We present a class of iterative fully distributed fixed point methods to solve a system of linear equations, such that each agent in the network holds one or several of the equations of the system. Under a generic directed, strongly connected network, we prove a convergence result analogous to the one for fixed point methods in the classical, centralized, framework: the proposed method converges to the solution of the system of linear equations at a linear rate. We further explicitly quantify the rate in terms of the linear system and network parameters. Next, we show that the algorithm provably works under time-varying directed networks provided that the underlying graph is connected over bounded iteration intervals, and we establish a linear convergence rate for this setting as well. A set of numerical results is presented, demonstrating practical benefits of the method over existing alternatives.

Distributed fixed point method for solving systems of linear algebraic equations / D. Jakovetic, N. Krejic, N. Krklec Jerinkic, G. Malaspina, A. Micheletti. - In: AUTOMATICA. - ISSN 0005-1098. - 134(2021 Dec), pp. 109924.1-109924.12. [10.1016/j.automatica.2021.109924]

Distributed fixed point method for solving systems of linear algebraic equations

A. Micheletti
2021

Abstract

We present a class of iterative fully distributed fixed point methods to solve a system of linear equations, such that each agent in the network holds one or several of the equations of the system. Under a generic directed, strongly connected network, we prove a convergence result analogous to the one for fixed point methods in the classical, centralized, framework: the proposed method converges to the solution of the system of linear equations at a linear rate. We further explicitly quantify the rate in terms of the linear system and network parameters. Next, we show that the algorithm provably works under time-varying directed networks provided that the underlying graph is connected over bounded iteration intervals, and we establish a linear convergence rate for this setting as well. A set of numerical results is presented, demonstrating practical benefits of the method over existing alternatives.
Consensus; Distributed optimization; Fixed point methods; Kriging; Systems of linear equations
Settore MAT/06 - Probabilita' e Statistica Matematica
Settore MAT/09 - Ricerca Operativa
Settore SECS-S/01 - Statistica
Big Data Challenges for Mathematics (BIGMATH)
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/875591
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