A regularization approach for solving the super-Gaussian Poisson-Boltzmann model with heterogeneous dielectric functions
A regularization approach for solving the super-Gaussian Poisson-Boltzmann model with heterogeneous dielectric functions
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DOI：
10.1016/j.jcp.2022.111340
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发表时间：
2022-05
期刊：
影响因子：
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通讯作者：
Siwen Wang;Yuanzhen Shao;E. Alexov;Shan Zhao
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文献类型：
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作者：
Siwen Wang;Yuanzhen Shao;E. Alexov;Shan Zhao
本文首次提出了一种正则化方法来处理非均匀介质Poisson-Boltzmann（PB）模型中的电荷奇异性。在现有的高斯和超高斯静电分析中，奇异源的狄拉克δ函数近似三线性插值，这引入了一个大的误差在潜在的解决方案，并依赖于误差消除提供可接受的估计的静电自由能。为了克服这一困难，在所提出的正则化中进行了电势和介电函数的双重分解，使得电荷奇异性可以被库仑电势解析地捕获，而反应场电势满足具有新的源项的正则化PB方程。严格的分析表明，在均匀介质中，原子中心周围的原子密度必须是超高斯密度，而不是高斯密度，这样才能保证原子中心周围的原子密度很小，从而使源项在分布意义上得到很好的定义.证明了正则化方程的适定性，阐明了弱解的正则性。在数值实现中，在计算源项时，尽可能使用解析微分而不是数值近似，给出了特殊的考虑。所提出的正则化的准确性，收敛性，效率和鲁棒性通过基准研究进行了数值验证。结果表明，正则化方法比三线性方法更精确，并且在能量估计方面具有更快的收敛性。此外，网格伪影或人工网格能量完全消除在本有限差分PB模型。
A regularization method is introduced for the first time in the literature for treating charge singularities in the heterogeneous dielectric Poisson-Boltzmann (PB) model. In the existing Gaussian and super-Gaussian electrostatic analysis, the singular sources in terms of Dirac delta functions are approximated by trilinear interpolation, which introduces a large error in potential solutions, and has to rely on an error cancellation for delivering acceptable estimates of the electrostatic free energy. To overcome this difficulty, a dual decomposition of potential and dielectric function is carried out in the proposed regularization, so that the charge singularities can be analytically captured by the Coulomb potential, while the reaction field potential satisfies a regularized PB equation with a new source term. A rigorous analysis has been conducted to show that a super-Gaussian density, instead of a Gaussian one, is required to guarantee a small neighborhood around each atom center with a nearly homogeneous dielectric medium, so that the source term can be well defined in the sense of distribution. Moreover, the well-posedness of the regularized formulation has been proved, and the regularity of the weak solution has been clarified. In the numerical implementation, special considerations are given in calculating the source term, by using analytical differentiations, instead of numerical approximations, whenever possible. The accuracy, convergence, efficiency, and robustness of the proposed regularization are numerically verified via benchmark studies. It is found that the regularization is more accurate than the trilinear method, as well as produces a faster convergence in energy estimation. Moreover, the grid artifact or artificial grid energy is completely eliminated in the present finite difference PB model.