The objective of this project is to develop and implement parallel algorithms for solving large sparse systems of linear equations by direct methods. The parallel algorithms we have developed use both geometric and nongeometric approaches to ordering sparse systems so that sparsity is preserved during subsequent factorization. The subsequent numeric phase is based on a multifrontal approach that maps naturally onto massively parallel target architectures. The resulting fully parallel sparse solver is portable across a wide variety of distributed parallel systems and is scalable over a wide range of numbers of processors. Current emphasis is on developing a more effective interface to make the solver easier to use and to facilitate its integration with application-specific software packages.

This research focuses on opportunities in optoelectronics that arise from the possibility of bandgap engineering. The goal is to simulate new forms of semiconductor heterolayers and thereby advise as to the optimum materials and structural forms that will lead to ultrahigh switching speeds in single devices and in optoelectronic integrated circuits. We investigate the dynamic equations of lasers, detectors, and modulators, and transient electronic transport in structures that are prototypical for optoelectronic devices. Our numerical approach, mostly based on Monte Carlo methods, goes far beyond conventional device modeling. We also simulate optoelectronic processes pertinent to femtosecond spectroscopy transients.

W e design numerical programs for the simulation of multidimensional quantum wells. The numerical model consists of Poisson's equation coupled with Schrödinger's equation. Poisson's equation is solved as an elliptic boundary value problem, while Schrödinger's equation defines an eigenvalue problem. The model is nonlinear and requires repeated solution of such eigenvalue problems. We compare the performance of Lanczos procedures with subspace iteration and examine the use of inverse iteration. Consistency of the two problems is obtained in an outer iteration that will be accelerated with a conjugate gradient procedure.

We cooperate on the development of a numerical program in which overshoot effects are included in the drift-diffusion model. The model is extended by the introduction of mobility functions for which the field dependence is computed by Monte Carlo simulation. The Scharfetter-Gummel discretization is adapted so that it can be applied to the modified model.

Knowing when to terminate an iterative method is equivalent to estimating the error accurately. In the case of the conjugate gradient method, we show that error norms appropriate for the method can be accurately estimated without additional work.

The numerical time evolution of systems of nonlinear equations requires effective time step procedures and in certain cases the enforcement of constraints. Time step methods and constraint enforcement are the topics of this project.

Novel iterative methods and preconditioning methods are being applied to the solution of dense complex linear systems arising from scattering problems.

Analysis in one dimension of the effectiveness of physically realistic preconditioning has been carried out for a boundary layer problem. Also, a suite of experiments on physically realistic preconditioners has been conducted for certain 2-D boundary layer problems.

A paper has been completed on a technique for the practical use of the Levinson algorithm in the symmetric indefinite case when the algorithm is unstable.

The objective is to find numerical algorithms suitable for large parallel computers that can much more efficiently model the dynamics of macromolecules such as proteins, DNA, and lipids. Emphasis is on the use of integration schemes, notably symplectic schemes, that can use large time steps to produce qualitatively correct simulations for long-time integrations. The goal is to obtain the desired information with the least computational effort, and the methodology is to use mathematical analysis and computational experiments on model problems. The techniques developed are to be tested on realistic molecular models. This work is closely tied to several projects of the Theoretical Biophysics Group, Beckman Institute.