Performance evaluation of a new parallel preconditioner

Gremban, K. D.; Zagha, Marco

doi:10.1109/ipps.1995.395915

Cited by 28 publications

(29 citation statements)

References 18 publications

(9 reference statements)

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“…Written in terms of m and n, our work bound is inferior. Nevertheless, we believe that the paper makes a contribution by presenting the first algorithm and analysis for arbitrary Laplacians based on the support-tree preconditioner approach of Gremban et al [15], which is the only approach to introduce nodes in the preconditioner that are not present in the graph. We also point out an interesting connection between preconditioners and the hierarchical decomposition trees of weighted undirected graphs, recently introduced by Räcke [21].…”

Section: Our Resultsmentioning

confidence: 99%

“…Gremban, Miller, and Zagha [15] and Gremban [14] considered a different kind of graph-based preconditioner. They demonstrated that the support graph of a good subgraph preconditioner need not be subgraphs of the graph represented by A.…”

Section: Previous Workmentioning

confidence: 99%

“…In order to approximate the topology of the graph, their trees are hierarchically constructed by recursively partitioning the graph. Promising experimentally results are described in [15]. All of the work described above has been superseded by two papers by Spielman and Teng [24,25] and a paper by Emek, Elkin, Spielman, and Teng [12].…”

Section: Previous Workmentioning

confidence: 99%

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Finding effective support-tree preconditioners

Maggs

Parekh

Ravi

et al. 2005

Proceedings of the Seventeenth Annual ACM Symposium on Parallelism in Algorithms and Architectures

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In 1995, Gremban, Miller, and Zagha introduced support-tree preconditioners and a parallel algorithm called support-tree conjugate gradient (STCG) for solving linear systems of the form Ax = b, where A is an n × n Laplacian matrix. A Laplacian is a symmetric matrix in which the off-diagonal entries are non-positive, and the row and column sums are zero. A Laplacian A with 2m non-zeros can be interpreted as an undirected positively-weighted graph G with n vertices and m edges, where there is an edge between two nodes i and j with weight c ((i, j)showed experimentally that STCG performs well on several classes of graphs commonly used in scientific computations. In his thesis, Gremban also proved upper bounds on the number of iterations required for STCG to converge for certain classes of graphs. In this paper, we present an algorithm for finding a preconditioner for an arbitrary graph G = (V, E) with n nodes, m edges, and a weight function c > 0 on the edges, where w.l.o.g.

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Section: Our Resultsmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

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Finding effective support-tree preconditioners

Maggs

Parekh

Ravi

et al. 2005

Proceedings of the Seventeenth Annual ACM Symposium on Parallelism in Algorithms and Architectures

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“…The basic support-tree preconditioners and the graph embedding tools used in bounding the condition number of the preconditioned system were extended and generalized by Miller and his students Gremban and Guattery [99,100,101]. The extensions by Miller and Gremban include projecting the matrix onto a larger space and building support trees using Steiner trees (a Steiner tree forms a spanning tree of a graph with possibly additional vertices and edges).…”

Section: Support Graph Preconditionersmentioning

confidence: 99%

Combinatorial Problems in Solving Linear Systems

Duff¹,

Uçar²

2012

Combinatorial Scientific Computing

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Abstract. 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.

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“…3 Prior work on SDD solvers and related graph theoretic problems Symmetric diagonally dominant systems are linear-time reducible to linear systems whose matrix is the Laplacian of a weighted graph via a construction known as double cover which only doubles the number of non-zero entries in the system [GMZ95,Gre96]. The one-to-one correspondence between graphs and their Laplacians allows us to focus on weighted graphs, and interchangeably use the words graph and Laplacian.…”

Section: Preliminariesmentioning

confidence: 99%

Approaching Optimality for Solving SDD Linear Systems

Koutis¹,

Miller²,

Peng³

2010

2010 IEEE 51st Annual Symposium on Foundations of Computer Science

143

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We present an algorithm that on input a graph G with n vertices and m + n − 1 edges and a value k, produces an incremental sparsifierĜ with n − 1 + m/k edges, such that the condition number of G withĜ is bounded above byÕ(k log 2 n), with probability 1 − p. The algorithm runs in timẽ O((m log n + n log 2 n) log(1/p)). 1As a result, we obtain an algorithm that on input an n × n symmetric diagonally dominant matrix A with m + n − 1 non-zero entries and a vector b, computes a vectorx satisfying ||x − AThe solver is based on a recursive application of the incremental sparsifier that produces a hierarchy of graphs which is then used to construct a recursive preconditioned Chebyshev iteration.

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Performance evaluation of a new parallel preconditioner

Abstract: The

Cited by 28 publications

References 18 publications

Finding effective support-tree preconditioners

Finding effective support-tree preconditioners

Combinatorial Problems in Solving Linear Systems

Approaching Optimality for Solving SDD Linear Systems

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