SCI Publications

GMRES(k) is widely used for solving non-symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES-like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications.

The Department of Energy and the National Science Foundation sponsored a series of workshops on data manipulation and visualization of large-scale scientific datasets. Three workshops were held in 1998, bringing together experts in high-performance computing, scientific visualization, emerging computer technologies, physics, chemistry, materials science, and engineering. These workshops were followed by two writing and review sessions, as well as numerous electronic collaborations, to synthesize the results. The results of these efforts are reported here. Across the government, mission agencies are charged with understanding scientific and engineering problems of unprecedented complexity. The DOE Accelerated Strategic Computing Initiative, for example, will soon be faced with the problem of understanding the enormous datasets created by teraops simulations, while NASA already has a severe problem in coping with the flood of data captured by earth observation satellites. Unfortunately, scientific visualization algorithms, and high-performance display hardware and software on which they depend, have not kept pace with the sheer size of emerging datasets, which threaten to overwhelm our ability to conduct research. Our capability to manipulate and explore large datasets is growing only slowly, while human cognitive and visual perception are an absolutely fixed resource. Thus, there is a pressing need for new methods of handling truly massive datasets, of exploring and visualizing them, and of communicating them over geographic distances. This report, written by representatives from academia, industry, national laboratories, and the government, is intended as a first step toward the timely creation of a comprehensive federal program in data manipulation and scientific visualization. There is, at this time, an exciting confluence of ideas on data handling, compression, telepresence, and scientific visualization. The combination of these new ideas, which we refer to as Da ta and Visualization Corridors (DVC), can raise scientific data understanding to new levels and will improve the way science is practiced

Richards' equation (RE) is often used to model flow in unsaturated porous media. This model captures physical effects, such as sharp fronts in fluid pressures and saturations, which are present in more complex models of multiphase flow. The numerical solution of RE is difficult not only because of these physical effects but also because of the mathematical problems that arise in dealing with the nonlinearities. The method of lines has been shown to be very effective for solving RE in one space dimension. When solving RE in two space dimensions, direct methods for solving the linearized problem for the Newton step are impractical. In this work, we show how the method of lines and Newton-iterative methods, which solve linear equations with iterative methods, can be applied to RE in two space dimensions. We present theoretical results on convergence and use that theory to design an adaptive method for computation of the linear tolerance. Numerical results show the method to be effective and robust compared with an existing approach.