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

Area of Science:

• Computational electrostatics
• Mathematical physics
• Computational chemistry and biology

Background:

• The nonlinear Poisson-Boltzmann equation (nPBE) is crucial for modeling electrostatic potential in biological and chemical systems.
• Quantifying solution uncertainty under coefficient variations is essential for reliable predictions.

Purpose of the Study:

• To establish the existence and uniqueness of solutions for the complexified nPBE.
• To demonstrate the analyticity of these solutions.
• To enable efficient statistical uncertainty quantification using numerical methods.

Main Methods:

• Establishing existence and uniqueness of complexified nPBE solutions via contraction mapping.
• Demonstrating analytic extensions of solutions in the complex hyperplane.
• Applying sparse grids for efficient approximation of high-dimensional integrals.

Main Results:

• Existence and uniqueness of complexified nPBE solutions are proven.
• Solutions admit analytic extensions, facilitating statistical analysis.
• Sparse grids achieve efficient, accurate approximations of relevant statistics.

Conclusions:

• The analyticity of complexified nPBE solutions allows for robust statistical uncertainty quantification.
• Sparse grid methods provide an efficient computational approach for these problems.
• Numerical experiments validate the theoretical error bounds.