Weighted Average Consensus-Based Optimization of Advection-Diffusion Systems

As a fundamental algorithm for collaborative processing over multi-agent systems, distributed consensus algorithm has been studied for optimizing its convergence rate. Due to the close analogy between the diffusion problem and the consensus algorithm, the previous trend in the literature is to transform the diffusion system from the spatially continuous domain into the spatially discrete one. In this transformation, the optimality is not necessarily preserved. In this paper, the reverse of this approach has been adopted, and it has been shown that the optimality can be preserved. This paper studies optimization of the Continuous-Time Consensus (CTC) problem on a weighted digraph with given average weight. Based on the detailed balance property, the CTC algorithm is converted into the weighted-average CTC algorithm. For the given distribution and average weight, a possible solution procedure has been provided. For finding the optimal weights corresponding to the weighted-average CTC algorithm with optimal convergence rate on a general graph. This solution procedure has been implemented based on the min-max theorem. For path topology, it is shown that the linearity of the drift term is the necessary and sufficient condition for the optimality of the consensus algorithm (and the corresponding diffusion system). Thus, the Pearson's class of discrete (continuous) distributions are optimal, where the closed-form formulas for the convergence rate, spectrum and other characteristics of the corresponding optimal consensus algorithm (diffusion system), i.e., the Hypergeometric types have been provided.