# Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory

## Abstract

Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-artmore »

- Authors:

- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States). Environmental Molecular Sciences Laboratory

- Publication Date:

- Research Org.:
- Lawrence Berkeley National Laboratory (LBNL), Berkeley, CA (United States). National Energy Research Scientific Computing Center (NERSC); Pacific Northwest National Lab. (PNNL), Richland, WA (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); Ministry of Education, Youth and Sports (Cambodia)

- OSTI Identifier:
- 1395271

- Report Number(s):
- PNNL-SA-114405

Journal ID: ISSN 0010-4655; PII: S0010465517302370

- Grant/Contract Number:
- AC05-76RL01830; AC02-05CH1123; AC02-05CH11231; KC-030106062653

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computer Physics Communications

- Additional Journal Information:
- Journal Volume: 221; Journal ID: ISSN 0010-4655

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Time dependent density functional theory; Linear response eigenvalue problem; Preconditioned eigensolvers

### Citation Formats

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao.
```*Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory*. United States: N. p., 2017.
Web. doi:10.1016/J.CPC.2017.07.017.

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, & Yang, Chao.
```*Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory*. United States. doi:10.1016/J.CPC.2017.07.017.

```
Vecharynski, Eugene, Brabec, Jiri, Shao, Meiyue, Govind, Niranjan, and Yang, Chao. Thu .
"Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory". United States.
doi:10.1016/J.CPC.2017.07.017.
```

```
@article{osti_1395271,
```

title = {Efficient block preconditioned eigensolvers for linear response time-dependent density functional theory},

author = {Vecharynski, Eugene and Brabec, Jiri and Shao, Meiyue and Govind, Niranjan and Yang, Chao},

abstractNote = {Within this paper, we present two efficient iterative algorithms for solving the linear response eigenvalue problem arising from the time dependent density functional theory. Although the matrix to be diagonalized is nonsymmetric, it has a special structure that can be exploited to save both memory and floating point operations. In particular, the nonsymmetric eigenvalue problem can be transformed into an eigenvalue problem that involves the product of two matrices M and K. We show that, because MK is self-adjoint with respect to the inner product induced by the matrix K, this product eigenvalue problem can be solved efficiently by a modified Davidson algorithm and a modified locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm that make use of the K-inner product. Additionally, the solution of the product eigenvalue problem yields one component of the eigenvector associated with the original eigenvalue problem. We show that the other component of the eigenvector can be easily recovered in an inexpensive postprocessing procedure. As a result, the algorithms we present here become more efficient than existing methods that try to approximate both components of the eigenvectors simultaneously. In particular, our numerical experiments demonstrate that the new algorithms presented here consistently outperform the existing state-of-the-art Davidson type solvers by a factor of two in both solution time and storage.},

doi = {10.1016/J.CPC.2017.07.017},

journal = {Computer Physics Communications},

number = ,

volume = 221,

place = {United States},

year = {Thu Aug 24 00:00:00 EDT 2017},

month = {Thu Aug 24 00:00:00 EDT 2017}

}

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