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. It can also be used to estimate the electrostatic
contribution to the free energy of a system. 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. 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 (AGMG) 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 (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
N
= 2
×
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.
