Abstract:
The Poisson-Boltzmann equation (PBE) is a fundamental implicit solvent continuum
model for calculating the electrostatic potential of large ionic solvated
biomolecules. However, its numerical solution encounters severe challenges arising
from its strong singularity and nonlinearity. In [1, 2], the effect of strong singularities
was eliminated by applying the range-separated (RS) canonical tensor format [3, 4] to
construct a solution decomposition scheme for the PBE. The RS tensor format allows
to derive a smooth approximation to the Dirac delta distribution in order to obtain a
regularized PBE (RPBE) model. However, solving the RPBE is still computationally
demanding due to its high dimension N, where N is always in the millions. In
this study, we propose to apply the reduced basis method (RBM) and the (discrete)
empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced
order model (ROM) of low dimension N N, whose solution accurately approximates
the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM
solution back to the N-space while the short-range potential is directly precomputed
analytically, thanks to the RS tensor format. The sum of both provides the total
electrostatic potential. The main computational benefit is the avoidance of computing
the numerical approximation of the singular electrostatic potential. We demonstrate
in the numerical experiments, the accuracy and efficacy of the reduced basis (RB)
approximation to the nonlinear RPBE (NRPBE) solution and the corresponding
computational savings over the classical nonlinear PBE (NPBE) as well as over the
RBM being applied to the classical NPBE
