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

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      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

      \[\label{PBE} -\vec{\nabla}.(\epsilon(x)\vec{\nabla}u(x)) + \bar{\kappa}^2(x) \sinh(u(x)) = \frac{4\pi e^2}{k_B T}\sum_{i=1}^{N_m}z_i\delta(x-x_i) \quad \textrm{in} \quad \Omega \in \mathbb{R}^3, \]
      \[\label{eq:Dirichlet} u(x) = g(x) \quad \textrm{on} \quad \partial{\Omega}, \]
      where $\epsilon(x)$ and $\bar{k}^2(x)$ 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 $u_{\textrm{s}}(x) = \alpha \sum_{i=1}^{Nm}z_i/ \lvert x-x_i \rvert$ 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, $I \in \bar{k}^2(x)$. 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).

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      [1 ]Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany
      [* ]Correspondence: kweyu@
      ScienceOpen Posters
      27 April 2018
      Copyright © 2018

      This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at

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