# Quasi-Optimal Elimination Trees for 2D Grids with Singularities

## Abstract

We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has cost $\mathcal{O}\left({N}_{e}\mathrm{log}\left({N}_{e}\right)\right)$ , where ${N}_{e}$ is the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.

- Authors:

- Jagiellonian University, 31007 Krakow, Poland
- AGH University of Science and Technology, 30059 Krakow, Poland
- Applied Mathematics & Computational Science, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
- Applied Mathematics & Computational Science, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia, Earth Science & Engineering and Center for Numerical Porous Media, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia
- Institute for Computational Engineering and Science, University of Texas, Austin, TX 78712-1229, USA

- Publication Date:

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1198427

- Resource Type:
- Journal Article: Published Article

- Journal Name:
- Scientific Programming

- Additional Journal Information:
- Journal Volume: 2015; Related Information: CHORUS Timestamp: 2016-08-18 13:07:40; Journal ID: ISSN 1058-9244

- Publisher:
- Hindawi Publishing Corporation

- Country of Publication:
- Egypt

- Language:
- English

### Citation Formats

```
Paszyńska, A., Paszyński, M., Jopek, K., Woźniak, M., Goik, D., Gurgul, P., AbouEisha, H., Moshkov, M., Calo, V. M., Lenharth, A., Nguyen, D., and Pingali, K.
```*Quasi-Optimal Elimination Trees for 2D Grids with Singularities*. Egypt: N. p., 2015.
Web. doi:10.1155/2015/303024.

```
Paszyńska, A., Paszyński, M., Jopek, K., Woźniak, M., Goik, D., Gurgul, P., AbouEisha, H., Moshkov, M., Calo, V. M., Lenharth, A., Nguyen, D., & Pingali, K.
```*Quasi-Optimal Elimination Trees for 2D Grids with Singularities*. Egypt. doi:10.1155/2015/303024.

```
Paszyńska, A., Paszyński, M., Jopek, K., Woźniak, M., Goik, D., Gurgul, P., AbouEisha, H., Moshkov, M., Calo, V. M., Lenharth, A., Nguyen, D., and Pingali, K. Thu .
"Quasi-Optimal Elimination Trees for 2D Grids with Singularities". Egypt.
doi:10.1155/2015/303024.
```

```
@article{osti_1198427,
```

title = {Quasi-Optimal Elimination Trees for 2D Grids with Singularities},

author = {Paszyńska, A. and Paszyński, M. and Jopek, K. and Woźniak, M. and Goik, D. and Gurgul, P. and AbouEisha, H. and Moshkov, M. and Calo, V. M. and Lenharth, A. and Nguyen, D. and Pingali, K.},

abstractNote = {We construct quasi-optimal elimination trees for 2D finite element meshes with singularities. These trees minimize the complexity of the solution of the discrete system. The computational cost estimates of the elimination process model the execution of the multifrontal algorithms in serial and in parallel shared-memory executions. Since the meshes considered are a subspace of all possible mesh partitions, we call these minimizers quasi-optimal. We minimize the cost functionals using dynamic programming. Finding these minimizers is more computationally expensive than solving the original algebraic system. Nevertheless, from the insights provided by the analysis of the dynamic programming minima, we propose a heuristic construction of the elimination trees that has cost O N e log N e , where N e is the number of elements in the mesh. We show that this heuristic ordering has similar computational cost to the quasi-optimal elimination trees found with dynamic programming and outperforms state-of-the-art alternatives in our numerical experiments.},

doi = {10.1155/2015/303024},

journal = {Scientific Programming},

number = ,

volume = 2015,

place = {Egypt},

year = {Thu Jan 01 00:00:00 EST 2015},

month = {Thu Jan 01 00:00:00 EST 2015}

}

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