Communications on Applied Mathematics and Computation >

2025 , Vol. 7 >Issue 2: 536 - 561

DOI: https://doi.org/10.1007/s42967-023-00352-z

Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations

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  • 1 School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China;
    2 School of Mathematical Sciences, MOE-LSC and CMA-Shanghai, Shanghai Jiao Tong University, Shanghai 200240, China;
    3 Department of Mathematics and Department of Computer Science, University of Maryland, College Park, MD 20742, USA

Received date: 2023-07-15

  Revised date: 2023-11-14

  Accepted date: 2023-11-20

  Online published: 2025-04-21

Supported by

Y. T. and Z. X. acknowledge the financial support from the National Natural Science Foundation of China (Grant Nos. 12071288 and 12325113), the Science and Technology Commission of Shanghai Municipality of China (Grant No. 21JC1403700), and Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA25010403). H. Y. thanks the support of the US National Science Foundation under awards DMS-2244988 and DMS-2206333.

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Abstract

The modified Poisson-Boltzmann (MPB) equations are often used to describe the equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve the MPB equations with the self Green’s function as the self-energy in three dimensions, where the solution of the self Green’s function poses a computational bottleneck due to the requirement of solving a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green’s function, building upon our previous result of two dimensions. This approach yields an algorithm with a complexity of O(N log N) by strategically leveraging the locality and low-rank characteristics of the corresponding operators. Additionally, the theoretical O(N) complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are conducted to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.

Key words： Selected inversion; Hierarchical interpolative factorization; Linear scaling; Self Green’s function; Modified Poisson-Boltzmann (MPB) equations

Cite this article

Yihui Tu, Zhenli Xu, Haizhao Yang . Hierarchical Interpolative Factorization for Self Green’s Function in 3D Modified Poisson-Boltzmann Equations[J]. Communications on Applied Mathematics and Computation, 2025 , 7(2) : 536 -561 . DOI: 10.1007/s42967-023-00352-z

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