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.
翻译:Poisson-Boltzmann 方程式(PBE)是计算大型离子溶解生物分子的电静电潜力的基本隐含溶剂连续模式。然而,其数字解决方案因其强异性和非线性而面临严峻的挑战。在[1,2]中,强异性效应的效应被消除,因为采用了范围分离的(RS)卡萨多尔格式[3,4]来为PBE构建一个溶解分解方案。RS 方格格式允许对Dirac 三角体分布进行平稳近似,以便获得一个非常规化的 PBE(RPBE) 模型。然而,RPBE的解算仍然在计算上要求很高,因为其高维性 $madcalcalcal {N} 美元, $malderalal=N} 美元。我们提议将降低的基础方法(RBMMMM) 和(drecreate) 实性内解算法方法(((D) (D) IM) 用于 RPBEal-alal dealal dealal dealalal) coal=nalalalalalalalalalalalal dal dal dal dal) 和(Oal dal dal dal) ral dal dal dal dal dal dal) ral) eal dreal) eal dal dal dal 将O, ealdal deal doldaldaldaldaldal dealdaldaldaldaldaldaldaldaldaldaldaldald 提供 提供 提供一种非的中, 提供 提供s 的中, 和(我们aldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldaldal rod rod rodaldaldaldaldaldaldaldaldaldaldaldaldaldaldal,我们算算算算算算