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

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 Calderon 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.

Automated summary

The Poisson-Boltzmann equation is a useful tool to study electrostatics in molecular systems. This paper presents a generalized expression of the boundary integral representation of the implicit solvent model, which leads to new ways of computing the electrostatic potential. It also proposes a strategy to build efficient preconditioners for the resulting systems, improving the convergence of the linear solver. The authors perform benchmark tests on different formulations and preconditioners, focusing on time to solution, matrix conditioning, and eigenvalue spectrum. The results suggest that the optimal choice depends on the problem size, with simpler direct formulations being faster for small molecules and more involved second-kind equations being better for larger problems. The paper also introduces a fast Calderon preconditioner for first-kind formulations, which shows promise for future analysis.