Consensus-based distribution estimation of 1-D diffusion phenomena

This paper presents an application of consensus problems to inverse problems of the diffusion equation. We consider a distributed inverse problem in order to estimate an initial distribution of concentration from time-series data measured by agents on a one dimensional (1-D) interval. The inverse problem of the 1-D diffusion equation can be formulated as a linear least squares problem by a difference approximation. To solve the linear least squares problem in a distributed way, we consider a multi-agent system where each agent has a difference approximation of a gradient of concentration as its state. In large-scale sensor networks, communication time-delays cannot be ignored due to large transfer distances under limited link capacity. In this paper, we propose a discrete-time consensus dynamics with communication time-delays which are common for all agents. We show that the multi-agent system can estimate the initial distribution of 1-D diffusion phenomena by the proposed consensus dynamics.