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

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(NlogN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(N\log N)$$\end{document} 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.