Towards optimal boundary integral formulations of the Poisson-Boltzmann equation for molecular electrostatics

TL;DR

This paper explores various boundary integral formulations of the Poisson-Boltzmann equation, introducing new forms and preconditioning strategies to optimize computational efficiency in molecular electrostatics simulations.

Contribution

It presents a generalized boundary integral expression, develops efficient preconditioners, and systematically benchmarks formulations to identify optimal choices based on problem size.

Findings

Eigenvalue clustering indicates matrix conditioning.

Preconditioners can be scaled to improve convergence.

Optimal formulation depends on molecule size.

Abstract

The Poisson-Boltzmann equation offers an efficient way to study electrostatics in molecular settings. Its numerical solution with the boundary element method is widely used, as the complicated molecular surface is accurately represented by the mesh, and the point charges are accounted for explicitly. In fact, there are several well-known boundary integral formulations available in the literature. This work presents a generalized expression of the boundary integral representation of the implicit solvent model, giving rise to new forms to compute the electrostatic potential. Moreover, it proposes a strategy to build efficient preconditioners for any of the resulting systems, improving the convergence of the linear solver. We perform systematic benchmarking of a set of formulations and preconditioners, focusing on the time to solution, matrix conditioning, and eigenvalue spectrum. We see that the eigenvalue clustering is a good indicator of the matrix conditioning, and show that they can be easily manipulated by scaling the preconditioner. Our results suggest that the optimal choice is problem-size dependent, where a simpler direct formulation is the fastest for small molecules, but more involved second-kind equations are better for larger problems. We also present a fast Calder\'on preconditioner for first-kind formulations, which shows promising behavior for future analysis. This work sets the basis towards choosing the most convenient boundary integral formulation of the Poisson-Boltzmann equation for a given problem.