L. Ridgway Scott

Load balancing on message passing architectures

Many natural processes are best modeled by dynamic, Monte Carlo type algorithms. When parallelizing these, several problems emerge. One potential problem is a low overall efficiency due to an imbalanced work load. This paper describes the implementation of a testbed for load balancing techniques. This testbed is used for different static and dynamic strategies for balancing the work load of an iPSC/2 implementation of a simple simulation of population evolution. One of the new techniques described here is a decentralized direct method, which joins advantages of local and global strategies. In making comparisons between the different balancing methods, a clear separation was made between the work load (the algorithm solving a given problem, here a population simulation) and the balancer. The feasibility of this separation implies that the burden of developing an appropriate load balancer for a given algorithm may be removed from the programmer. The experience gained regarding load balancing for this class of problems will help guide the development of automated techniques for load balancing, either by an operating system or by a run-time system for a highlevel language.

Implicit spectral methods for wave propagation problems

Abstract The numerical solution of a non-linear wave equation can be obtained by using spectral methods to resolve the unknown in space and the standard Crank-Nicolson differencing scheme to advance the solution in time. We have analyzed iterative techniques for solving the non-linear equations that arise from such implicit time-stepping schemes for the K-dV and the Kue5f8P equations. We derived predictor—corrector method that retain the full accuracy of the implicit method with minimal stability restrictions on the size of the time step. Some numerical examples show the propagation of interacting solitons.

Error control and mesh optimization for high-order finite element approximation of incompressible viscous flow

Abstract We discuss the use of a posteriori error estimates for high-order finite element methods during simulation of the flow of incompressible viscous fluids. The correlation between the error estimator and actual error is used as a criterion for the error analysis efficiency. We show how to use the error estimator for mesh optimization which improves computational efficiency for both steady-state and unsteady flows. The method is applied to two-dimensional problems with known analytical solutions (Jeffrey-Hamel flow) and more complex flows around a body, both in a channel and in an open domain.

Implementing and using high-order finite element methods

Abstract We present theoretical analyses of and detailed timings for two programs which use high-order finite element methods to solve the Navier- Strokes equations in two and three dimensions. The analyses show that algorithms popular in low-order finite element implementations are not always appropriate for high-order methods. The timings show that with the proper algorithms high-order finite element methods are viable for solving the Navier-Stokes equations. We show that it is more efficient, both in time and storage, not to precompute element matrices as the degree of approximating functions increases. We also study the cost of assembling the stiffness matrix versus directly evaluating bilinear forms in two and three dimensions. We show that it is more efficient not to assemble the full stiffness matrix for high-order methods in some cases. We consider the computational issues with regard to both Euclidean and isoparametric elements. We show that isoparametric elements may be used with higher-order elements without increasing the order of computational complexity.

Parallel Linear Stationary Iterative Methods

A parallel linear stationary iterative method, defined by domain partitioning and referred to as the JSOR method, is analyzed in this paper. Basic JSOR convergence theorems, including one concerning the optimal relaxation parameter, are presented. JSOR is shown to have a much faster convergence rate than Jacobi and the same efficiency of interprocessor-data communication as Jacobi. Since JSOR contains the classic SOR and damped-Jacobi methods as its two extreme cases, the JSOR analysis can lead to a general linear stationary iteration theory, and imply both SOR and damped-Jacobi theories directly. Numerical results are presented to demonstrate the parallel performance of JSOR on a MIMD parallel computer Finally, the development and application of JSOR are discussed.

Finite Element Multigrid Methods

The multigrid method provides an optimal order algorithm for solving elliptic boundary value problems. The error bounds of the approximate solution obtained from the full multigrid algorithm are comparable to the theoretical bounds of the error in the finite element method, while the amount of computational work involved is proportional only to the number of unknowns in the discretized equations.

Correctness and determinism of parallel Monte Carlo processes

Abstract Many natural processes are best modeled by Monte Carlo type algorithms. When parallelizing these, several problems emerge. Some difficulties are well known from parallelizing algorithms in general, such as data coherence or load balancing. Other problems are specific to this kind of algorithm, namely code correctness and deterministic behavior of the code. We illustrate these two problems by describing and iPSC/2 implementation of a simple simulation of population evolution. This paper describes two techniques, the unrolling of an algorithm and the usage of decision dependent random seeds, which may be useful for adapting other dynamic algorithms to parallel computers.

Theoretical Issues Arising in the Modeling of Viscous Free-Surface Flows

This paper discusses theoretical and computational issues regarding viscous flows which have a free surface. A number of mathematical models for a particular flow are described and compared, both with one another and with some physical experiments. We consider some approximate models based both on lubrication theory and finite element methods. The importance of the choice of boundary conditions in modeling practical flow phenomena is discussed, and some related open theoretical questions regarding the well-posedness of mathematical models for such phenomena are presented. The discussion also touches upon the role that surface tension has so far played in the mathematical theory of free-surface flows and in many numerical calculations. Briefly outlined is some preliminary work related to convergence estimates for finite-element methods for free-boundary problems.

Overlapping and Short-Cutting Techniques in Loosely Synchronous Irregular Problems

We introduce short-cutting and overlapping techniques that, separately and in combination show promise of speedup for parallel processing of problems with irregular or asymmetric computation. Methodology is developed and demonstrated on an example problem. Experiments on an IBM SP-2 and a workstation cluster are presented.

Loop Splitting for High Performance Computers

This paper discusses program transformations and algorithm modifications that reduce execution time of iterative methods for solv ing partial differential equations on high performance computers. Tech niques typically associated with par allel computers turn out to be essen tial to obtain optimal performance on current superscalar uniprocessors.