A Distributed Algorithm for Solving a Linear Algebraic Equation

Mou, Shaoshuai; Liu, Ji; Morse, A. Stephen

doi:10.1109/tac.2015.2414771

Cited by 266 publications

(277 citation statements)

References 43 publications

(100 reference statements)

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“…Using this and the fact that the graph of S ′ is strongly connected, one can conclude that (PS(τ )P ) p < 1, p ≥ (m − 1) 2 This is a direct consequence of Proposition 2 of [19]. Thus B p (τ ) < 1, p ≥ (m − 1) 2 (25) 1) N is constant: In this case B(τ ) is constant so it is sufficient to choose q so that Ã B q (τ ) ≤ λ.…”

Section: B Mixed Matrix Normmentioning

confidence: 65%

A Distributed Observer for a Discrete-Time Linear System

Wang

Liu

Morse

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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A simply structured distributed observer is described for estimating the state of a discrete-time, jointly observable, input-free, linear system whose sensed outputs are distributed across a time-varying network. It is explained how to construct the local estimators which comprise the observer so that their state estimation errors all converge exponentially fast to zero at a fixed, but arbitrarily chosen rate provided the network's graph is strongly connected for all time. This is accomplished by exploiting several well-known properties of invariant subspaces plus several kinds of suitably defined matrix norms.

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Section: B Mixed Matrix Normmentioning

confidence: 65%

A Distributed Observer for a Discrete-Time Linear System

Wang

Liu

Morse

et al. 2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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“…Each agent recursively updates its estimate of the solution using the current estimates from its neighbors. Recently several solutions under different sufficient conditions have been proposed [28]- [30], and in particular in [30], the sequence of the neighbor relationship graphs G(k) is required to be repeated jointly strongly connected. We show that a much weaker condition is sufficient to solve the problem almost surely, namely the algorithm in [30] works if there exists a fixed length such that any subsequence of {G(k)} at this length is jointly strongly connected with positive probability.…”

Section: Introductionmentioning

confidence: 99%

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

Qin

Cao

Anderson

2020

IEEE Trans. Automat. Contr.

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This paper presents new sufficient conditions for convergence and asymptotic or exponential stability of a stochastic discrete-time system, under which the constructed Lyapunov function always decreases in expectation along the system's solutions after a finite number of steps, but without necessarily strict decrease at every step, in contrast to the classical stochastic Lyapunov theory. As the first application of this new Lyapunov criterion, we look at the product of any random sequence of stochastic matrices, including those with zero diagonal entries, and obtain sufficient conditions to ensure the product almost surely converges to a matrix with identical rows; we also show that the rate of convergence can be exponential under additional conditions. As the second application, we study a distributed network algorithm for solving linear algebraic equations. We relax existing conditions on the network structures, while still guaranteeing the equations are solved asymptotically.

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“…The Banach-Picard method and its Krasnosel'skiȋ-Mann (KM) variant have been leveraged to establish convergence of a number of iterative algorithmic frameworks for solving convex optimization problems as well as problems associated with (non)linear systems [1]- [5]. Focusing on the KM method, recall that an operator T : D → D, where D is a nonempty convex subset of a finite-dimensional Hilbert space H with a given norm · , is non-expansive if it is 1-Lipschitz in H; that is, ∀ x, y ∈ D one has that T(x) − T(y) ≤ x − y .…”

Section: Introduction and Problem Formulationmentioning

confidence: 99%

On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

Dall’Anese

Simonetto

Bernstein

2019

IEEE Control Syst. Lett.

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This paper leverages a framework based on averaged operators to tackle the problem of tracking fixed points associated with maps that evolve over time. In particular, the paper considers the Krasnosel'skiȋ-Mann method in a settings where: (i) the underlying map may change at each step of the algorithm, thus leading to a "running" implementation of the Krasnosel'skiȋ-Mann method; and, (ii) an imperfect information of the map may be available. An imperfect knowledge of the maps can capture cases where processors feature a finite precision or quantization errors, or the case where (part of) the map is obtained from measurements. The analytical results are applicable to inexact running algorithms for solving optimization problems, whenever the algorithmic steps can be written in the form of (a composition of) averaged operators; examples are provided for inexact running gradient methods and the forward-backward splitting method. Convergence of the average fixed-point residual is investigated for the nonexpansive case; linear convergence to a unique fixed-point trajectory is showed in the case of inexact running algorithms emerging from contractive operators.

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A Distributed Algorithm for Solving a Linear Algebraic Equation

Cited by 266 publications

References 43 publications

A Distributed Observer for a Discrete-Time Linear System

A Distributed Observer for a Discrete-Time Linear System

Lyapunov Criterion for Stochastic Systems and Its Applications in Distributed Computation

On the Convergence of the Inexact Running Krasnosel’skiĭ–Mann Method

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