  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 $\mathcal{N}$, where $\mathcal{N}$ 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 $N \ll \mathcal{N}$, whose solution
accurately approximates the nonlinear RPBE. The long-range potential can be
obtained by lifting the ROM solution back to the $\mathcal{N}$-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.
