Paul E. Plassmann

A parallel graph coloring heuristic

The problem of computing good graph colorings arises in many diverse applications, such as in the estimation of sparse Jacobians and in the development of efficient, parallel iterative methods for solving sparse linear systems. This paper presents an asynchronous graph coloring heuristic well suited to distributed memory parallel computers. Experimental results obtained on an Intel iPSC/860 are presented, which demonstrate that, for graphs arising from finite element applications, the heuristic exhibits scalable performance and generates colorings usually within three or four colors of the best-known linear time sequential heuristics. For bounded degree graphs, it is shown that the expected running time of the heuristic under the P-RAM computation model is bounded by

An efficient parallel algorithm for mesh smoothing

EO(\log (n)/\log \log (n))

An improved incomplete Cholesky factorization

. This bound is an improvement over the previously known best upper bound for the expected running time of a random heuristic for the graph coloring problem.

Scalable iterative solution of sparse linear systems

Automatic mesh generation and adaptive refinement methods have proven to be very successful tools for the efficient solution of complex finite element applications. A problem with these methods is that they can produce poorly shaped elements; such elements are undesirable because they introduce numerical difficulties in the solution process. However, the shape of the elements can be improved through the determination of new geometric locations for mesh vertices by using a mesh smoothing algorithm. In this paper the authors present a new parallel algorithm for mesh smoothing that has a fast parallel runtime both in theory and in practice. The authors present an efficient implementation of the algorithm that uses non-smooth optimization techniques to find the new location of each vertex. Finally, they present experimental results obtained on the IBM SP system demonstrating the efficiency of this approach.

Parallel Algorithms for Adaptive Mesh Refinement

Incomplete factorization has been shown to be a good preconditioner for the conjugate gradient method on a wide variety of problems. It is well known that allowing some fill-in during the incomplete factorization can significantly reduce the number of iterations needed for convergence. Allowing fill-in, however, increases the time for the factorization and for the triangular system solutions. Additionally, it is difficult to predict a priori how much fill-in to allow and how to allow it. The unpredictability of the required storage/work and the unknown benefits of the additional fill-in make such strategies impractical to use in many situations. In this article we motivate, and then present, two “black-box” strategies that significantly increase the effectiveness of incomplete Cholesky factorization as a preconditioner. These strategies require no parameters from the user and do not increase the cost of the triangular system solutions. Efficient implementations for these algorithms are described. These algorithms are shown to be successful for a variety of problems from the Harwell-Boeing sparse matrix collection.

Adaptive refinement of unstructured finite-element meshes

Abstract The efficiency of a parallel implementation of the conjugate gradient method preconditioned by an incomplete Cholesky factorization can very dramatically depending on the column ordering chosen. One method to minimize the number of major parallel steps is to choose an ordering based on a coloring of the symmetric graph representing the nonzero adjacency structure of the matrix. In this paper, we compare the performance of the preconditioned conjugate gradient method using these coloring orderings with a number of standard orderings on matrices arising from finite element models. Because optimal colorings for these systems may not be known a priori, we employ a graph coloring heuristic to obtain consistent colorings. Based on lower bounds obtained from the local structure of these systems, we find that the colorings determined by the heuristic are nearly optimal. For these problems, we find that the increase in parallelism afforded by the coloring-based orderings more than offsets any increase in the number of iterations required for the convergence of the conjugate gradient algorithm. We also demonstrate that the performance of this parallel preconditioner is scalable. We give results from the Intel iPSC/860 to support our claims.

Parallel Heuristics for Improved, Balanced Graph Colorings

Computational methods based on the use of adaptively constructed nonuniform meshes reduce the amount of computation and storage necessary to perform many scientific calculations. The adaptive construction of such nonuniform meshes is an important part of these methods. In this paper, we present a parallel algorithm for adaptive mesh refinement that is suitable for implementation on distributed-memory parallel computers. Experimental results obtained on the Intel DELTA are presented to demonstrate that for scientific computations involving the finite element method, the algorithm exhibits scalable performance and has a small run time in comparison with other aspects of the scientific computations examined. It is also shown that the algorithm has a fast expected running time under the parallel random access machine (PRAM) computation model.

The Efficient Parallel Iterative Solution of Large Sparse Linear Systems

Abstract The finite-element method used in conjunction with adaptive mesh refinement algorithms can be an efficient tool in many scientific and engineering applications. In this paper we review algorithms for the adaptive refinement of unstructured simplicial meshes (triangulations and tetra hedralizations). We discuss bounds on the quality of the meshes resulting from these refinement algorithms. Unrefinement and refinement along curved surfaces are also discussed. Finally, we give an overview of recent developments in parallel refinement algorithms.

Solution of large, sparse systems of linear equations in massively parallel applications

The computation of good, balanced graph colorings is an essential part of many algorithms required in scientific and engineering applications. Motivated by an effective sequential heuristic, we introduce a new parallel heuristic, PLF, and show that this heuristic has the same expected runtime under the PRAM computational model as the scalable coloring heuristic introduced by Jones and Plassmann. We present experimental results performed on the Intel DELTA that demonstrate that this new heuristic consistently generates better colorings and requires only slightly more time than the JP heuristic. In the second part of the paper we introduce two new parallel color-balancing heuristics, PDR(k) and PLF(k). We show that these heuristics have the desirable property that they do not increase the number of colors used by an initial coloring during the balancing process. We present experimental results that show that these heuristics are very effective in obtaining balanced colorings and, in addition, exhibit scalable performance.

Collaborative Virtual Environments Used in the Design of Pollution Control Systems

The development of efficient, general-purpose software for the iterative solution of sparse linear systems on parallel MIMD computers depends on recent results from a wide variety of research areas. Parallel graph heuristics, convergence analysis, and basic linear algebra implementation issues must all be considered.