Self-stabilizing Numerical Iterative Computation

TL;DR

This paper introduces a novel self-stabilizing iterative algorithm for solving linear systems in dynamic sensor networks, ensuring convergence from any initial state and adapting to changing environments.

Contribution

It presents extsyncAlg, a self-stabilizing variant of linear iterative methods, with proofs of convergence and extensions to asynchronous settings.

Findings

The extsyncAlg converges under mild conditions.

The algorithm is applicable to sensor calibration problems.

Simulation results demonstrate effectiveness in dynamic environments.

Abstract

Many challenging tasks in sensor networks, including sensor calibration, ranking of nodes, monitoring, event region detection, collaborative filtering, collaborative signal processing, {\em etc.}, can be formulated as a problem of solving a linear system of equations. Several recent works propose different distributed algorithms for solving these problems, usually by using linear iterative numerical methods. In this work, we extend the settings of the above approaches, by adding another dimension to the problem. Specifically, we are interested in {\em self-stabilizing} algorithms, that continuously run and converge to a solution from any initial state. This aspect of the problem is highly important due to the dynamic nature of the network and the frequent changes in the measured environment. In this paper, we link together algorithms from two different domains. On the one hand, we use the rich linear algebra literature of linear iterative methods for solving systems of linear equations, which are naturally distributed with rapid convergence properties. On the other hand, we are interested in self-stabilizing algorithms, where the input to the computation is constantly changing, and we would like the algorithms to converge from any initial state. We propose a simple novel method called \syncAlg as a self-stabilizing variant of the linear iterative methods. We prove that under mild conditions the self-stabilizing algorithm converges to a desired result. We further extend these results to handle the asynchronous case. As a case study, we discuss the sensor calibration problem and provide simulation results to support the applicability of our approach.