Regularization of Poisson--Boltzmann Type Equations with Singular Source Terms Using the Range-Separated Tensor Format

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.

Reduced Higher Order SVD: ubiquitous rank-reduction method in tensor-based scientific computing

“…The smooth long-range part is calculated by solving the PBE with the modified right-hand side by using the RS decomposition of the Dirac delta, so that now it does not contain singularities. We refer to papers [4,38] describing the approach in details.…”

“…At present there is an active research toward further progress of tensor numerical methods in scientific computing [59,60,36,40,50]. In particular, there are considerable achievements of tensor-based approaches in computational chemistry [26,27,34,12,25], in bio-molecular modeling [3,4,31,38], in optimal control problems (including the case of fractional control) [22,11,13,53], and in many other fields [2,5,15,42,7,41].…”

“…The RHOSVD is the basic tool for the construction of the range-separated (RS) tensor format introduced in [3] for the low-rank tensor representation of the bio-molecular long-range electrostatic potentials. Recent example on the RS representation of the multi-centered Dirac delta function [31] paves the way for efficient solution decomposition scheme introduced for the Poisson-Boltzmann equation [4,38].…”

Range-separated tensor decomposition of the discretized Dirac delta and elliptic operator inverse

“…The above representation was also applied in [4]. Figure 3.2 shows the cross-sections of the Newton kernel and of the Dirac delta computed at the plane z = 0 using n × n × n 3D Cartesian grids with n = 1024.…”

“…The latter equation may be incorporated into the more complicated elliptic equation with jumping coefficients across the irregular interface and with possible nonlinearity (see Fig. 1.1 and the related example of the Poisson-Boltzmann equation in §3.3 also considered in [4]). The RS decomposition of the multiparticle Dirac delta introduced in this paper provides the efficient way for regularization of the solution scheme for the arising elliptic boundary value problem.…”