Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format

TL;DR

This paper introduces an extension of the range-separated tensor decomposition method to efficiently solve the nonlinear Poisson-Boltzmann equation, improving accuracy and handling singularities in biomolecular electrostatics calculations.

Contribution

It develops a novel regularization approach for the nonlinear PBE using tensor decomposition, enabling better treatment of singularities and continuity at the solute-solvent interface.

Findings

Enhanced accuracy of nonlinear PBE solutions

Automatic maintenance of boundary condition continuity

Effective handling of singularities in biomolecular electrostatics

Abstract

The Poisson-Boltzmann equation (PBE) is an implicit solvent continuum model for calculating the electrostatic potential and energies of ionic solvated biomolecules. However, its numerical solution remains a significant challenge due strong singularities and nonlinearity caused by the singular source terms and the exponential nonlinear terms, respectively. An efficient method for the treatment of singularities in the linear PBE was introduced in \cite{BeKKKS:18}, that is based on the RS tensor decomposition for both electrostatic potential and the discretized Dirac delta distribution. In this paper, we extend this regularization method to the nonlinear PBE. We apply the PBE only to the regular part of the solution corresponding to the modified right-hand side via extraction of the long-range part in the discretized Dirac delta distribution. The total electrostatic potential is obtained by adding the long-range solution to the directly precomputed short-range potential. The main computational benefit of the approach is the automatic maintaining of the continuity in the Cauchy data on the solute-solvent interface. The boundary conditions are also obtained from the long-range component of the precomputed canonical tensor representation of the Newton kernel. In the numerical experiments, we illustrate the accuracy of the nonlinear regularized PBE (NRPBE) over the classical variant.