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The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE that arisesin biomolecular modeling and is a fundamental tool for structural biology. Itis used to calculate electrostatic potentials around an ensemble of fixedcharges immersed in an ionic solution. Efficient numerical computation of thePBE yields a high number of degrees of freedom in the resultant algebraicsystem of equations. Coupled with the fact that in most cases the PBE requiresto be solved multiple times for a large number of system configurations, thisposes great computational challenges to conventional numerical techniques. Toaccelerate such computations, we here present the reduced basis method (RBM)which greatly reduces this computational complexity by constructing a reducedorder model of typically low dimension. In this study, we employ a simpleversion of the PBE for proof of concept and discretize the linearized PBE(LPBE) with a centered finite difference scheme. The resultant linear system issolved by the aggregation-based algebraic multigrid method at different samplesof ionic strength on a three-dimensional Cartesian grid. The discretized LPBE,which we call the high-fidelity full order model (FOM), yields solution asaccurate as other LPBE solvers. We then apply the RBM to FOM. The discreteempirical interpolation method (DEIM) is applied to the Dirichlet boundaryconditions which are nonaffine with the parameter, to reduce the complexity ofthe reduced order model (ROM). From the numerical results, we notice that theRBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ atan accuracy of $10^{-9}$ and reduces computational time by a factor ofapproximately $7,600$. DEIM, on the other hand, is also used in theoffline-online phase of solving the ROM for different values of parameterswhich provides a speed-up of $20$ for a single iteration of the greedyalgorithm.

Fast Solution of the Linearized Poisson-Boltzmann Equation with nonaffine Parametrized Boundary Conditions Using the Reduced Basis Method