Gremban, K. D., Miller, G. L. and Zagha, M. (1995) Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium 65-69. IEEE.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....two previous relations may 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 ....

K.D. Gremban, G.L. Miller, and M. Zagha. Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium, pages 65--69, Santa Barbara, April 1995. IEEE.

....linear systems involving a generalized definition of Laplacians. They also defined the support tree preconditioner, and used the support number bounds to prove properties about the quality of these preconditioners. Gremban, Miller, and Zagha have evaluated the performance of these techniques [9]. The construction of the embedding we use below is related to the e#ective resistances between pairs of vertices in the graph when the graph is viewed as a network of unit conductances. Chandra et al. [6] have defined the maximum such value taken over all distinct pairs as the electrical ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner,in 9th International Parallel Processing Symposium, Santa Barbara, April 1995, IEEE, pp. 65--69.

....analyzing preconditioners called support graph preconditioning. The technique was first proposed and used by Pravin Vaidya [11] who described it in a talk in 1991, but did not publish a paper. Vaidya used the technique to design a family of novel preconditioners. Later, Gremban, Miller, and Zagha [5, 6] extended the technique and used it to construct another family of preconditioners. This paper explains the technique, extends it further, and uses it to analyze two classes of known preconditioners for model problems. Specifically, we use the extended technique to analyze certain ....

....and BPX preconditioners are explicit, in the sense that the construction gives B 1 directly. But we can also view them as augmented preconditioners on a larger linear space that contains representations for all the coarse meshes, much like the augmented preconditioner B # of Gremban and Miller [5, 6]. When viewed as 23 an augmented preconditioner, the construction for an MDS preconditioner actually gives its triangular decomposition. Suppose that we apply an MDS preconditioner by applying all the restriction operators in a sequence to obtain all the coarse representations of the vector that ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner, in 9th International Parallel Processing Symposium, Santa Barbara, April 1995, IEEE, pp. 65--69.

....analyzing preconditioners called support graph preconditioning. The technique was first proposed and used by Pravin Vaidya [11] who described it in a talk in 1991, but did not publish a paper. Vaidya used the technique to design a family of novel preconditioners. Later, Gremban, Miller, and Zagha [5, 6] extended the technique and used it to construct another family of preconditioners. This paper explains the technique, extends it further, and uses it to analyze two classes of known preconditioners for model problems. Specifically, we use the extended technique to analyze certain ....

....preconditioners are explicit, in the sense that the construction gives B Gamma1 directly. But we can also view them as augmented preconditioners on a larger linear space that contains representations for all the coarse meshes, much like the augmented preconditioner B 0 of Gremban and Miller [5, 6]. When viewed as 23 an augmented preconditioner, the construction for an MDS preconditioner actually gives its triangular decomposition. Suppose that we apply an MDS preconditioner by applying all the restriction operators in a sequence to obtain all the coarse representations of the vector that ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner, in 9th International Parallel Processing Symposium, Santa Barbara, April 1995, IEEE, pp. 65--69.

....definition of Laplacians. This work is reviewed in Section 4 below. He also defined the support tree preconditioner, and used the support number bounds to prove properties about the quality of these preconditioners. Gremban, Miller, and Zagha have evaluated the performance of these techniques [10]. 3. Notation and Terminology. 3.1. Matrices. All matrices considered in this paper are real matrices. We use capital letters (e.g. A) to represent matrices and bold lower case letters to represent vectors (e.g. x) A matrix A is diagonally dominant if all diagonal entries are positive (a ii ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner,in 9th International Parallel Processing Symposium, Santa Barbara, April 1995, IEEE, pp. 65--69.

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K. Gremban, G. Miller, and M. Zagha. Performance evaluation of a parallel preconditioner. CS CMU-CS-94-205, CMU, October 1994.

....of this author was supported in part by NSF grant CCR 9505472, DARPA contract N00014 95 1246, and Army contract DAAH04 95 1 0607. 703 704 STEPHEN GUATTERY AND GARY L. MILLER of iterative linear system solvers to estimate rates of convergence [18] and to analyze the quality of preconditioners [4, 13]. Bounds on # 2 are useful in the analysis of spectral partitioning, both because # 2 occurs in bounds on cut quality [24] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [16, 28] The eigenvalue # 2 has been related to expansion ....

....linear systems involving a generalized definition of Laplacians. They also defined the support tree preconditioner and used the support number bounds to prove properties about the quality of these preconditioners. Gremban, Miller, and Zagha have evaluated the performance of these techniques [13]. The construction of the embedding we use below is related to the e#ective resistances between pairs of vertices in the graph when the graph is viewed as a network of unit conductances. Chandra et al. 8] have defined the maximum such value taken over all distinct pairs as the electrical ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner, in Proceedings 9th International Parallel Processing Symposium, Santa Barbara, CA, 1995, pp. 65--69.

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K.D. Gremban, G.L. Miller, and M. Zagha. Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium, pages 65--69, Santa Barbara, April 1995. IEEE.

....have other important applications. Since the matrices are symmetric, their extreme eigenvalues can be used in computing their condition numbers, which are used in the study of iterative linear system solvers to estimate rates of convergence [18] and to analyze the quality of preconditioners [4, 13]. Bounds on # 2 are useful in the analysis of spectral partitioning, both because # 2 occurs in bounds on cut quality [24] and because they can be used in isolating # Institute for Computer Applications in Science and Engineering, Mail Stop 403, NASA Langley Research Center, Hampton, VA 23681 ....

....linear systems involving a generalized definition of Laplacians. He also defined the support tree preconditioner, and used the support number bounds to prove properties about the quality of these preconditioners. Gremban, Miller, and Zagha have evaluated the performance of these techniques [13]. The construction of the embedding we use below is related to the e#ective resistances between pairs of vertices in the graph when the graph is viewed as a network of unit conductances. Chandra et al. [8] have defined the maximum such value taken over all distinct pairs as the electrical ....

K. Gremban, G. Miller, and M. Zagha, Performance evaluation of a parallel preconditioner,in 9th International Parallel Processing Symposium, Santa Barbara, April 1995, IEEE, pp. 65--69.

....have other important applications. Since the matrices are symmetric, their extreme eigenvalues can be used in computing their condition numbers, which are used in the study of iterative linear system solvers to estimate rates of convergence [HY81] and to analyze the quality of preconditioners [Axe92, GMZ95]. Bounds on # 2 are useful in the analysis of spectral partitioning, both because # 2 occurs in bounds on cut quality [Moh89] and because they can be used in isolating structural properties of the eigenvectors used in making the cuts [GM95, ST96] The eigenvalue # 2 has been related to expansion ....

K.D. Gremban, G.L. Miller, and M. Zagha. Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium, pages 65--69, Santa Barbara, April 1995. IEEE.

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Gremban, K. D., Miller, G. L. and Zagha, M. (1995) Performance evaluation of a parallel preconditioner. In 9th International Parallel Processing Symposium 65-69. IEEE.

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