Computing electrostatic potentials using regularization based on the range-separated tensor format

TL;DR

This paper introduces a novel regularization scheme for solving the Poisson-Boltzmann equation in biomolecular electrostatics using range-separated tensor formats, enabling efficient and accurate computation of electrostatic potentials.

Contribution

The authors develop a new regularization method based on RS tensor formats for the PBE, improving computational efficiency and accuracy in biomolecular electrostatics simulations.

Findings

Efficient RS tensor-based regularization improves solution accuracy.

Localization of the source term simplifies the numerical solution.

Finer grid computations are enabled with the proposed method.

Abstract

In this paper, we apply the range-separated (RS) tensor format [6] for the construction of new regularization scheme for the Poisson-Boltzmann equation (PBE) describing the electrostatic potential in biomolecules. In our approach, we use the RS tensor representation to the discretized Dirac delta [21] to construct an efficient RS splitting of the PBE solution in the solute (molecular) region. The PBE then needs to be solved with a regularized source term, and thus black-box solvers can be applied. The main computational benefits are due to the localization of the modified right-hand side within the molecular region and automatic maintaining of the continuity in the Cauchy data on the interface. Moreover, this computational scheme only includes solving a single system of FDM/FEM equations for the smooth long-range (i.e., regularized) part of the collective potential represented by a low-rank RS-tensor with a controllable precision. The total potential is obtained by adding this solution to the directly precomputed rank-structured tensor representation for the short-range contribution. Enabling finer grids in PBE computations is another advantage of the proposed techniques. In the numerical experiments, we consider only the free space electrostatic potential for proof of concept. We illustrate that the classical Poisson equation (PE) model does not accurately capture the solution singularities in the numerical approximation as compared to the new approach by the RS tensor format.