Abstract:
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 of biomacromolecules in liquid solutions. To efficiently compute
the electrostatic potential numerically, a very large domain is required to sufficiently accommodate
both the biomacromolecule and the electrolyte. This yields high degrees of freedom in
the resultant algebraic system of equations ranging from several hundred thousands to a few
millions. This poses great computational challenges to conventional numerical techniques, especially
when many simulations for varying parameters, for instance, the ionic strength, are
to be run. The reduced basis method (RBM) 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 differences scheme and solve the resultant linear system by
the preconditioned conjugate gradient (PCG) method with algebraic multigrid as the preconditioner.
We then apply the RBM to the high-fidelity full order model (FOM) and present the
numerical results. We notice that the RBM reduces the model order from N = 1; 614; 177 to
N = 6 at an accuracy of 109 and reduces computational time by a factor of approximately
over 1300.