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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
12/2019

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

Peter Benner, Venera Khoromskaia, Boris N. Khoromskij, Cleophas Kweyu and Matthias Stein

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.

Received:
29.01.19
Published:
29.01.19
MSC Codes:
65F30, 65F50, 65N35, 65F10
Keywords:
Poisson-Boltzmann equation, coulomb potential, collective electrostatic potential, long-range many-particle interactions, low-rank tensor decompositions, range-separated tensor format

Related publications

inJournal
2021 Repository Open Access
Peter Benner, Venera Khoromskaia, Boris N. Khoromskij, Cleophas Kweyu and Matthias Stein

Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format

In: SIAM journal on scientific computing, 43 (2021) 1, A415-A445