Journal of the Korea Institute of Information and Communication Engineering (한국정보통신학회논문지)

The Korea Institute of Information and Commucation Engineering (한국정보통신학회)
권호 표지
DOI QR코드

DOI QR Code

An algebraic multigrids based prediction of a numerical solution of Poisson-Boltzmann equation for a generation of deep learning samples

딥러닝 샘플 생성을 위한 포아즌-볼츠만 방정식의 대수적 멀티그리드를 사용한 수치 예측

  • Shin, Kwang-Seong (Department of Digital Content Engineering, Wonkwang University) ;
  • Jo, Gwanghyun (Department of Mathematics, Kunsan National University)
  • Received : 2021.11.29
  • Accepted : 2021.12.13
  • Published : 2022.02.28
Download PDF
⟨ Previous Next ⟩

Abstract

Poisson-Boltzmann equation (PBE) is used to model problems arising from various disciplinary including bio-pysics and colloid chemistry. Therefore, to predict a numerical solution of PBE is an important issue. The authors proposed deep learning based methods to solve PBE while the computational time to generate finite element method (FEM) solutions were bottlenecks of the algorithms. In this work, we shorten the generation time of FEM solutions in two directions. First, we experimentally find certain penalty parameter in a bilinear form. Second, we applied algebraic multigrids methods to the algebraic system so that condition number is bounded regardless of the meshsize. In conclusion, we have reduced computation times to solve algebraic systems for PBE. We expect that algebraic multigrids methods can be further employed in various disciplinary to generate deep learning samples.

포아즌 볼츠만 방정식 (Poisson-Boltzmann equation, PBE)은 생물물리, 콜로이드 화학 등에서 등장하는 문제들을 모델링하는데 사용되는 방정식이다. 따라서 PBE의 수치해를 효율적으로 예측하는 것은 중요한 이슈이다. 저자들은 기존의 연구에서 PBE를 풀기위한 딥러닝 방법을 제안하였으나, 딥러닝을 훈련하기 위한 샘플을 생성하는 시간이 컸다는 어려움이 있었다. 본 논문에서는 FEM 수치해를 생성하는데 걸리는 시간을 줄이는 두가지 방안을 마련하였다. 첫째로 대수 방정식을 만들 때 bilinar form에 포함되는 penalty 파라메터를 실험적으로 조정하였다. 두 번째로, 대수적멀티그리드 기법을 활용하여 대수 방정식의 컨디션 넘버를 meshsize와 무관하게 만들었다. 따라서 PBE 방정식의 대수 방정식을 풀 때 계산 시간을 효과적으로 줄였다. 이러한 대수적 멀티그리드를 사용한 방법은 다양한 분야에서 딥러닝의 샘플을 생성하는데 효과적으로 활용될 수 있을 것으로 기대된다.

Keywords

Acknowledgement

The first author is supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (NO. 2020R 1C1C1A01005396) and second author is supported by NRF grant funded by MSIT (NO. NRF-2019R1G1A1087290).

References

  1. G. Borlesk and Y. C. Zhou, "Enriched gradient recovery for interface solutions of the Poisson-Boltzmann equation," Journal of Computational Physics, vol. 421, Article ID: 109725, Nov. 2020.
  2. V. Ramm, J. H. Chaudry, and C. D. Cooper, "Efficient mesh refinement for the Poisson-Boltzmann equation with boundary elements," Journal of Computational Chemistry, vol. 42, no. 12, pp. 855-869, Mar. 2021. https://doi.org/10.1002/jcc.26506
  3. S. Wang, E. Alexov, and S. Zhao, "On regularization of charge singularities in solving the Poisson-Boltzmann equation with a smooth solute-solvent boundary," Mathematical Biosciences and Engineering, vol. 18, no. 2, pp. 1370-1405, Jun. 2021. https://doi.org/10.3934/mbe.2021072
  4. I. Kwon and D. Y. Kwak, "Discontinuous bubble immersed finite element method for Poisson-Boltzmann equation," Communications in Computational Physics, vol. 25, no. 3, pp. 928-946, Aug. 2019.
  5. I. Kwon, D. Y. Kwak, and G. Jo, "Discontinuous bubble immersed finite element method for Poisson-Boltzmann-Nernst-Plank model," Journal of Computational Physics, vol. 438, Article ID: 110370, Aug. 2021.
  6. K. In, G. Jo, and K. S. Shin, "A deep neural netwok based on ResNet for predicting solutions of Poisson-Boltzmann eequation," Electronics, vol. 10, no. 21, pp. 2627, Oct. 2021. https://doi.org/10.3390/electronics10212627
  7. J. Xu and L. Zikatanov, "Algebraic multigrid methods," Acta numerica, vol. 26, pp. 591-721, May. 2017. https://doi.org/10.1017/S0962492917000083
  8. S. H. Chou, D. Y. Kwak, and K. T. Wee, "Optimal convergence analysis of an immersed interface finite element method," Advances in Computational Mathematics, vol. 33, no. 2, pp. 149-168, Mar. 2009. https://doi.org/10.1007/s10444-009-9122-y