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

TL;DR

This paper introduces a fast, scalable algorithm for computing the self Green's function in 3D modified Poisson-Boltzmann equations, significantly improving efficiency by leveraging hierarchical interpolative factorization and low-rank properties.

Contribution

It extends hierarchical interpolative factorization techniques to 3D for the self Green's function, achieving near-linear complexity in solving MPB equations.

Findings

Achieves $O(N ext{log}N)$ computational complexity.

Demonstrates high accuracy and efficiency in 3D numerical experiments.

Reduces dimensionality effectively using cubic edge skeletonization.

Abstract

The modified Poisson-Boltzmann (MPB) equations are often used to describe equilibrium particle distribution of ionic systems. In this paper, we propose a fast algorithm to solve 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 need to solve a high-dimensional partial differential equation. Our algorithm combines the selected inversion with hierarchical interpolative factorization for the self Green's function by extending our previous result of two dimensions. This leads to an $O(N\log N)$ algorithm through the strategical utilization of locality and low-rank characteristics of the corresponding operators. Furthermore, the estimated $O(N)$ complexity is obtained by applying cubic edge skeletonization at each level for thorough dimensionality reduction. Extensive numerical results are performed to demonstrate the accuracy and efficiency of the proposed algorithm for problems in three dimensions.