The Poisson-Boltzmann equation (PBE) is an implicit solvent continuum model for calculating the electrostatic potential and energies of ionic solvated biomolecules. However, its numerical solution remains a significant challenge due strong singularities and nonlinearity caused by the singular source terms and the exponential nonlinear terms, respectively. An efficient method for the treatment of singularities in the linear PBE was introduced in \cite{BeKKKS:18}, that is based on the RS tensor decomposition for both electrostatic potential and the discretized Dirac delta distribution. In this paper, we extend this regularization method to the nonlinear PBE. We apply the PBE only to the regular part of the solution corresponding to the modified right-hand side via extraction of the long-range part in the discretized Dirac delta distribution. The total electrostatic potential is obtained by adding the long-range solution to the directly precomputed short-range potential. The main computational benefit of the approach is the automatic maintaining of the continuity in the Cauchy data on the solute-solvent interface. The boundary conditions are also obtained from the long-range component of the precomputed canonical tensor representation of the Newton kernel. In the numerical experiments, we illustrate the accuracy of the nonlinear regularized PBE (NRPBE) over the classical variant.

翻译：泊松-玻尔兹曼方程（PBE）是一种用于计算离子溶剂化生物分子静电势和能量的隐式溶剂连续介质模型。然而，由于奇异源项和指数非线性项分别导致的强奇异性与非线性特性，其数值求解仍面临重大挑战。针对线性PBE中奇异性处理的有效方法已在文献\cite{BeKKKS:18}中提出，该方法基于静电势和离散化狄拉克δ分布的张量RS分解。本文将该正则化方法推广至非线性PBE。通过从离散化狄拉克δ分布中提取长程部分，仅对对应修正右端项的正则解部分应用PBE。总静电势由长程解与直接预计算的短程势叠加获得。该方法的主要计算优势在于自动保持溶质-溶剂界面柯西数据的连续性。边界条件亦从牛顿核预计算规范张量表示的长程分量中提取。数值实验验证了非线性正则化PBE（NRPBE）相较于经典变体的精度优势。