M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. Submitted to SIAM J. Matrix Anal. Appl., 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....schemes in the Bunch Parlett sparse supernodal left looking or right looking (MA47) linear solvers for saddle point matrices. Recently, an interesting class of preconditioners constructed from spanning trees of matrix graphs or their extensions and based on work of P. Vaidya was introduced [3], 4] Construction of the preconditioners needs to test linear independence of sets of structurally simple vectors having thus connecting links to our work. 6. Acknowledgment. The author gratefully acknowledges the hospitality provided by Emilio Spedicato at the University of Bergamo. He would ....

M. Bern, J.R. Gilbert, B. Hendrickson N. Nguyen and S. Toledo. Support graph preconditioners, Sandia Labs Technical Report, submitted to SIAM J. Matrix Anal. Appl., 2000

....rate of CG is determined by how well the eigenspectrum of the preconditioned system J can be fit by a polynomial [28] Effective preconditioners produce smoother, flatter eigenspectra which are accurately fit by a lower order polynomial. Several authors have recently rediscovered [36], analyzed [37] and extended [38, 39] a technique called support graph theory which provides powerful methods for constructing effective preconditioners from maximum weight spanning trees. Support graph theory is especially interesting because it provides a set of results guaranteeing the ....

....tractable embedded subgraphs, we have not provided a procedure for choosing these subgraphs. Our results indicate that there are important interactions among cut edges, suggesting that simple methods (e.g. maximal spanning trees) may not provide the best performance. Although support graph theory [36, 37] provides some guarantees for embedded trees, extending these methods to more general subgraphs is an important open problem. The multiscale modeling example of Section I A suggests that adding a small number of edges to tree structured graphs may greatly increase their effectiveness, in ....

M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support--graph preconditioners. Submitted to SIAM Journal on Matrix Analysis and Applications, January 2001.

....[6] have defined the maximum such value taken over all distinct pairs as the electrical resistance of a graph, and have used that quantity to bound the commute and cover times of random walks on the graph. Independently, papers by Guattery [11] and Bern, Gilbert, Hendrickson, Nguyen, and Toledo [2] have given extensions to matrices that are not diagonally dominant. 3. Terminology, Notation, and Background Results. We assume that the reader is familiar with the basic definitions of graph theory (in particular, for undirected graphs) and with the basic definitions and results of matrix ....

M. Bern, J. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo, Support-graph preconditioners, in Copper Mountain Conference on Iterative Methods, Copper Mountain, Colorado, 1998, pp. 1--7.

....positive de nite. There is no simple extension to the complex symmetric case. Other previous related work includes another combinatorial preconditioner for weighted node arc adjacency matrices by Guo and Skeel [7] previous versions of support tree preconditioners by Vaidya [10] and Bern, et al. [2], and work by Vuik, Segal, and Meijerink [11] on a related mathematical problem arising in di usion modeling using an explicit eigenvector projection. The problem analyzed by Vuik et al. involves a real, symmetric, positive de nite matrix that is highly ill conditioned due to a large contrast in ....

M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo, Support-graph preconditioners, (preprint). 27

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

.... Several core ideas in support graph theory can be traced back to Beauwens [5] and were rediscovered by Vaidya who used them to study spanning tree preconditioners [30] The techniques were extended and applied to multilevel methods by Gremban [13] Gremban et al. 14] Reif [26] and Bern et al. [6]. The resulting methods have been applied to the analysis of incomplete Cholesky factorization by Guattery [15] and by Bern et al. 6] and to multilevel diagonal scaling [6] Unfortunately, support graph theory is fairly limited in its applicability. It only applies to symmetric, positive ....

.... spanning tree preconditioners [30] The techniques were extended and applied to multilevel methods by Gremban [13] Gremban et al. 14] Reif [26] and Bern et al. 6] The resulting methods have been applied to the analysis of incomplete Cholesky factorization by Guattery [15] and by Bern et al. [6] and to multilevel diagonal scaling [6] Unfortunately, support graph theory is fairly limited in its applicability. It only applies to symmetric, positive semidefinite, diagonally dominant M matrices (a subset of Stieltjes matrices) and, in some cases, to all spsd diagonally dominant matrices. ....

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

....has a physical justification [4, Section 3.4. 2] More importantly, it has been shown that when A represents a regular 5 point discretization of Laplace s equation with Dirichlet boundary conditions, a no fill modified incomplete factorization reduced the condition number from O(n) to O( # n) see [4, 6, 16] and their reference) An unmodified factorization reduces the condition number by a constant factor, but not asymptotically [4, 16] 1.5. REORDERING MATRICES FOR SPARSITY 9 1.4. Matrices And Graphs We normally think of matrices as two dimensional arrays of numbers or as representations of linear ....

....t = n, where n is the dimension of A, leads to M = A. Fill in the factors slows down the factorization of M and slows down each iteration. A 16 2.1. INTRODUCTION 17 small t on the other hand, yields a higher condition number but sparser factors. Vaidya stated (without a proof; for a proof, see [6]) that for any n by n M matrix with m nonzeros, t = 1 yields a condition number # = O(mn) The construction of such a preconditioner costs only O(m n log n) work and its factorization costs only O(m) work. For general sparse M matrices with a constant bound on the number of nonzeros per row, the ....

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Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

....not reported here. Thanks to Haim Kaplan for referring us to Tarjan s paper on augmented union find data structures. 6 CHAPTER 1 Background This chapter provides background material for the next two chapters, which describe original research results. This chapter is based on material from [4] [5], 10] 12] and from Sivan Toledo s lecture notes on high performance computing. 1.1. Iterative Solvers The term iterative methods refers in this thesis to a wide range of techniques that use successive approximations to obtain more accurate solutions to a linear system Ax = b at each step. ....

....be written as a linear system A h u h = b h , the vector u h having as its components the values u h (P ) P ## h . Notice that the matrix A h is a sparse M matrix. 1.7. Support Graph Theory This section describes the basic linear algebra tools that Gremban et al. 14, 15] and Bern et al. [5] developed to analyze Vaidya s preconditioners. These preconditioners are for M matrices, i.e. matrices which are symmetric, diagonally dominant (the sum of each row is non negative) whose o# diagonals are all nonpositive. When a preconditioner B is used in the Conjugate Gradient method, the ....

Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. In Proceedings of the Copper Mountain Conference On Iterative Methods, page 7 unnumbered pages, Copper Mountain, Colorado, 1998.

....of M . In particular, t = n, where n is the dimension of A, leads to M = A. Fill in the factors slows down the factorization of M and slows down each iteration. A small t on the other hand, yields a higher condition number but sparser factors. Vaidya stated (without a proof; for a proof, see [6]) that for any n by n SPDDD matrix with m nonzeros, t = 1 yields a condition number # = O(mn) The construction of such a preconditioner costs only O(m n log n) work and its factorization costs only O(m) work. For general sparse SPDDD matrices with a constant bound on the number of nonzeros per ....

....in M and in A are identical. The preconditioner M is the matrix whose underlying graph is GM . We denote by M t the Vaidya preconditioner constructed with the parameter t. We have Mn = A. The preconditioner M 1 consists solely of a maximum weight spanning tree with no added edges. Bern et al. [6] show that the condition number of this M 1 is O(mn) In general, Bern et al. show that the condition number of Vaidya s preconditioner is O(n 2 k 2 ) where k is the number of subgraphs that T is actually split into. They also analyze the cost of factoring M when GA is a bounded degree ....

M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo, Support-graph preconditioners, tech. report, School of Computer Science, Tel-Aviv University, 2001.

....of M . In particular, t = n, where n is the dimension of A, leads to M = A. Fill in the factors slows down the factorization of M and slows down each iteration. A small t on the other hand, yields a higher condition number but sparser factors. Vaidya stated (without a proof; for a proof, see [3]) that for any n by n M matrix with m nonzeros, t = 1 yields a condition number # = O(mn) The construction of such a preconditioner costs only O(m n log n) work and its factorization costs only O(m) work. For general sparse M matrices with a constant bound on the number of nonzeros per row, the ....

....in M and in A are identical. The preconditioner M is the matrix whose underlying graph is GM . We denote by M t the Vaidya s preconditioner constructed with the parameter t. We have M n = A. The preconditioner M 1 consists solely of a maximum weight spanning tree with no added edges. Bern et al. [3] show that the condition number of this M 1 is O(mn) In general, Bern et al. show that the condition number of Vaidya s preconditioner is O(n 2 k 2 ) where k is the number of subgraphs that T is actually split into. They also analyze the cost of factoring M when GA is a bounded degree ....

Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

.... 1 2 AM 1 2 . Perhaps the most remarkable aspect of Vaidya s preconditioners is that for many classes of Mmatrices, the condition number of the preconditioned matrix depends only on the size of A and is independent of the condition number #(A) of A. Vaidya stated (without proofs; for proofs, see [1]) several theoretical condition number bounds for his preconditioners. We omit them from this abstract due to lack of space. We have evaluated Vaidya s preconditioners by experimentally comparing their performance to that of droptolerance incomplete Cholesky preconditioners (ICC) in the context ....

Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

....of M . In particular, t = n, where n is the dimention of A, leads to M = A. Fill in the factors slows down the factorization of M and slows down each iteration. A small t on the other hand, yields a higher condition number but sparser factors. Vaidya stated (without a proof; for a proof, see [3]) that for any n by n M matrix with m nonzeros, t = 1 yields a condition number # = O(mn) The construction of such a preconditioner costs only O(m n log n) work and its factorization costs only O(m) work. For general sparse M matrices with a constant bound on the number of nonzeros per row, the ....

....in M and in A are identical. The preconditioner M is the matrix whose underlying graph is GM . We denote by M t the Vaidya s preconditioner constructed with the parameter t. We have Mn = A. The preconditioner M 1 consists solely of a maximum weight spanning tree with no added edges. Bern et al. [3] show that the condition number of this M 1 is O(mn) In general, Bern et al. show that the condition number of Vaidya s preconditioner is O(n 2 k 2 ) where k is the number of subgraphs that T is actually split into. They also analyze the cost of factoring M when GA is a bounded degree ....

Marshall Bern, John R. Gilbert, Bruce Hendrickson, Nhat Nguyen, and Sivan Toledo. Support-graph preconditioners. Technical report, School of Computer Science, Tel-Aviv University, 2001.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. Submitted to SIAM J. Matrix Anal. Appl., 2001.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. SIAM Journal on Matrix Analysis and Applications, 2002. Submitted.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. Submitted to SIAM J. Matrix Anal. Appl., 2001.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support graph preconditioners. Unpublished manuscript. Available at support-graph.ps.gz.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support graph preconditioners. Unpublished manuscript. Available at support-graph.ps.gz.

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M. Bern, J. R. Gilbert, B. Hendrickson, N. Nguyen, and S. Toledo. Support-graph preconditioners. SIAM J. Matrix Anal. Appl., 2002. to appear.

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