Semi-linear elliptic Partial Differential Equations (PDEs) such as the non-linear Poisson Boltzmann Equation (nPBE) is highly relevant for non-linear electrostatics in computational biology and chemistry. It is of particular importance for modeling potential fields from molecules in solvents or plasmas with stochastic fluctuations. The extensive applications include ones in condensed matter and solid state physics, chemical physics, electrochemistry, biochemistry, thermodynamics, statistical mechanics, and materials science, among others. In this paper we study the complex analytic properties of semi-linear elliptic Partial Differential Equations with respect to random fluctuations on the domain. We first prove the existence and uniqueness of the nPBE on a bounded domain in $\mathbb{R}^3$. This proof relies on the application of a contraction mapping reasoning, as the standard convex optimization argument for the deterministic nPBE no longer applies. Using the existence and uniqueness result we subsequently show that solution to the nPBE admits an analytic extension onto a well defined region in the complex hyperplane with respect to the number of stochastic variables. Due to the analytic extension, stochastic collocation theory for sparse grids predict algebraic to sub-exponential convergence rates with respect to the number of knots. A series of numerical experiments with sparse grids is consistent with this prediction and the analyticity result. Finally, this approach readily extends to a wide class of semi-linear elliptic PDEs.

翻译：半线性椭圆型偏微分方程（如非线性泊松-玻尔兹曼方程（nPBE））在计算生物学和化学的非线性静电学中具有高度相关性，尤其适用于模拟溶剂或等离子体中具有随机涨落的分子势场。其广泛应用涵盖凝聚态物理、固体物理、化学物理、电化学、生物化学、热力学、统计力学及材料科学等领域。本文研究域上随机涨落下半线性椭圆型偏微分方程的复解析性质。首先，我们证明$\mathbb{R}^3$中有界域上nPBE解的存在唯一性。该证明依赖于压缩映射推理的应用，因为确定性nPBE的标准凸优化论证不再适用。基于存在唯一性结果，我们进一步证明nPBE解关于随机变量数目在复超平面的明确定义区域内存在解析延拓。由于解析延拓性质，稀疏网格随机配置理论预测解的收敛速率关于网格节点数呈代数阶至次指数阶。一系列基于稀疏网格的数值实验与该预测及解析性结果一致。最后，该方法可自然推广至更广泛的半线性椭圆型偏微分方程类。