# An *O*( *N*) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles

## Abstract

Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires *O*( *N* ^{2}) operations, where *N* is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method (FMM) to evaluate the integrals in *O*( *N*) operations, with *O*( *N*) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the context of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. Lastly, the results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accuratemore »

- Authors:

- Argonne National Lab. (ANL),Lemont, IL (United States)
- Univ. of Chicago, Chicago, IL (United States)
- Argonne National Lab. (ANL), Lemont, IL (United States)
- Stanford Univ., Stanford, CA (United States)
- Argonne National Lab. (ANL), Lemont, IL (United States); Univ. of Chicago, Chicago, IL (United States); KCG Holdings, Inc. (United States)
- Univ. of Chicago, Chicago, IL (United States); Univ. Nacional de Colombia-Medellin, Medellin (Columbia)
- Argonne National Lab. (ANL),Lemont, IL (United States); Univ. of Chicago, Chicago, IL (United States)
- Argonne National Lab. (ANL),Lemont, IL (United States); Northwestern-Argonne Institute for Science and Engineering, Evanston, IL (United States)

- Publication Date:

- Research Org.:
- Argonne National Lab. (ANL), Argonne, IL (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences and Engineering Division; USDOE

- OSTI Identifier:
- 1339573

- Alternate Identifier(s):
- OSTI ID: 1420630

- Grant/Contract Number:
- AC02-06CH11357

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Journal of Chemical Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 6; Journal ID: ISSN 0021-9606

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS

### Citation Formats

```
Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., and Heinonen, Olle.
```*An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles*. United States: N. p., 2016.
Web. doi:10.1063/1.4960436.

```
Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., & Heinonen, Olle.
```*An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles*. United States. doi:10.1063/1.4960436.

```
Jiang, Xikai, Li, Jiyuan, Zhao, Xujun, Qin, Jian, Karpeev, Dmitry, Hernandez-Ortiz, Juan, de Pablo, Juan J., and Heinonen, Olle. Wed .
"An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles". United States.
doi:10.1063/1.4960436. https://www.osti.gov/servlets/purl/1339573.
```

```
@article{osti_1339573,
```

title = {An O(N) and parallel approach to integral problems by a kernel-independent fast multipole method: Application to polarization and magnetization of interacting particles},

author = {Jiang, Xikai and Li, Jiyuan and Zhao, Xujun and Qin, Jian and Karpeev, Dmitry and Hernandez-Ortiz, Juan and de Pablo, Juan J. and Heinonen, Olle},

abstractNote = {Large classes of materials systems in physics and engineering are governed by magnetic and electrostatic interactions. Continuum or mesoscale descriptions of such systems can be cast in terms of integral equations, whose direct computational evaluation requires O(N2) operations, where N is the number of unknowns. Such a scaling, which arises from the many-body nature of the relevant Green's function, has precluded wide-spread adoption of integral methods for solution of large-scale scientific and engineering problems. In this work, a parallel computational approach is presented that relies on using scalable open source libraries and utilizes a kernel-independent Fast Multipole Method (FMM) to evaluate the integrals in O(N) operations, with O(N) memory cost, thereby substantially improving the scalability and efficiency of computational integral methods. We demonstrate the accuracy, efficiency, and scalability of our approach in the context of two examples. In the first, we solve a boundary value problem for a ferroelectric/ferromagnetic volume in free space. In the second, we solve an electrostatic problem involving polarizable dielectric bodies in an unbounded dielectric medium. Lastly, the results from these test cases show that our proposed parallel approach, which is built on a kernel-independent FMM, can enable highly efficient and accurate simulations and allow for considerable flexibility in a broad range of applications.},

doi = {10.1063/1.4960436},

journal = {Journal of Chemical Physics},

number = 6,

volume = 145,

place = {United States},

year = {Wed Aug 10 00:00:00 EDT 2016},

month = {Wed Aug 10 00:00:00 EDT 2016}

}

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