Abstract:
In this paper, we apply the range-separated (RS) tensor format [6] for the construc-
tion of new regularization scheme for the Poisson-Boltzmann equation (PBE) describing
the electrostatic potential in biomolecules. In our approach, we use the RS tensor rep-
resentation 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 ac-
curately capture the solution singularities in the numerical approximation as compared
to the new approach by the RS tensor format.
