This work proposes a fast iterative method for local steric Poisson--Boltzmann (PB) theories, in which the electrostatic potential is governed by the Poisson's equation and ionic concentrations satisfy equilibrium conditions. To present the method, we focus on a local steric PB theory derived from a lattice-gas model, as an example. The advantages of the proposed method in efficiency are achieved by treating ionic concentrations as scalar implicit functions of the electrostatic potential, though such functions are only numerically achievable. The existence, uniqueness, boundness, and smoothness of such functions are rigorously established. A Newton iteration method with truncation is proposed to solve a nonlinear system discretized from the generalized PB equations. The existence and uniqueness of the solution to the discretized nonlinear system are established by showing that it is a unique minimizer of a constructed convex energy. Thanks to the boundness of ionic concentrations, truncation bounds for the potential are obtained by using the extremum principle. The truncation step in iterations is shown to be energy and error decreasing. To further speed-up computations, we propose a novel precomputing-interpolation strategy, which is applicable to other local steric PB theories and makes the proposed methods for solving steric PB theories as efficient as for solving the classical PB theory. Analysis on the Newton iteration method with truncation shows local quadratic convergence for the proposed numerical methods. Applications to realistic biomolecular solvation systems reveal that counterions with steric hindrance stratify in an order prescribed by the parameter of ionic valence-to-volume ratio. Finally, we remark that the proposed iterative methods for local steric PB theories can be readily incorporated in well-known classical PB solvers.

翻译：本文提出了一种用于局部空间位阻Poisson-Boltzmann（PB）理论的快速迭代方法，其中静电势由泊松方程控制，离子浓度满足平衡条件。为呈现该方法，我们以源自晶格气体模型的局部空间位阻PB理论为例。所提方法的高效性优势源于将离子浓度视为静电势的标量隐函数，尽管此类函数仅能通过数值方式实现。我们严格证明了此类函数的存在性、唯一性、有界性和光滑性。提出一种带截断的牛顿迭代法求解广义PB方程离散后的非线性系统。通过证明离散非线性系统解是构造凸能量泛函的唯一极小化子，确立了其存在性与唯一性。基于离子浓度的有界性，利用极值原理获得了势函数的截断界。截断迭代步骤被证明能降低能量与误差。为进一步加速计算，我们提出一种新颖的预计算-插值策略，该策略可适用于其他局部空间位阻PB理论，并使求解空间位阻PB理论的方法与求解经典PB理论同样高效。对带截断牛顿迭代法的分析表明，所提数值方法具有局部二次收敛性。应用于实际生物分子溶剂化系统时发现，具有空间位阻效应的反离子按照离子价态-体积比参数规定的顺序分层排列。最后，我们指出所提局部空间位阻PB理论的迭代方法可便捷地集成至现有经典PB求解器中。