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

In this paper, we present a new regularization scheme for the linearized Poisson--Boltzmann equation (PBE) which models the electrostatic potential of biomolecules in a solvent. This scheme is based on the splitting of the target potential into the short- and long-range components localized in the molecular region by using the range-separated (RS) tensor format P. Benner, V. Khoromskaia, and B. N. Khoromskij, SIAM J. Comput., 2 (2018), pp. A1034--A1062 for representation of the discretized multiparticle Dirac delta B. N. Khoromskij, J. Comput. Phys., 401 (2020), 108998 constituting the highly singular right-hand side in the PBE. From the computational point of view our regularization approach requires only the modification of the right-hand side in the PBE so that it can be implemented within any open-source grid-based software package for solving PBE that already includes some FEM/FDM disretization scheme for elliptic PDE and solver for the arising linear system of equations. The main computational benefits are twofold. First, one applies the chosen PBE solver only for the smooth long-range (regularized) part of the collective potential with the regular right-hand side represented by a low-rank RS tensor with a controllable precision. Thus, we eliminate the numerical treatment of the singularities in the right-hand side and do not change the interface and boundary conditions. And second, the elliptic PDE need not be solved for the singular part in the right-hand side at all, since the short-range part of the target potential of the biomolecule is precomputed independently on a computational grid by simple one-dimensional tensor operations. The total potential is then obtained by adding the numerical solution of the PBE for the smooth long-range part to the directly precomputed tensor representation for the short-range contribution. Numerical tests illustrate that the new regularization scheme, implemented by a simple modification of the right-hand side in the chosen PBE solver, improves the accuracy of the approximate solution on rather coarse grids. The scheme also demonstrates good convergence behavior on a sequence of refined grids.