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36
Preconditioning techniques for large linear systems: A survey
 J. COMPUT. PHYS
, 2002
"... This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization i ..."
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Cited by 164 (5 self)
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This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An extensive bibliography completes the paper.
hypre: a Library of High Performance Preconditioners
 Preconditioners,” Lecture Notes in Computer Science
, 2002
"... 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 ..."
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Cited by 72 (3 self)
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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 eciency of the library.
What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specific ..."
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Cited by 56 (9 self)
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Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
The design and implementation of hypre, a library of parallel high performance preconditioners
 Numerical solution of Partial Differential Equations on Parallel Computers, Lect. Notes Comput. Sci. Eng
, 2006
"... 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 natura ..."
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Cited by 30 (2 self)
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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. 1
MSP: a class of parallel multistep successive sparse approximate inverse preconditioning strategies
 SIAM J. Sci. Comput
, 2002
"... Abstract. We develop a class of parallel multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques. The key idea is to compute a series of simple sparse matrices to approximate the inverse of the original ma ..."
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Cited by 11 (5 self)
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Abstract. We develop a class of parallel multistep successive preconditioning strategies to enhance efficiency and robustness of standard sparse approximate inverse preconditioning techniques. The key idea is to compute a series of simple sparse matrices to approximate the inverse of the original matrix. Studies are conducted to show the advantages of such an approach in terms of both improving preconditioning accuracy and reducing computational cost, compared to the standard sparse approximate inverse preconditioners. Numerical experiments using one prototype implementation to solve a few sparse matrices on a distributed memory parallel computer are reported.
Computational experience with sequential and parallel, preconditioned Jacobi–Davidson for large, sparse symmetric matrices
, 2003
"... ..."
DISTRIBUTEDMEMORY PARALLEL ALGORITHMS FOR DISTANCE2 COLORING AND THEIR APPLICATION TO DERIVATIVE COMPUTATION
, 2010
"... The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. ..."
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Cited by 8 (6 self)
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The distance2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than two from each other. Its applications include derivative computation in numerical optimization and channel assignment in radio networks. We present efficient, distributedmemory, parallel heuristic algorithms for this NPhard problem as well as for two related problems used in the computation of Jacobians and Hessians. Parallel speedup is achieved through graph partitioning, speculative (iterative) coloring, and a BSPlike organization of parallel computation. Results from experiments conducted on a PC cluster employing up to 96 processors and using largesize realworld as well as synthetically generated test graphs show that the algorithms are scalable. In terms of quality of solution, the algorithms perform remarkably well—the number of colors used by the parallel algorithms was observed to be very close to the number used by the sequential counterparts, which in turn are quite often near optimal. Moreover, the experimental results show that the parallel distance2 coloring algorithm compares favorably with the alternative approach of solving the distance2 coloring problem on a graph G by first constructing the square graph G² and then applying a parallel distance1 coloring algorithm on G2. Implementations of the algorithms are made available via the Zoltan loadbalancing library.
Combinatorial problems in solving linear systems
, 2009
"... Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. ..."
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Cited by 7 (3 self)
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Numerical linear algebra and combinatorial optimization are vast subjects; as is their interaction. In virtually all cases there should be a notion of sparsity for a combinatorial problem to arise. Sparse matrices therefore form the basis of the interaction of these two seemingly disparate subjects. As the core of many of today’s numerical linear algebra computations consists of the solution of sparse linear system by direct or iterative methods, we survey some combinatorial problems, ideas, and algorithms relating to these computations. On the direct methods side, we discuss issues such as matrix ordering; bipartite matching and matrix scaling for better pivoting; task assignment and scheduling for parallel multifrontal solvers. On the iterative method side, we discuss preconditioning techniques including incomplete factorization preconditioners, support graph preconditioners, and algebraic multigrid. In a separate part, we discuss the block triangular form of sparse matrices.
ShyLU: A hybrid–hybrid solver for multicore platforms
 IN PROC. OF 26TH IEEE INTL. PARALLEL AND DISTRIBUTED PROCESSING SYMP. (IPDPS’12). IEEE
, 2012
"... With the ubiquity of multicore processors, it is crucial that solvers adapt to the hierarchical structure of modern architectures. We present ShyLU, a “hybridhybrid” solver for general sparse linear systems that is hybrid in two ways: First, it combines direct and iterative methods. The iterative p ..."
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Cited by 6 (3 self)
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With the ubiquity of multicore processors, it is crucial that solvers adapt to the hierarchical structure of modern architectures. We present ShyLU, a “hybridhybrid” solver for general sparse linear systems that is hybrid in two ways: First, it combines direct and iterative methods. The iterative part is based on approximate Schur complements where we compute the approximate Schur complement using a valuebased dropping strategy or structurebased probing strategy. Second, the solver uses two levels of parallelism via hybrid programming (MPI+threads). ShyLU is useful both in sharedmemory environments and on large parallel computers with distributed memory. In the latter case, it should be used as a subdomain solver. We argue that with the increasing complexity of compute nodes, it is important to exploit multiple levels of parallelism even within a single compute node. We show the robustness of ShyLU against other algebraic preconditioners. ShyLU scales well up to 384 cores for a given problem size. We also study the MPIonly performance of ShyLU against a hybrid implementation and conclude that on present multicore nodes MPIonly implementation is better. However, for future multicore machines (96 or more cores) hybrid / hierarchical algorithms and implementations are important for sustained performance.