Reduced basis method for poisson-boltzmann equation

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 10􀀀9 and reduces computational time by a factor of approximately over 1300.