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Towards optimal boundary integral formulations of the Poisson–Boltzmann equation for molecular electrostatics
Author(s) -
Search Stefan D.,
Cooper Christopher D.,
Wout Elwin
Publication year - 2022
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.26825
Subject(s) - preconditioner , integral equation , solver , eigenvalues and eigenvectors , boundary (topology) , mathematics , poisson's equation , convergence (economics) , electrostatics , mathematical optimization , mathematical analysis , physics , linear system , quantum mechanics , economics , economic growth
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 Calderón 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.