Numerical Methods

We develop subspace decomposition methods for the numerical solution of the nonlinear Poisson-Boltzmann equation, a non-linear partial differential equation, describes the electrostatic potential of large complex biomolecules in solvent. These important problems have several interesting features impacting numerical algorithms, including discontinuous coefficients, rapid nonlinearities, and three spatial dimensions. We develop linear multilevel and domain decomposition methods for the linearized problem, and extend the methods to the nonlinear case with global inexact-Newton methods. Both theoretical numerical analysis and empirical evidence suggests that these methods are among the most robust and efficient methods available for the this class of problems.

Implementation

The program implementing the numerical methods described above is known as MG. The original version of MG (Multigrid Solver) was implemented in FORTRAN 77 as part of Michael Holst's Phd thesis "Multilevel Methods for the Poisson-Boltzmann Equation" at the University of Illinois at Urbana-Champaign (UIUC).

A parallel implementation (PMG) of the multilevel-based methods for the fully nonlinear 3D problem has been developed using Compositional C++ (CC++). This parallel implementation allows for the simulation of extremely large and complex biological systems, (on the order of one hundred million unknowns), due to both the large total memory avaiable on massively parallel distributed computers, and due to the additional computational efficiency gained by parallelization of an already extrememly efficient linear complexity (in both computation and memory) sequential algorithm.

This parallel implementation was accomplished via a collaboration between Michael Holst in Applied Mathematics at Caltech and John Garnett in the Computational Biology Group at Caltech.

Presentations

• Solving Elliptic Equations in Biophysics: Numerical Methods and CC++

Applications

• The Nonlinear Poisson-Boltzmann Equation.

  • The potential isosurface of an SOD enzyme.
  
  

• Drift-Diffusion Semiconductor Modeling.

  
  

• Nonlinear Elliptic Systems in General Relativity.