Elias N. Houstis

THE PERFORMANCE OF THE COLLOCATION AND GALERKIN METHODS WITH HERMITE BI-CUBICS*

This paper presents a study of the performance of the collocation and Galerkin methods using Hermite bi-cubic basis functions. It is a sequel to the studies of Houstis et al. [6] and Weiser et al. [15]. The two methods have the linear systems solved by direct methods, band Gauss elimination or Cholesky factorization. The problem domain consists of linear, self-adjoint elliptic equations on two-dimensional rectangular domains. The measures of performance are computer time and memory needed to achieve moderate accuracy. The earlier study comparing finite element and finite difference methods observes that collocation uses less computer time than Galerkin. The second study gave detailed operation counts which support this observation, but also gave substantial experimental evidence to the contrary. We use a new implementation of the collocation method by E. N. Houstis which is tailored for rectangular domains (the one used in Houstis et al. [6] was designed for general domains). We use the same Galerkin impl...

Mapping algorithms and software environment for data parallel PDE iterative solvers

We consider computations associated with data parallel iterative solvers used for the numerical solution of Partial Di erential Equations (PDEs). The mapping of such computations into load balanced tasks requiring minimum synchronization and communication is a di cult combinatorial optimization problem. Its optimal solution is essential for the e cient parallel processing of PDE computations. Determining data mappings that optimize a number of criteria, like workload balance, synchronization and local communication, often involves the solution of an NP-Complete problem. Although data mapping algorithms have been known for a few years there is lack of qualitative and quantitative comparisons based on the actual performance of the parallel computation. In this paper we present two new data mapping algorithms and evaluate them together with a large number of existing ones using the actual performance of data parallel iterative PDE solvers on the nCUBE II. Comparisons on the performance of data parallel iterative PDE solvers on medium and large scale problems demonstrate that some computationally inexpensive data block partitioning algorithms are as e ective as the computationally expensive deterministic optimization algorithms. Also, these comparisons demonstrate that the existing approach in solving the data partitioning problem is ine cient for large scale problems. Finally, a software environment for the solution of the partitioning problem of data parallel iterative solvers is presented.

Network Based Problem Solving Environments for Partial Differential Equation Computing

The Internet offers scientists round the world access to high-powered problem-solving environments. With Purdue’s Net PELLPACK, they can solve complex partial differential eqautions with common Web browsers that support Java applets. This chapter presents the architecture of a PSE server and identifies the associated research issues.