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.

Almost Block Diagonal Linear Systems: Sequential and Parallel Solution Techniques, and Applications

Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in numerical methods for two-point boundary value problems for ordinary differential equations and in related partial differential equation problems. The stable, efficient sequential solution of ABDs has received much attention over the last fifteen years and the parallel solution more recently. We survey the fields of application with emphasis on how ABDs and bordered ABDs (BABDs) arise. We outline most known direct solution techniques, both sequential and parallel, and discuss the comparative efficiency of the parallel methods. Finally, we examine parallel iterative methods for solving BABD systems. Copyright (C) 2000 John Wiley & Sons, Ltd

Unified Heat Kernel Regression for Diffusion, Kernel Smoothing and Wavelets on Manifolds and Its Application to Mandible Growth Modeling in CT Images

We present a novel kernel regression framework for smoothing scalar surface data using the Laplace-Beltrami eigenfunctions. Starting with the heat kernel constructed from the eigenfunctions, we formulate a new bivariate kernel regression framework as a weighted eigenfunction expansion with the heat kernel as the weights. The new kernel regression is mathematically equivalent to isotropic heat diffusion, kernel smoothing and recently popular diffusion wavelets. Unlike many previous partial differential equation based approaches involving diffusion, our approach represents the solution of diffusion analytically, reducing numerical inaccuracy and slow convergence. The numerical implementation is validated on a unit sphere using spherical harmonics. As an illustration, we have applied the method in characterizing the localized growth pattern of mandible surfaces obtained in CT images from subjects between ages 0 and 20 years by regressing the length of displacement vectors with respect to the template surface.Comment: Accepted in Medical Image Analysi

Investigation and development of implicit numerical methods for building energy simulation

A variety of building energy analysis and simulation tools are increasingly used to determine peak heating and cooling loads, size thermal plant, anticipate annual energy consumption and analyse thermal comfort. Numerical solution techniques are considered the most flexible for building energy simulation. When applied to the differential equations modelling energy flows in buildings, they give rise to a system of non-linear algebraic (difference) equations. In order to evaluate numerical methods for building energy simulation, the problem has been characterized mathematically and comprehensive test problems (equation sets) with these characteristics have been prepared. The principal attribute of the problem was found to be a stifiess ratio of the order of lo4. Candidate methods have been programmed and their outputs compared, in numerical experiments, with highly accurate (converged) solutions for the test problems. The accepted validation methods, empirical validation, analytical verification and inter-modal comparison were considered inappropriate. The first estimates total and not just numerical error, the second is too confined and the third lacks an absolute standard. The main evaluation parameter used was computational efficiency which is defined as accuracy attained per unit (computational) effort expended. An improved difference equation solver has been proposed and compared with the one used in the European reference model (ESP) and elsewhere. It was found to produce 27% less error than the currently used method. A fundamental method for estimating the pre-conditioning period of a building has been put forward in this part of the work. The trapezoidal rule (TR) is currently used in a number of building energy simulation packages including ESP. A known instability associated with the method is described and an implicit member of the Runge-Kutta family, possessing the necessary strong stability, has been shown, using the test problems, to be more efficient than TR by a factor of 4.27

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

The Sixth Copper Mountain Conference on Multigrid Methods, part 2

The Sixth Copper Mountain Conference on Multigrid Methods was held on April 4-9, 1993, at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection clearly shows its rapid trend to further diversity and depth