正则化非线性Poisson-Boltzmann方程的约化基方法
Reduced basis method for the nonlinear Poisson-Boltzmann equation regularized by the range-separated canonical tensor format
摘要：
Poisson-Boltzmann方程（PBE）是一个基本的隐式溶剂连续模型，用于计算大离子溶剂化生物分子的静电势。然而，由于其强奇异性和非线性，其数值解面临着严峻的挑战。在文献[1，2]中，强奇异性的影响通过应用距离分离（RS）正则张量格式[3，4]来构造PBE的解分解格式而被消除. RS张量格式允许导出狄拉克δ分布的平滑近似，以获得正则化PBE（RPBE）模型。然而，求解RPBE由于其高维数$\mathcal{N}$（其中$\mathcal{N}$总是以百万计）而仍然需要计算。在这项研究中，我们提出了应用减少基方法（RBM）和（离散）经验插值方法（（D）EIM）的RPBE，以建立一个降阶模型（ROM）的低维$N \ll \mathcal{N}$，其解决方案精确逼近的非线性RPBE。长程势可以通过将ROM解提升回$\mathcal{N}$ -空间来获得，而短程势由于RS张量格式而直接解析地预先计算。两者之和提供总静电势。主要的计算好处是避免了计算奇异静电势的数值逼近。我们证明了在数值实验中，减少的基础（RB）近似的非线性RPBE（NRPBE）的解决方案和相应的计算节省了经典的非线性PBE（NPBE），以及在RBM被施加到经典NPBE的准确性和有效性。
英文摘要：
The Poisson-Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In [1,2], the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format [3,4] to construct a solution decomposition scheme for the PBE. The RS tensor format allows to derive a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension $\mathcal{N}$, where $\mathcal{N}$ is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension $N \ll \mathcal{N}$, whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the $\mathcal{N}$-space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.
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