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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 8 (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.
Contributions au partitionnement de graphes parallèle multiniveaux (Contributions to parallel multilevel graph partitioning)
, 2009
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ALGORITHMS FOR VERTEXWEIGHTED MATCHING IN GRAPHS
, 2009
"... A matching M in a graph is a subset of edges such that no two edges in M are incident on the same vertex. Matching is a fundamental combinatorial problem that has applications in many contexts: highperformance computing, bioinformatics, network switch design, web technologies, etc. Examples in th ..."
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Cited by 3 (0 self)
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A matching M in a graph is a subset of edges such that no two edges in M are incident on the same vertex. Matching is a fundamental combinatorial problem that has applications in many contexts: highperformance computing, bioinformatics, network switch design, web technologies, etc. Examples in the first context include sparse linear systems of equations, where matchings are used to place large matrix elements on or close to the diagonal, to compute the block triangular decomposition of sparse matrices, to construct sparse bases for the null space or column space of underdetermined matrices, and to coarsen graphs in multilevel graph partitioning algorithms. In the first part of this thesis, we develop exact and approximation algorithms for vertex weighted matchings, an understudied variant of the weighted matching problem. We propose three exact algorithms, three half approximation algorithms, and a twothird approximation algorithm. We exploit inherent properties of this problem such as lexicographical orders, decomposition into subproblems, and the reachability property, not only to design efficient algorithms, but also to provide simple proofs of correctness of the proposed algorithms. In the second part
HYPERGRAPHBASED COMBINATORIAL OPTIMIZATION OF MATRIXVECTOR MULTIPLICATION
, 2009
"... Combinatorial scientific computing plays an important enabling role in computational science, particularly in high performance scientific computing. In this thesis, we will describe our work on optimizing matrixvector multiplication using combinatorial techniques. Our research has focused on two di ..."
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Cited by 2 (0 self)
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Combinatorial scientific computing plays an important enabling role in computational science, particularly in high performance scientific computing. In this thesis, we will describe our work on optimizing matrixvector multiplication using combinatorial techniques. Our research has focused on two different problems in combinatorial scientific computing, both involving matrixvector multiplication, and both are solved using hypergraph models. For both of these problems, the cost of the combinatorial optimization process can be effectively amortized over many matrixvector products. The first problem we address is optimization of serial matrixvector multiplication for relatively small, dense matrices that arise in finite element assembly. Previous work showed that combinatorial optimization of matrixvector multiplication can lead to faster assembly of finite element stiffness matrices by eliminating redundant operations. Based on a graph model characterizing row relationships, a more efficient set of operations can be generated to perform matrixvector multiplication. We improved this graph model by extending the
Combinatorial algorithms enabling computational science: tales from the front
 Journal of Physics: Conference Series
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Experiments on Instance Preconditioning for Combinatorial Solvers
, 2008
"... Preconditioning of matrices, with the objective to solve large systems of linear equations more efficiently, is an active area of research. In contrast, there is no comparable systematic effort to precondition graphbased instances before solving them with a combinatorial solver. This paper asks t ..."
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Preconditioning of matrices, with the objective to solve large systems of linear equations more efficiently, is an active area of research. In contrast, there is no comparable systematic effort to precondition graphbased instances before solving them with a combinatorial solver. This paper asks the question: Does an existing preconditioning technique with known merits in solving systems of linear equations also improves the efficiency and effectiveness for a class of solvers on instances of combinatorial problems? We propose an experimental approach to evaluate merits of the Fiedler permutation when solving instances of the maximal independent set (MaxIS) problem with several solvers. Preliminary results are not only encouraging, they also demonstrate the value of Fiedler permutation when characterizing fundamental structural properties of graph instances themselves.
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"... On domain decomposition with space filling curves for the parallel solution of the coupled Maxwell/Vlasov equations ..."
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On domain decomposition with space filling curves for the parallel solution of the coupled Maxwell/Vlasov equations