<i>m</i>-Step Preconditioned Conjugate Gradient Methods

Adams, Loyce M.

doi:10.1137/0906032

Cited by 74 publications

(24 citation statements)

References 7 publications

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“…For , notice that since the set of edges contained in and are disjoint, the walk-sets and are disjoint. Therefore, from Section II-C, (32) Every walk can be decomposed uniquely as , where , , and . The unique decomposition property is a consequence of and being disjoint, and the walk in being restricted to a length-1 hop.…”

Section: B Proof Of Propositionmentioning

confidence: 99%

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Estimation in Gaussian Graphical Models Using Tractable Subgraphs: A Walk-Sum Analysis

Chandrasekaran

Johnson

Willsky

2008

IEEE Trans. Signal Process.

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Abstract-Graphical models provide a powerful formalism for statistical signal processing. Due to their sophisticated modeling capabilities, they have found applications in a variety of fields such as computer vision, image processing, and distributed sensor networks. In this paper, we present a general class of algorithms for estimation in Gaussian graphical models with arbitrary structure. These algorithms involve a sequence of inference problems on tractable subgraphs over subsets of variables. This framework includes parallel iterations such as embedded trees, serial iterations such as block Gauss-Seidel, and hybrid versions of these iterations. We also discuss a method that uses local memory at each node to overcome temporary communication failures that may arise in distributed sensor network applications. We analyze these algorithms based on the recently developed walk-sum interpretation of Gaussian inference. We describe the walks "computed" by the algorithms using walk-sum diagrams, and show that for iterations based on a very large and flexible set of sequences of subgraphs, convergence is guaranteed in walk-summable models. Consequently, we are free to choose spanning trees and subsets of variables adaptively at each iteration. This leads to efficient methods for optimizing the next iteration step to achieve maximum reduction in error. Simulation results demonstrate that these nonstationary algorithms provide a significant speedup in convergence over traditional one-tree and two-tree iterations.Index Terms-Distributed estimation, Gauss-Markov random fields, graphical models, maximum walk-sum block, maximum walk-sum tree, subgraph preconditioners, walk-sum diagrams, walk-sums.

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Section: B Proof Of Propositionmentioning

confidence: 99%

“…Hence, we can focus on nodes in . For , (32) and (33) The last equality is from the inductive hypothesis because . Next, we have that where the second equality is from (17), and the third from (15).…”

Section: Proof Of Propositionmentioning

confidence: 99%

Estimation in Gaussian Graphical Models Using Tractable Subgraphs: A Walk-Sum Analysis

Chandrasekaran

Johnson

Willsky

2008

IEEE Trans. Signal Process.

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“…In this proof we will omit the superscripts (/). Part (1) follows from the fact that, for all specified …”

Section: F W Wubs Ed De Goede /An Explicit-implicit Methods For Pdesmentioning

confidence: 99%

“…Approximating the inverse of a matrix using truncated Neumann series is commonly applied in the construction of iterative algorithms on vector computers (see [1,3,5,18]). Assume that Dk is simply the diagonal of Tk, then we have the following result due to Heller [7].…”

Section: Approximation Of the Solution Of The Reducedmentioning

confidence: 99%

An explicit-implicit method for a class of time-dependent partial differential equations

Wubs

Goede²

1992

Applied Numerical Mathematics

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Wubs, F.W. and E.D. de Goede, An explicit-implicit method for a class of time-dependent partial differential equations, Applied Numerical Mathematics 9 (1992) 157-181.For the integration of partial differential equations, we distinguish explicit and implicit methods. In this paper, we consider an explicit-implicit method, which follows from the truncation of the solution process of a fully implicit method. Such a method is of interest because not only better vectorizing properties can be obtained by increasing the explicit part, but the method also fits well in a domain-decomposition approach. In this paper, we focus on the feasibility of such methods by studying their stability and accuracy properties. Nevertheless, we also did some experiments on vectorcomputers to show that for a sufficient degree of explicitness our method is more efficient than fully implicit methods.

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“…subdivide the region of interest into elements, 2. choose basis functions spanning the space in which the approximate solution lles, 3. integrate the basis functions over each element to determine its contribution, 4.…”

Section: Machine Design Issuesmentioning

confidence: 99%