Convergence of sequential and asynchronous nonlinear paracontractions

Elsner, Ludwig; Koltracht, I.; Neumann, Michael

doi:10.1007/bf01396232

Cited by 96 publications

(67 citation statements)

References 7 publications

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“…This algorithmic model of block iterations is a special case of asynchronous iterations, see, e.g., Frommer and Szyld [26] and Elsner, Koltracht and Neumann [23]. Those were called in early days chaotic relaxation by Chazan and Miranker [16].…”

Section: The Block-iterative Dropmentioning

confidence: 99%

On Diagonally Relaxed Orthogonal Projection Methods

Censor¹,

Elfving²,

Herman³

et al. 2008

SIAM J. Sci. Comput.

135

117

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Abstract. We propose and study a block-iterative projections method for solving linear equations and/or inequalities. The method allows diagonal component-wise relaxation in conjunction with orthogonal projections onto the individual hyperplanes of the system, and is thus called diagonallyrelaxed orthogonal projections (DROP). Diagonal relaxation has proven useful in accelerating the initial convergence of simultaneous and block-iterative projection algorithms but until now it was available only in conjunction with generalized oblique projections in which there is a special relation between the weighting and the oblique projections. DROP has been used by practitioners and in this paper a contribution to its convergence theory is provided. The mathematical analysis is complemented by some experiments in image reconstruction from projections which illustrate the performance of DROP.

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Section: The Block-iterative Dropmentioning

confidence: 99%

On Diagonally Relaxed Orthogonal Projection Methods

Censor¹,

Elfving²,

Herman³

et al. 2008

SIAM J. Sci. Comput.

135

117

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“…For a sequence of stochastic matrices , consider the linear system (26) which may be interpreted as a consensus algorithm over a digraph of nodes. The edges of are uniquely determined by the nonzero off-diagonal elements in .…”

Section: B Linear Systems Governed By Compatible Nonnegative Matricesmentioning

confidence: 99%

“…We note that there has existed an extensive literature (see [40] and references therein) on ergodicity of backward products of stochastic matrices. In particular, for analyzing inhomogeneous backward products, Wolfowitz's ergodicity theorem [20], [36], [45] and paracontraction [13], [25], [26] are well known powerful tools. Also, for the iterations of a nonnegative matrix with stationary delays, multiplicative ergodicity was studied via Lyaponov exponents in [14].…”

mentioning

confidence: 99%

Stochastic Approximation for Consensus: A New Approach via Ergodic Backward Products

Huang

2012

IEEE Trans. Automat. Contr.

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Abstract-This paper considers both synchronous and asynchronous consensus algorithms with noisy measurements. For stochastic approximation based consensus algorithms, the existing convergence analysis with dynamic topologies heavily relies on quadratic Lyapunov functions, whose existence, however, may be difficult to ensure for switching directed graphs. Our main contribution is to introduce a new analytical approach. We first show a fundamental role of ergodic backward products for mean square consensus in algorithms with additive noise. Subsequently, we develop ergodicity results for backward products of degenerating stochastic matrices converging to a 0-1 matrix via a discrete time dynamical system approach. These results complement the existing ergodic theory of stochastic matrices and provide an effective tool for analyzing consensus problems. Under a joint connectivity assumption, our approach may deal with switching topologies, delayed measurements and correlated noises, and it does not require the double stochasticity condition typically assumed for the existence of quadratic Lyapunov functions.Index Terms-Backward product, consensus, delay, ergodicity, mean square convergence, measurement noise, stochastic approximation, synchronous and asynchronous algorithms.

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“…(3) Aharoni and Censor [10], Flam and Zowe [11], Tseng [12], Eisner et al [13]: If X is finite dimensional, then the random product converges in norm to a point in C.…”

Section: Introductionmentioning

confidence: 99%