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Mathematics, Numerical Analysis, math.NA
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.
