Parallel Low-Storage Runge-Kutta Solvers for ODE Systems with Limited Access Distance

We consider the solution of initial value problems (IVPs) of large systems of ordinary differential equations (ODEs) for which memory space requirements determine the choice of the integration method. In particular, we discuss the space-efficient sequential and parallel implementation of embedded Runge—Kutta (RK) methods. Our focus is on the exploitation of a special structure of commonly appearing ODE systems, referred to as ‘‘limited access distance,’’ to improve scalability and memory usage. Such systems may arise, for example, from the semi-discretization of partial differential equations (PDEs). The storage space required by classical RK methods is directly proportional to the dimension n of the ODE system and the number of stages s of the method. We propose an implementation strategy based on a pipelined processing of the stages of the RK method and show how the memory usage of this computation scheme can be reduced to less than three storage registers by an overlapping of vectors without compromising the choice of method coefficients or the potential for efficient stepsize control. We analyze and compare the scalability of different parallel implementation strategies in detailed runtime experiments on different modern parallel architectures. </jats:p

Diagonal - implicity iterated Runge-Kutta methods on distributed memory multiprocessors

We investigate the parallel implementation of the diagonal-implicitly iterated Ruge-Kutta (DIIRK) method, an iteration method based on a predictor-corrector scheme. This method is appropriate for the solution of stiff systems of ordinary differential equations (ODEs) and provides embedded formulae to control the stepsize. We discuss different strategies for the implementation of the DIIRK method on distributed memory multiprocessors which mainly differ in the order of independent computations and the data distribution. In particular, we consider a consecutive implementation that executes the steps of each corrector iteration in sequential order and distributes the resulting equation systems among all available processors, and a group implementation that executes the steps in parallel by independent groups of processors. The performance of these implementations depends on the right hand side of the ODE system: For sparse functions, the group implementations is superior and achieves medium range seedup values. For dense functions, the consecutive implementation is better and achieves good speedup values.

Parallel iterated Runge-Kutta methods and applications

The iterated Runge-Kutta (IRK) method is an iteration scheme for the numerical solution of initial value problems (IVP) of ordinary differential equations (ODEs) that is based on a predictor-corrector method with an Runge-Kutta (RK) method as corrector. Embedded approxination formulae are used to control stepsize. We present different parallel algorithms of the IRK method on distributed memory multiprocessors for the solution of systems of ODEs. The parallel algorithms are given in an SPMD (single-program multipledata) programming style where data exchanges are described with appropriate communication primitives. A theoretical performance analysis and a runtime simulation allow to value the presented algorithms. The implementation on the Intel iPSC/860 confirms the predicted runtimes. The speedup values strongly depend on the particular system of ODEs to be solved. The parallel IRK method is applied to a typical discretization problem, the discretized Brusselator equation. Application specific modifications of the general parallel ODE solver are developped which result in a considerable reduction of the parallel execution time.

Using PostScript Programming Language in an Undergraduate Computer Graphics Course

Abstract. We report about the experiences of using the PostScript programming language in an undergraduate computer science and computer engineering course as a complementary tool besides OpenGL to teach basic concepts of computer graphics, especially affine transformations and hierarchical modeling using a transformation matrix stack mechanism. We can conclude that once the somewhat cryptic syntax of this stack-oriented language has been overcome, a natural computer graphics programming interface is available which permits a rapid understanding of essential concepts in graphics which can then easily be extrapolated to a 3-D interface like OpenGL. We would like to emphasize that the use of PostScript is not intended as an alternative to the standard graphics programming languages, but as an enrichment of the students programming skills in a completely distinct programming paradigm

Decision Manifolds: Classification Inspired by Self-Organization

We present a classifier algorithm that approximates the decision surface of labeled data by a patchwork of separating hyperplanes. The hyperplanes are arranged in a way inspired by how Self-Organizing Maps are trained. We take advantage of the fact that the boundaries can often be approximated by linear ones connected by a low-dimensional nonlinear manifold. The resulting classifier allows for a voting scheme that averages over the classifiction results of neighboring hyperplanes. Our algorithm is computationally efficient both in terms of training and classification. Further, we present a model selection framework for estimation of the paratmeters of the classification boundary, and show results for artificial and real-world data sets