Abstract. hypre is a software library for the solution of large, sparse linear systems on massively parallel computers. Its emphasis is on modern powerful and scalable preconditioners. hypre provides various conceptual interfaces to enable application users to access the library in the way they naturally think about their problems. This paper presents the conceptual interfaces in hypre. An overview of the preconditioners that are available in hypre is given, including some numerical results that show the efficiency of the library.

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Summary. 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.

Abstract. Algebraic multigrid (AMG) is a very efficient iterative solver and preconditioner for large unstructured linear systems. Traditional coarsening schemes for AMG can, however, lead to computational complexity growth as problem size increases, resulting in increased memory use and execution time, and diminished scalability. Two new parallel AMG coarsening schemes are proposed, that are based on solely enforcing a maximum independent set property, resulting in sparser coarse grids. The new coarsening techniques remedy memory and execution time complexity growth for various large three-dimensional (3D) problems. If used within AMG as a preconditioner for Krylov subspace methods, the resulting iterative methods tend to converge fast. This paper discusses complexity issues that can arise in AMG, describes the new coarsening schemes and examines the performance of the new preconditioners for various large 3D problems. Key words.AMS subject classifications.1. Introduction. The Algebraic Multigrid (AMG) algorithm [1,11,12,2] is one of the most efficient algorithms for solving large unstructured sparse linear systems that arise in a wide range of science and engineering applications. One of AMG's most desirable properties, especially in the context of large scale problems and massively parallel computing, is its potential for algorithmic scalability: for a matrix problem with n unknowns, the number of iterative V-cycles required for convergence is ideally independent of the problem size n (resulting from error reduction per cycle with convergence factors bounded away from one that are constant in terms of the problem size n), and the work in the setup phase and in each V-cycle is, in the ideal case, linearly proportional to the problem size n. A brief overview of the basic AMG algorithm is given in Sec. 2. Familiarity with the basic AMG algorithm is assumed in the remainder of this introductory section. For real-life problems, the AMG algorithm has shown to deliver consistent near-optimal algorithmic scalability for a wide range of applications [5,12]. Various parallel versions of AMG have been developed [4,6,9,8].Traditional coarsening schemes for AMG that are based on the coarsening heuristics originally proposed by Ruge and Stueben (RS) [11], tend to work well for problems that arise from the discretization of elliptic partial differential equations (PDEs) in two spatial dimensions (2D). For many 2D problems, a solver with optimal scalability can be obtained, with the number of iterations that is required for convergence independent of the problem size n, and memory use, setup time, and solution time per iteration linearly proportional to n. However, when traditional AMG algorithms are applied to three-dimensional (3D) problems, numerical tests show that in many cases scalability is lost: while the number of iterations required may remain constant, computational complexities, particularly stencil size, may grow significantly leading to increased execution times and memory usage. This loss of scalability is alr...

Abstract. This paper investigates the properties of smoothers in the context of algebraic multigrid (AMG) running on parallel computers with potentially millions of processors. The development of multigrid smoothers in this case is challenging, because some of the best relaxation schemes, such as the Gauss-Seidel (GS) algorithm, are inherently sequential. Based on the sharp two-grid multigrid theory from [22,23] we characterize the smoothing properties of a number of practical candidates for parallel smoothers, including several C-F , polynomial, and hybrid schemes. We show, in particular, that the popular hybrid GS algorithm has multigrid smoothing properties which are independent of the number of processors in many practical applications, provided that the problem size per processor is large enough. This is encouraging news for the scalability of AMG on ultra-parallel computers. We also introduce the more robust 1 smoothers, which are always convergent and have already proven essential for the parallel solution of some electromagnetic problems [29].