## Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations

## Abstract

The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale twodimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. In conclusion, employing 55 296more »

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Divisio; Univ. of California, Berkeley, CA (United States). Dept. of Mathematics
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Divisio
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States). Physics Division

- Publication Date:

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

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

- OSTI Identifier:
- 1438776

- Alternate Identifier(s):
- OSTI ID: 1420728; OSTI ID: 1456961

- Report Number(s):
- LLNL-JRNL-735618

Journal ID: ISSN 0021-9606; TRN: US1900523

- Grant/Contract Number:
- AC52-07NA27344; AC52- 07NA27344; AC02-05CH11231

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 15; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; 97 MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE

### Citation Formats

```
Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, and Pask, John E. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations. United States: N. p., 2016.
Web. doi:10.1063/1.4964861.
```

```
Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, & Pask, John E. Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations. United States. doi:10.1063/1.4964861.
```

```
Banerjee, Amartya S., Lin, Lin, Hu, Wei, Yang, Chao, and Pask, John E. Fri .
"Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations". United States. doi:10.1063/1.4964861. https://www.osti.gov/servlets/purl/1438776.
```

```
@article{osti_1438776,
```

title = {Chebyshev polynomial filtered subspace iteration in the discontinuous Galerkin method for large-scale electronic structure calculations},

author = {Banerjee, Amartya S. and Lin, Lin and Hu, Wei and Yang, Chao and Pask, John E.},

abstractNote = {The Discontinuous Galerkin (DG) electronic structure method employs an adaptive local basis (ALB) set to solve the Kohn-Sham equations of density functional theory in a discontinuous Galerkin framework. The adaptive local basis is generated on-the-fly to capture the local material physics and can systematically attain chemical accuracy with only a few tens of degrees of freedom per atom. A central issue for large-scale calculations, however, is the computation of the electron density (and subsequently, ground state properties) from the discretized Hamiltonian in an efficient and scalable manner. We show in this work how Chebyshev polynomial filtered subspace iteration (CheFSI) can be used to address this issue and push the envelope in large-scale materials simulations in a discontinuous Galerkin framework. We describe how the subspace filtering steps can be performed in an efficient and scalable manner using a two-dimensional parallelization scheme, thanks to the orthogonality of the DG basis set and block-sparse structure of the DG Hamiltonian matrix. The on-the-fly nature of the ALB functions requires additional care in carrying out the subspace iterations. We demonstrate the parallel scalability of the DG-CheFSI approach in calculations of large-scale twodimensional graphene sheets and bulk three-dimensional lithium-ion electrolyte systems. In conclusion, employing 55 296 computational cores, the time per self-consistent field iteration for a sample of the bulk 3D electrolyte containing 8586 atoms is 90 s, and the time for a graphene sheet containing 11 520 atoms is 75 s.},

doi = {10.1063/1.4964861},

journal = {Journal of Chemical Physics},

number = 15,

volume = 145,

place = {United States},

year = {2016},

month = {10}

}

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