Fast Solution of the Nonlinear Poisson-Boltzmann Equation Using the Reduced Basis Method and Range-Separated Tensor Format

摘要

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that is ubiquitous in biomolecular modeling. It determines a dimensionless electrostatic potential around a biomolecule immersed in an ionic solution cite{Chen}. For a monovalent electrolyte (i.e. a symmetric 1:1 ionic solution) it is given by$$label{PBE} -ec{abla}.(epsilon(x)ec{abla}u(x)) + ar{kappa}^2(x) sinh(u(x)) = rac{4pi e^2}{k_B T}sum_{i=1}^{N_m}z_idelta(x-x_i) quad extrm{in} quad Omega in mathbb{R}^3, $$$$label{eq:Dirichlet} u(x) = g(x) quad extrm{on} quad partial{Omega}, $$where$epsilon(x)$and$ar{k}^2(x)$are discontinuous functions at the interface between the charged biomolecule and the solvent, respectively.$delta(x-x_i)$is the Dirac delta distribution at point$x_i$. In this study, we treat the PBE as an interface problem by employing the recently developed range-separated tensor format as a solution decomposition technique cite{BKK_RS:16}. This is aimed at separating efficiently the singular part of the solution, which is associated with$delta(x-x_i)$, from the regular (or smooth) part. It avoids building numerical approximations to the highly singular part because its analytical solution, in the form of$u_{extrm{s}}(x) = lpha sum_{i=1}^{Nm}z_i/ lvert x-x_i vert$exists, hence increasing the overall accuracy of the PBE solution.On the other hand, numerical computation of eqref{PBE} yields a high-fidelity full order model (FOM) with dimension of$mathcal{O}(10^5)$$sim$$mathcal{O}(10^6)$, which is computationally expensive to solve on modern computer architectures for parameters with varying values, for example, the ionic strength,$I in ar{k}^2(x)$. Reduced basis methods are able to circumvent this issue by constructing a highly accurate yet small-sized reduced order model (ROM) which inherits all of the parametric properties of the original FOM cite{morRozHP08}. This greatly reduces the computational complexity of the system, thereby enabling fast simulations in a many-query context. We show numerical results where the RBM reduces the model order by a factor of approximately $350,000$ and computational time by $7,000$ at an accuracy of$mathcal{O}(10^{-8})$. This shows the potential of the RBM to be incorporated in the available software packages, for example, the adaptive Poisson-Boltzmann software (APBS).