Asynchronous global-local non-invasive coupling for linear elliptic problems

This paper presents the first asynchronous version of the Global/Local non-invasive coupling, capable of dealing efficiently with multiple, possibly adjacent, patches. We give a new interpretation of the coupling in terms of primal domain decomposition method, and we prove the convergence of the relaxed asynchronous iteration. The asynchronous paradigm lifts many bottlenecks of the Global/Local coupling performance. We illustrate the method on several linear elliptic problems as encountered in thermal and elasticity studies

Distributed Algorithms for Computing a Fixed Point of Multi-Agent Nonexpansive Operators

This paper investigates the problem of finding a fixed point for a global nonexpansive operator under time-varying communication graphs in real Hilbert spaces, where the global operator is separable and composed of an aggregate sum of local nonexpansive operators. Each local operator is only privately accessible to each agent, and all agents constitute a network. To seek a fixed point of the global operator, it is indispensable for agents to exchange local information and update their solution cooperatively. To solve the problem, two algorithms are developed, called distributed Krasnosel'ski\u{\i}-Mann (D-KM) and distributed block-coordinate Krasnosel'ski\u{\i}-Mann (D-BKM) iterations, for which the D-BKM iteration is a block-coordinate version of the D-KM iteration in the sense of randomly choosing and computing only one block-coordinate of local operators at each time for each agent. It is shown that the proposed two algorithms can both converge weakly to a fixed point of the global operator. Meanwhile, the designed algorithms are applied to recover the classical distributed gradient descent (DGD) algorithm, devise a new block-coordinate DGD algorithm, handle a distributed shortest distance problem in the Hilbert space for the first time, and solve linear algebraic equations in a novel distributed approach. Finally, the theoretical results are corroborated by a few numerical examples

Recurrent Averaging Inequalities in Multi-Agent Control and Social Dynamics Modeling

Many multi-agent control algorithms and dynamic agent-based models arising in natural and social sciences are based on the principle of iterative averaging. Each agent is associated to a value of interest, which may represent, for instance, the opinion of an individual in a social group, the velocity vector of a mobile robot in a flock, or the measurement of a sensor within a sensor network. This value is updated, at each iteration, to a weighted average of itself and of the values of the adjacent agents. It is well known that, under natural assumptions on the network's graph connectivity, this local averaging procedure eventually leads to global consensus, or synchronization of the values at all nodes. Applications of iterative averaging include, but are not limited to, algorithms for distributed optimization, for solution of linear and nonlinear equations, for multi-robot coordination and for opinion formation in social groups. Although these algorithms have similar structures, the mathematical techniques used for their analysis are diverse, and conditions for their convergence and differ from case to case. In this paper, we review many of these algorithms and we show that their properties can be analyzed in a unified way by using a novel tool based on recurrent averaging inequalities (RAIs). We develop a theory of RAIs and apply it to the analysis of several important multi-agent algorithms recently proposed in the literature