Jim E. Jones

The Design and Implementation of hypre, a Library of Parallel High Performance Preconditioners

The hypre software library provides high performance preconditioners and solvers for the solution of large, sparse linear systems on massively parallel computers. One of its attractive features is the provision of conceptual interfaces. These interfaces give application users a more natural means for describing their linear systems, and provide access to methods such as geometric multigrid which require additional information beyond just the matrix. This chapter discusses the design of the conceptual interfaces in hypre and illustrates their use with various examples. We discuss the data structures and parallel implementation of these interfaces. A brief overview of the solvers and preconditioners available through the interfaces is also given.

Robustness and Scalability of Algebraic Multigrid

Algebraic multigrid (AMG) is currently undergoing a resurgence in popularity, due in part to the dramatic increase in the need to solve physical problems posed on very large, unstructured grids. While AMG has proved its usefulness on various problem types, it is not commonly understood how wide a range of applicability the method has. In this study, we demonstrate that range of applicability, while describing some of the recent advances in AMG technology. Moreover, in light of the imperatives of modern computer environments, we also examine AMG in terms of algorithmic scalability. Finally, we show some of the situations in which standard AMG does not work well and indicate the current directions taken by AMG researchers to alleviate these difficulties.

Semicoarsening Multigrid on Distributed Memory Machines

This paper presents the results of a scalability study for a three-dimensional semicoarsening multigrid solver on a distributed memory computer. In particular, we are interested in the scalability of the solver---how the solution time varies as both problem size and number of processors are increased. For an iterative linear solver, scalability involves both algorithmic issues and implementation issues. We examine the scalability of the solver theoretically by constructing a simple parallel model and experimentally by results obtained on an IBM SP. The results are compared with those obtained for other solvers on the same computer.

AMG E Based on Element Agglomeration

This paper contains the main ideas for an AMGe (algebraic multigrid for finite elements) method based on element agglomeration. In the method, coarse grid elements are formed by agglomerating fine grid elements. Compatible interpolation operators are constructed which yield coarse grid basis functions with a minimal energy property. Heuristics based on interpolation quality measures are used to guide the agglomeration procedure. The performance of the resulting method is demonstrated in two-level numerical experiments.

Pursuing scalability for hypre 's conceptual interfaces

The software library hypre provides high-performance preconditioners and solvers for the solution of large, sparse linear systems on massively parallel computers as well as conceptual interfaces that allow users to access the library in the way they naturally think about their problems. These interfaces include a stencil-based structured interface (Struct); a semistructured interface (semiStruct), which is appropriate for applications that are mostly structured, for example, block structured grids, composite grids in structured adaptive mesh refinement applications, and overset grids; and a finite element interface (FEI) for unstructured problems, as well as a conventional linear-algebraic interface (IJ). It is extremely important to provide an efficient, scalable implementation of these interfaces in order to support the scalable solvers of the library, especially when using tens of thousands of processors. This article describes the data structures, parallel implementation, and resulting performance of the IJ, Struct and semiStruct interfaces. It investigates their scalability, presents successes as well as pitfalls of some of the approaches and suggests ways of dealing with them.

Coarse-Grid Selection for Parallel Algebraic Multigrid

The need to solve linear systems arising from problems posed on extremely large, unstructured grids has sparked great interest in parallelizing algebraic multigrid (AMG). To date, however, no parallel AMG algorithms exist. We introduce a parallel algorithm for the selection of coarse-grid points, a crucial component of AMG, based on modifications of certain parallel independent set algorithms and the application of heuristics designed to insure the quality of the coarse grids. A prototype serial version of the algorithm is implemented, and tests are conducted to determine its effect on multigrid convergence, and AMG complexity.

Multigrid on Massively Parallel Architectures

The scalable implementation of multigrid methods for machines with several thousands of processors is investigated. Parallel performance models are presented for three different structured-grid multigrid algorithms, and a description is given of how these models can be used to guide implementation. Potential pitfalls are illustrated when moving from moderate-sized parallelism to large-scale parallelism, and results are given from existing multigrid codes to support the discussion. Finally, the use of mixed programming models is investigated for multigrid codes on clusters of SMPs.

Spectral Element Agglomerate AMGe

The purpose of this note is to describe an algorithm resulting from the uniting of two ideas introduced and applied elsewhere. For many problems, AMG has always been difficult due to complexities whose natures are difficult to discern from the entries of matrix A alone. Element-based interpolation has been shown to be an effective method for some of these problems, but it requires access to the element matrices on all levels. One way to obtain these has been to perform element agglomeration to form coarse elements, but in complicated situations defining the coarse degrees of freedom (dofs) is not easy. The spectral approach to coarse dof selection is very attractive due to its elegance and simplicity. The algorithm presented here combines the robustness of element interpolation, the ease of coarsening by element agglomeration, and the simplicity of defining coarse dofs through the spectral approach. As demonstrated in the numerical results, the method does yield a reasonable solver for the problems described. It can, however, be an expensive method due to the number and cost of the local, small dense linear algebra problems; making it a generally competitive method remains an area for further research.

Conceptual interfaces in hypre

The hypre software library is being developed with the aim of providing scalable solvers for the solution of large, sparse linear systems on massively parallel computers. To this end, the notion of conceptual interfaces was introduced. These interfaces give applications users a more natural means for describing their linear systems, and provide access to methods such as geometric multigrid which require additional information beyond just the matrix. This paper discusses the design of the conceptual interfaces in hypre and illustrates their use with various examples. A brief overview of the solvers and preconditioners available through these interfaces is also given.

A Multigrid Method for Variable Coefficient Maxwell's Equations

This paper presents a multigrid method for solving variable coefficient Maxwells equations. The novelty in this method is the use of interpolation operators that do not produce multilevel commutativity complexes that lead to multilevel exactness. Rather, the effects of multilevel exactness are built into the level equations themselves---on the finest level using a discrete