Krylov-subspace recycling; proper orthogonal decomposition; augmented Krylov methods; model reduction; conjugate-gradient method

### Citation Formats

```
Carlberg, Kevin, Forstall, Virginia, and Tuminaro, Ray.
```*Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method*. United States: N. p., 2016.
Web. doi:10.1137/16M1057693.

```
Carlberg, Kevin, Forstall, Virginia, & Tuminaro, Ray.
```*Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method*. United States. doi:10.1137/16M1057693.

```
Carlberg, Kevin, Forstall, Virginia, and Tuminaro, Ray. Fri .
"Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method". United States. doi:10.1137/16M1057693. https://www.osti.gov/servlets/purl/1251146.
```

```
@article{osti_1251146,
```

title = {Krylov-Subspace Recycling via the POD-Augmented Conjugate-Gradient Method},

author = {Carlberg, Kevin and Forstall, Virginia and Tuminaro, Ray},

abstractNote = {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.},

doi = {10.1137/16M1057693},

journal = {SIAM Journal on Matrix Analysis and Applications},

issn = {0895-4798},

number = 3,

volume = 37,

place = {United States},

year = {2016},

month = {1}

}

*Citation information provided by*

Web of Science

Web of Science