Reduced basis method for the nonlinear Poisson-Boltzmann equation regularized by the range-separated canonical tensor format

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 (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, "Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format," SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415-A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, "Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format," arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, "Range-separated tensor format for many-particle modeling," SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034-A1062, 2018; B. N. Khoromskij, "Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,"I. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving 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 Ar, where Al 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 Ar-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.