# Efficient scalable algorithms for hierarchically semiseparable matrices

## Abstract

Hierarchically semiseparable (HSS) matrix algorithms are emerging techniques in constructing the superfast direct solvers for both dense and sparse linear systems. Here, we develope a set of novel parallel algorithms for the key HSS operations that are used for solving large linear systems. These include the parallel rank-revealing QR factorization, the HSS constructions with hierarchical compression, the ULV HSS factorization, and the HSS solutions. The HSS tree based parallelism is fully exploited at the coarse level. The BLACS and ScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the ne-grained level. We have appplied our new parallel HSS-embedded multifrontal solver to the anisotropic Helmholtz equations for seismic imaging, and were able to solve a linear system with 6.4 billion unknowns using 4096 processors, in about 20 minutes. The classical multifrontal solver simply failed due to high demand of memory. To our knowledge, this is the first successful demonstration of employing the HSS algorithms in solving the truly large-scale real-world problems. Our parallel strategies can be easily adapted to the parallelization of the other rank structured methods.

- Authors:

- Publication Date:

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

- Sponsoring Org.:
- Computational Research Division

- OSTI Identifier:
- 1052181

- Report Number(s):
- LBNL-5238E

- DOE Contract Number:
- DE-AC02-05CH11231

- Resource Type:
- Journal Article

- Journal Name:
- SIAM J. Scientific Computing

- Additional Journal Information:
- Journal Name: SIAM J. Scientific Computing

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 01 COAL, LIGNITE, AND PEAT; HSS matrix; parallel HSS algorithm; low-rank property; compression; HSS construction; direct solver

### Citation Formats

```
Wang, Shen, Xia, Jianlin, Situ, Yingchong, and Hoop, Maarten V. de.
```*Efficient scalable algorithms for hierarchically semiseparable matrices*. United States: N. p., 2011.
Web.

```
Wang, Shen, Xia, Jianlin, Situ, Yingchong, & Hoop, Maarten V. de.
```*Efficient scalable algorithms for hierarchically semiseparable matrices*. United States.

```
Wang, Shen, Xia, Jianlin, Situ, Yingchong, and Hoop, Maarten V. de. Wed .
"Efficient scalable algorithms for hierarchically semiseparable matrices". United States. https://www.osti.gov/servlets/purl/1052181.
```

```
@article{osti_1052181,
```

title = {Efficient scalable algorithms for hierarchically semiseparable matrices},

author = {Wang, Shen and Xia, Jianlin and Situ, Yingchong and Hoop, Maarten V. de},

abstractNote = {Hierarchically semiseparable (HSS) matrix algorithms are emerging techniques in constructing the superfast direct solvers for both dense and sparse linear systems. Here, we develope a set of novel parallel algorithms for the key HSS operations that are used for solving large linear systems. These include the parallel rank-revealing QR factorization, the HSS constructions with hierarchical compression, the ULV HSS factorization, and the HSS solutions. The HSS tree based parallelism is fully exploited at the coarse level. The BLACS and ScaLAPACK libraries are used to facilitate the parallel dense kernel operations at the ne-grained level. We have appplied our new parallel HSS-embedded multifrontal solver to the anisotropic Helmholtz equations for seismic imaging, and were able to solve a linear system with 6.4 billion unknowns using 4096 processors, in about 20 minutes. The classical multifrontal solver simply failed due to high demand of memory. To our knowledge, this is the first successful demonstration of employing the HSS algorithms in solving the truly large-scale real-world problems. Our parallel strategies can be easily adapted to the parallelization of the other rank structured methods.},

doi = {},

journal = {SIAM J. Scientific Computing},

number = ,

volume = ,

place = {United States},

year = {2011},

month = {9}

}