  The Poisson-Boltzmann equation (PBE) is a nonlinear elliptic PDE 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. 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. In this study, we employ a simple
version of the PBE for proof of concept and discretize the linearized PBE
(LPBE) with a centered finite difference scheme. The resultant linear system is
solved by the aggregation-based algebraic multigrid method at different samples
of ionic strength on a three-dimensional Cartesian grid. The discretized LPBE,
which we call the high-fidelity full order model (FOM), yields solution as
accurate as other LPBE solvers. We then apply the RBM to FOM. The discrete
empirical interpolation method (DEIM) is applied to the Dirichlet boundary
conditions which are nonaffine with the parameter, 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 $10^{-9}$ and reduces computational time by a factor of
approximately $7,600$. DEIM, on the other hand, is also used in the
offline-online phase of solving the ROM for different values of parameters
which provides a speed-up of $20$ for a single iteration of the greedy
algorithm.
