Analytic regularity of strong solutions for the complexified stochastic nonlinear Poisson-Boltzmann Equation

The nonlinear Poisson-Boltzmann equation (nPBE) is a fundamental partial differential equation (PDE) in electrostatics, widely used in computational biology and chemistry to model potential fields in solvents or plasmas. In this paper, we consider the problem of quantifying the statistical uncertainty of the stochastic nPBE solution under random variations in its coefficients. We establish the existence and uniqueness of solutions of the complexified nPBE using a contraction mapping argument, as conventional convex optimization techniques for the real-valued nPBE do not naturally extend to the complex setting. Using the existence and uniqueness result, we demonstrate that the solutions admit analytic extensions over a well-defined region in the complex hyperplane The analyticity makes the computation for statistics of real-valued quantities of interest amenable to numerical techniques such as sparse grids. Sparse grids are applied to uniformly approximate analytic functions with algebraic to sub-exponential error with respect to the number of knots, thus allowing for efficient approximations of high-dimensional integrals. Our numerical experiments confirm the predicted error behavior.