# Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals

## Abstract

A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. Here, the method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Finally, numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.

- Authors:

- Sandia National Lab. (SNL-CA), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

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

- OSTI Identifier:
- 1429500

- Report Number(s):
- SAND2018-2510J

Journal ID: ISSN 0045-7825; 661273

- Grant/Contract Number:
- AC04-94AL85000; FG02-12ER46875

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Volume: 336; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

### Citation Formats

```
Rai, Prashant, Sargsyan, Khachik, and Najm, Habib N.
```*Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals*. United States: N. p., 2018.
Web. doi:10.1016/j.cma.2018.02.026.

```
Rai, Prashant, Sargsyan, Khachik, & Najm, Habib N.
```*Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals*. United States. doi:10.1016/j.cma.2018.02.026.

```
Rai, Prashant, Sargsyan, Khachik, and Najm, Habib N. Tue .
"Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals". United States. doi:10.1016/j.cma.2018.02.026. https://www.osti.gov/servlets/purl/1429500.
```

```
@article{osti_1429500,
```

title = {Compressed sparse tensor based quadrature for vibrational quantum mechanics integrals},

author = {Rai, Prashant and Sargsyan, Khachik and Najm, Habib N.},

abstractNote = {A new method for fast evaluation of high dimensional integrals arising in quantum mechanics is proposed. Here, the method is based on sparse approximation of a high dimensional function followed by a low-rank compression. In the first step, we interpret the high dimensional integrand as a tensor in a suitable tensor product space and determine its entries by a compressed sensing based algorithm using only a few function evaluations. Secondly, we implement a rank reduction strategy to compress this tensor in a suitable low-rank tensor format using standard tensor compression tools. This allows representing a high dimensional integrand function as a small sum of products of low dimensional functions. Finally, a low dimensional Gauss–Hermite quadrature rule is used to integrate this low-rank representation, thus alleviating the curse of dimensionality. Finally, numerical tests on synthetic functions, as well as on energy correction integrals for water and formaldehyde molecules demonstrate the efficiency of this method using very few function evaluations as compared to other integration strategies.},

doi = {10.1016/j.cma.2018.02.026},

journal = {Computer Methods in Applied Mechanics and Engineering},

issn = {0045-7825},

number = ,

volume = 336,

place = {United States},

year = {2018},

month = {3}

}