A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements

A parallel fast direct solver for rank-compressible block tridiagonal linear systems is presented. Algorithmic synergies between Cyclic Reduction and Hierarchical matrix arithmetic operations result in a solver with $O(N \log^2 N)$ arithmetic complexity and $O(N \log N)$ memory footprint. We provide a baseline for performance and applicability by comparing with well known implementations of the $\mathcal{H}$-LU factorization and algebraic multigrid with a parallel implementation that leverages the concurrency features of the method. Numerical experiments reveal that this method is comparable with other fast direct solvers based on Hierarchical Matrices such as $\mathcal{H}$-LU and that it can tackle problems where algebraic multigrid fails to converge

Implementing techniques for elliptic problems on vector processors

To provide the arithmetic power required by large-scale numerical simulations, the fastest computers today incorporate vector processing. Two types of vector architecture are defined, and the variation in performance that can occur on a vector processor as a function of algorithm and implementation, the consequences of this variation, and the performance of some basic operators on the two classes of vector architecture are discussed. The performance of some higher-level operators that should be used with caution is also considered. Then the implementation of techniques for elliptic problems using the operators discussed previously is reviewed. Included are Fast Poisson solvers, dissection, and point, line, block, and conjugant gradient schemes. Finally, some areas of research are noted. 1 figure. (RWR

Learning about supercomputers on a microcomputer with no keyboard: a science museum exhibit

A microcomputer exhibit was developed to acquaint visitors of the Los Alamos National Laboratory's Bradbury Science Museum with supercomputers and computer-graphics applications. The exhibit is highly interactive, yet the visitor uses only the touch panel of the CD 110 microcomputer. The museum environment presented many constraints to the development team, yet the five minute exhibit has been extremely popular with visitors. Design details of how each constraint was dealt with to produce a motivating and instructional exhibit are provided. Although the program itself deals with a subject area primarily applicable to Los Alamos, the design features are transferrable to other courseware where motivational and learning aspects are of equal importance

On Block Relaxation Techniques

In connection with efforts to utilize the CRAY-1 computer efficiently, we present some methods of analysis of rates of convergence for block iterative methods applied to the model problem. One of the more interesting methods involves relaxing on p x p blocks of points. A Cholesky decomposition is used for that smaller problem. One of the basic methods of analysis is a modification of a method discussed earlier by Parter. This analysis easily extends to more general second order elliptic problems

Department of Energy research in utilization of high-performance computers

Department of Energy (DOE) and other Government research laboratories depend on high-performance computer systems to accomplish their programmatic goals. As the most powerful computer systems become available, they are acquired by these laboratories so that advances can be made in their disciplines. These advances are often the result of added sophistication to numerical models, the execution of which is made possible by high-performance computer systems. However, high-performance computer systems have become increasingly complex, and consequently it has become increasingly difficult to realize their potential performance. The result is a need for research on issues related to the utilization of these systems. This report gives a brief description of high-performance computers, and then addresses the use of and future needs for high-performance computers within DOE, the growing complexity of applications within DOE, and areas of high-performance computer systems warranting research. 1 figure