Computing electrostatic potentials using regularization based on the range-separated tensor format
Peter Benner, Venera Khoromskaia, Boris N. Khoromskij, Cleophas Kweyu, and Matthias Stein
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Submission date: 29. Jan. 2019
MSC-Numbers: 65F30, 65F50, 65N35, 65F10
Keywords and phrases: Poisson-Boltzmann equation, coulomb potential, collective electrostatic potential, long-range many-particle interactions, low-rank tensor decompositions, range-separated tensor format
Link to arXiv:See the arXiv entry of this preprint.
In this paper, we apply the range-separated (RS) tensor format  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  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.