## PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case

## Abstract

We describe an efficient parallel implementation of the selected inversion algorithm for distributed memory computer systems, which we call PSelInv. The PSelInv method computes selected elements of a general sparse matrix Athat can be decomposed as A= LU, where L is lower triangular and U is upper triangular. The implementation described in this article focuses on the case of sparse symmetric matrices. It contains an interface that is compatible with the distributed memory parallel sparse direct factorization SuperLU-DIST. Yet, the underlying data structure and design of PSelInv allows it to be easily combined with other factorization routines, such as PARDISO. We discuss general parallelization strategies such as data and task distribution schemes. Specifically, we describe how to exploit the concurrency exposed by the elimination tree associated with the LU factorization of A. We demonstrate the efficiency and accuracy of PSelInv by presenting several numerical experiments. In particular, we show that PSelInv can run efficiently on more than 4,000 cores for a modestly sized matrix. We also show how PSelInv can be used to accelerate large-scale electronic structure calculations.

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Univ. of California, Berkeley, CA (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21). Scientific Discovery through Advanced Computing (SciDAC); USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1525182

- Grant/Contract Number:
- AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- ACM Transactions on Mathematical Software

- Additional Journal Information:
- Journal Volume: 43; Journal Issue: 3; Journal ID: ISSN 0098-3500

- Publisher:
- Association for Computing Machinery

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; selected inversion; sparse direct method; distributed memory parallel algorithm; high performance computation; electronic structure theory

### Citation Formats

```
Jacquelin, Mathias, Lin, Lin, and Yang, Chao. PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case. United States: N. p., 2017.
Web. doi:10.1145/2786977.
```

```
Jacquelin, Mathias, Lin, Lin, & Yang, Chao. PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case. United States. doi:10.1145/2786977.
```

```
Jacquelin, Mathias, Lin, Lin, and Yang, Chao. Mon .
"PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case". United States. doi:10.1145/2786977. https://www.osti.gov/servlets/purl/1525182.
```

```
@article{osti_1525182,
```

title = {PSelInv—A Distributed Memory Parallel Algorithm for Selected Inversion: The Symmetric Case},

author = {Jacquelin, Mathias and Lin, Lin and Yang, Chao},

abstractNote = {We describe an efficient parallel implementation of the selected inversion algorithm for distributed memory computer systems, which we call PSelInv. The PSelInv method computes selected elements of a general sparse matrix Athat can be decomposed as A= LU, where L is lower triangular and U is upper triangular. The implementation described in this article focuses on the case of sparse symmetric matrices. It contains an interface that is compatible with the distributed memory parallel sparse direct factorization SuperLU-DIST. Yet, the underlying data structure and design of PSelInv allows it to be easily combined with other factorization routines, such as PARDISO. We discuss general parallelization strategies such as data and task distribution schemes. Specifically, we describe how to exploit the concurrency exposed by the elimination tree associated with the LU factorization of A. We demonstrate the efficiency and accuracy of PSelInv by presenting several numerical experiments. In particular, we show that PSelInv can run efficiently on more than 4,000 cores for a modestly sized matrix. We also show how PSelInv can be used to accelerate large-scale electronic structure calculations.},

doi = {10.1145/2786977},

journal = {ACM Transactions on Mathematical Software},

number = 3,

volume = 43,

place = {United States},

year = {2017},

month = {1}

}

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##
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