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Abstract

The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic parametrized partial differential equation that arises in biomolecular modeling and is a fundamental tool for structural biology. It is used to calculate electrostatic potentials around an ensemble of fixed charges immersed in an ionic solution. Efficient numerical computation of the PBE yields a high number of degrees of freedom in the resultant algebraic system of equations, ranging from several hundred thousands to millions. Coupled with the fact that in most cases the PBE requires to be solved multiple times for a large number of system configurations, this poses great computational challenges to conventional numerical techniques. To accelerate such computations, we here present the reduced basis method (RBM) which greatly reduces this computational complexity by constructing a reduced order model of typically low dimension. We discretize the linearized PBE (LPBE) with a centered finite difference scheme and solve the resultant linear system by the preconditioned conjugate gradient (PCG) method with an algebraic multigrid (AMG) V-cycle as preconditioner at different samples of ionic strength on a three-dimensional Cartesian grid. We then apply the RBM to the high-fidelity full order model (FOM). The discrete empirical interpolation method (DEIM) is applied to the Dirichlet boundary conditions which are nonaffine in one parameter (the ionic strength) to reduce the complexity of the reduced order model (ROM). From the numerical results, we notice that the RBM reduces the model order from $\mathcal{N} = 2\times 10^{6}$ to $N = 6$ at an accuracy of $10^{-10}$ and reduces computational time by a factor of approximately $8,000$. DEIM, on the other hand, provides a speed-up of $20$ in the online phase at a single iteration of the greedy algorithm.