OPTIMIZING THE DIFFUSION SYSTEM BASED ON CONTINUOUS-TIME CONSENSUS ALGORITHM

Due to beneficial insights on PDEs, recently, the reverse of this approach is implemented where a spatially discrete system is approximated by a spatially continuous one, governed by linear PDEs forming diffusion equations. In the case of distributed consensus algorithms, this approach is adapted to enhance its convergence rate to the equilibrium. In previous studies within this context, constant diffusion coefficient is considered for obtaining the diffusion equations. This is equivalent to assigning a constant weight to all edges of the underlying graph in the consensus algorithm. Here, by relaxing this restricting assumption, a spatially variable diffusion coefficient is considered and by optimizing the obtained system, it is shown that significant improvements are achievable in terms of the convergence rate of the obtained spatially continuous system. As a result of approximation, the system is divided into two sections, namely, the spatially continuous path branches and the lattice core which connects these branches at one end. The optimized weights and diffusion coefficient for each of these sections are optimal individually, but considering the whole system, they are suboptimal. It is shown that the symmetric star topology is an exception and the obtained results for this topology are globally optimal. Furthermore, through a variational method, the results obtained for the symmetric star topology are validated and it is shown that the variable diffusion coefficient improves the robustness of the system too.