Regularization of poisson–boltzmann type equations with singular source terms using the range-separated tensor format

Abstract

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 compo- nents 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 represen- tation 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 lin- ear 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 regu- lar right-hand side represented by a low-rank RS tensor with a controllable precision. Thus, we elimi- nate 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 precom- puted 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.