Uncertainty quantification and complex analyticity of the nonlinear Poisson-Boltzmann equation for the interface problem with random domains

TL;DR

This paper proves the analyticity of solutions to the nonlinear Poisson-Boltzmann equation with respect to domain perturbations, enabling efficient uncertainty quantification methods for complex, high-dimensional random domains.

Contribution

It establishes the first analyticity results for nonlinear elliptic PDEs like the NPBE under domain perturbations, extending previous linear results and providing bounds on the analyticity region.

Findings

Analyticity of NPBE solutions with respect to domain perturbations is proven.

A priori bounds on the analyticity region are derived.

Application to trypsin molecule confirms the theoretical convergence rates.

Abstract

The nonlinear Poisson-Boltzmann equation (NPBE) is an elliptic partial differential equation used in applications such as protein interactions and biophysical chemistry (among many others). It describes the nonlinear electrostatic potential of charged bodies submerged in an ionic solution. The kinetic presence of the solvent molecules introduces randomness to the shape of a protein, and thus a more accurate model that incorporates these random perturbations of the domain is analyzed to compute the statistics of quantities of interest of the solution. When the parameterization of the random perturbations is high-dimensional, this calculation is intractable as it is subject to the curse of dimensionality. However, if the solution of the NPBE varies analytically with respect to the random parameters, the problem becomes amenable to techniques such as sparse grids and deep neural networks. In this paper, we show analyticity of the solution of the NPBE with respect to analytic perturbations of the domain by using the analytic implicit function theorem and the domain mapping method. Previous works have shown analyticity of solutions to linear elliptic equations but not for nonlinear problems. We further show how to derive \emph{a priori} bounds on the size of the region of analyticity. This method is applied to the trypsin molecule to demonstrate that the convergence rates of the quantity of interest are consistent with the analyticity result. Furthermore, the approach developed here is sufficiently general enough to be applied to other nonlinear problems in uncertainty quantification.