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

Peter Benner, Venera Khoromskaia, Boris Khoromskij, Cleophas Kweyu, Matthias Stein

TL;DR

The paper addresses the challenge of computing biomolecular electrostatics by solving the Poisson–Boltzmann equation with singular Dirac sources. It introduces the range-separated (RS) tensor format to decompose the total potential into a long-range low-rank part and a short-range localized part, enabling a regularized right-hand side and a single coarse-grid solve for the smooth component while combining a precomputed short-range RS tensor. The authors develop a splitting scheme based on a RS discretization of the Dirac delta, demonstrating that the long-range piece can be solved independently with a localized RHS and that the total potential is obtained by adding the short-range RS contribution; numerical tests on Born ions and biomolecules show improved accuracy over classical Poisson–Boltzmann treatments, especially near singularities. The approach supports finer grids and offers potential extensions to nonlinear PBE and multiple molecular orientations, with practical implications for fast, accurate electrostatics in large biomolecular systems.

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.

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

TL;DR

The paper addresses the challenge of computing biomolecular electrostatics by solving the Poisson–Boltzmann equation with singular Dirac sources. It introduces the range-separated (RS) tensor format to decompose the total potential into a long-range low-rank part and a short-range localized part, enabling a regularized right-hand side and a single coarse-grid solve for the smooth component while combining a precomputed short-range RS tensor. The authors develop a splitting scheme based on a RS discretization of the Dirac delta, demonstrating that the long-range piece can be solved independently with a localized RHS and that the total potential is obtained by adding the short-range RS contribution; numerical tests on Born ions and biomolecules show improved accuracy over classical Poisson–Boltzmann treatments, especially near singularities. The approach supports finer grids and offers potential extensions to nonlinear PBE and multiple molecular orientations, with practical implications for fast, accurate electrostatics in large biomolecular systems.

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.

Paper Structure

This paper contains 13 sections, 1 theorem, 54 equations, 13 figures, 3 tables.

Table of Contents

  1. Introduction
  2. On existing solution decomposition techniques
  3. Rank-structured approximation of electrostatic potentials
  4. Canonical tensor approximation of the Newton kernel
  5. Properties of the RS tensor format
  6. Application of RS tensor format for solving PBE
  7. Tensor based regularization scheme for the PBE
  8. The new RS tensor based splitting scheme
  9. Discussion of the computational scheme
  10. Numerical Tests
  11. Solutions to the modified Poisson equation on a sequence of fine grids
  12. Accurate representation of the long-range electrostatic potential by the RS tensor
  13. Conclusions

Key Result

Proposition 4.1

Let the effective support of the short-range components in the reference potential ${\bf P}_R$ be chosen not larger than $\gamma/2$. Then the interface conditions in the regularized formulation of the PBE in (eqn:Regul_PBE) depend only on the low-rank long-range component in the free-space electrost

Figures (13)

  • Figure 2.1: Solute and solvent regions in the computational domain for the PBE.
  • Figure 3.1: The full free space electrostatic potential of a system with 782 particles (left) and the sum of their short range contributions (right).
  • Figure 3.2: The low-rank tensor representation of the long-range part in the electrostatic potential of 782 charged particles (left) and the error of the canonical rank reduction.
  • Figure 4.1: The long-range part of the Dirac delta $\boldsymbol{\delta}_l$ on an $n^{\otimes 3}$ 3D grid, $n=256$.
  • Figure 5.1: The free space potential for the Born ion computed by the APBS (left), the FDM solver (middle) and the corresponding error (right).
  • ...and 8 more figures

Theorems & Definitions (1)

  • Proposition 4.1