Faisal Saied

Multigrid solution of the Poisson-Boltzmann equation

A multigrid method is presented for the numerical solution of the linearized Poisson–Boltzmann equation arising in molecular biophysics. The equation is discretized with the finite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for twodimensional interface problems occurring in reactor physics. A detailed analysis of the resulting method is presented for several computer architectures, including comparisons to diagonally scaled CG, ICCG, vectorized ICCG and MICCG, and to SOR provided with an optimal relaxation parameter. Our results indicate that the multigrid method is superior to the preconditioned CG methods and SOR and that the advantage of multigrid grows with the problem size.

Numerical solution of the nonlinear Poisson–Boltzmann equation: Developing more robust and efficient methods

We present a robust and efficient numerical method for solution of the nonlinear Poisson‐Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact‐Newton method, combined with linear multilevel techniques we have described in an article appearing previously in this journal. A detailed analysis of the resulting method is presented, with comparisons to other methods that have been proposed in the literature, including the classical nonlinear multigrid method, the nonlinear conjugate gradient method, and nonlinear relaxation methods such as successive overrelaxation. Both theoretical and numerical evidence suggests that this method will converge in the case of molecules for which many of the existing methods will not. In addition, for problems which the other methods are able to solve, numerical experiments show that the new method is substantially more efficient, and the superiority of this method grows with the problem size. The method is easy to implement once a linear multilevel solver is available and can also easily be used in conjunction with linear methods other than multigrid.

Protein Electrostatics: Rapid Multigrid-Based Newton Algorithm for Solution of the Full Nonlinear Poisson-Boltzmann Equation

A new method for solving the full nonlinear Poisson-Boltzmann equation is outlined. This method is robust and efficient, and uses a combination of the multigrid and inexact Newton algorithms. The novelty of this approach lies in the appropriate combination of the two methods, neither of which by themselves are capable of solving the nonlinear problem accurately. Features of the Poisson-Boltzmann equation are fully exploited by each component of the hybrid algorithm to provide robustness and speed. The advantages inherent in this method increase with the size of the problem. The efficacy of the method is illustrated by calculations of the electrostatic potential around the enzyme Superoxide Dismutase. The CPU time required to solve the full nonlinear equation is less than half that needed for a conjugate gradient solution of the corresponding linearized Poisson-Boltzmann equation. The solutions reveal that the field around the active sites is significantly reduced as compared to that obtained by solving the corresponding linearized Poisson-Boltzmann equation. This new method for the nonlinear Poisson-Boltzmann equation will enable fast and accurate solutions of large protein electrostatics problems.

Supercomputers in computational ocean acoustics

In this paper, we report on some computational experience in solving ocean acoustic propagation problems in three dimensions on supercomputers. The underlying Helmholtz equation is transformed into a parabolic-type equation in the Lee-Saad-Schultz model [5], which has a natural alternating direction implicit (ADI) implementation. We give estimates of the computing power required to solve problems with realistic sound velocity profiles. We then give performance results for the CRAY X-MP and for the computational kernel on the Intel hypercube (iPSC/2). We conclude with some remarks about architectural enhancements that would be beneficial to our application.

Solving coupled 3-D paraxial wave and thermal diffusion equations with mixed-mode parallel computations

The application of parallel computational methods to the modeling of laser light interactions with a thermally self-induced inhomogeneous medium is described. In these processes, laser light dynamically changes the local refractive index through heat absorption, which in-turn alters the beam profile. The mathematical model couples the 3-D heat diffusion and paraxial wave equations resulting in a system of non-linear equations. An explicit, finite-difference method (FDM) for solving the heat diffusion equation is coupled with an implicit FDM solution of the paraxial wave equation. The reasons for choosing these schemes are presented, along with a mixed-mode parallelization method in which OpenMP is used for the explicit solver, and MPI for the implicit solver. Finally, convergence, stability, accuracy and code performance is presented.

Multiscale Optimal Control of In Situ Bioremediation

Multiscale methods have been demonstrated to be highly efficient techniques for solving partial differential equations. In this paper, the idea of applying multiscale computation to an optimal control model of groundwater in situ bioremediation is investigated. The optimal control model, which was developed in previous work, uses an optimal control method called successive approximation linear quadratic regulator to identify optimal well locations and pumping rates to minimize pumping costs. The model can be used as an aid in designing more cost-effective aerobic in situ bioremediation, where injection wells are used to supply oxygen and extraction wells are used to contain the contaminant plume. The goal of this research is to improve the computational efficiency of the model so that complex field sites can be addressed. A spatial multiscale approach is presented in this paper. The spatial multiscale concept comes from discretization of the model domain with different mesh sizes. By solving the optimization on different numerical meshes and using bilinear interpolation operator to switch from the coarser mesh to finest mesh, significant computational savings can be gained. Both the convergence behavior and CPU time are presented for a case study under homogeneous conditions. The impact and choice of penalty weight when applying the multiscale approach are also discussed.