Linear-Scaling Selected Inversion based on Hierarchical Interpolative Factorization for Self Green's Function for Modified Poisson-Boltzmann Equation in Two Dimensions

TL;DR

This paper introduces a fast, linear-scaling algorithm combining selected inversion and hierarchical interpolative factorization to efficiently compute the Green's function diagonal for modified Poisson-Boltzmann equations, significantly reducing computational cost.

Contribution

The paper presents a novel linear-scaling selected inversion method based on hierarchical interpolative factorization for efficiently solving MPB equations.

Findings

Achieves linear computational complexity for Green's function diagonal evaluation.

Demonstrates high accuracy and efficiency through extensive numerical tests.

Reduces the computational bottleneck in solving modified Poisson-Boltzmann equations.

Abstract

This paper studies an efficient numerical method for solving modified Poisson-Boltzmann (MPB) equations with the self Green's function as a state equation to describe electrostatic correlations in ionic systems. Previously, the most expensive point of the MPB solver is the evaluation of Green's function. The evaluation of Green's function requires solving high-dimensional partial differential equations, which is the computational bottleneck for solving MPB equations. Numerically, the MPB solver only requires the evaluation of Green's function as the diagonal part of the inverse of the discrete elliptic differential operator of the Debye-H\"uckel equation. Therefore, we develop a fast algorithm by a coupling of the selected inversion and hierarchical interpolative factorization. By the interpolative factorization, our new selected inverse algorithm achieves linear scaling to compute the diagonal of the inverse of this discrete operator. The accuracy and efficiency of the proposed algorithm will be demonstrated by extensive numerical results for solving MPB equations.