Search Stefan D, Cooper Christopher D, Van't Wout Elwin
Department of Mechanical Engineering, Universidad Técnica Federico Santa María, Valparaíso, Chile.
Centro Científico Tecnológico de Valparaíso, Universidad Técnica Federico Santa María, Valparaíso, Chile.
J Comput Chem. 2022 Apr 15;43(10):674-691. doi: 10.1002/jcc.26825. Epub 2022 Feb 24.
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.
泊松-玻尔兹曼方程为研究分子环境中的静电学提供了一种有效方法。用边界元法对其进行数值求解被广泛应用，因为复杂的分子表面能被网格精确表示，且点电荷能被明确考虑在内。事实上，文献中有几种知名的边界积分公式。这项工作给出了隐式溶剂模型边界积分表示的广义表达式，从而产生了计算静电势的新形式。此外，它还提出了一种为任何所得系统构建高效预条件器的策略，以改善线性求解器的收敛性。我们对一组公式和预条件器进行了系统的基准测试，重点关注求解时间、矩阵条件数和特征值谱。我们发现特征值聚类是矩阵条件数的一个良好指标，并表明通过缩放预条件器可以轻松控制它们。我们的结果表明，最优选择取决于问题规模，对于小分子，更简单的直接公式求解速度最快，但对于较大问题，更复杂的第二类方程效果更好。我们还为第一类公式提出了一种快速卡尔德隆预条件器，它在未来分析中表现出良好的前景。这项工作为针对给定问题选择泊松-玻尔兹曼方程最方便的边界积分公式奠定了基础。
