Asynchronous Distributed Algorithms for Solving Linear Algebraic Equations.

Two asynchronous distributed algorithms are presented for solving a linear equation of the form $Ax=b$ with at least one solution. The equation is simultaneously and asynchronously solved by $m$ agents assuming that each agent knows only a subset of the rows of the partitioned matrix $[A\ \ b]$, the estimates of the equation's solution generated by its neighbors, and nothing more. Neighbor relationships are characterized by a time-dependent directed graph whose vertices correspond to agents and whose arcs depict neighbor relationships. Each agent recursively updates its estimate of a solution at its own event times by utilizing estimates generated by its neighbors which are transmitted with delays. The event time sequences of different agents are not assumed to be synchronized. It is shown that for any matrix-vector pair $(A, b)$ for which the equation has a solution and any repeatedly jointly strongly connected sequence of neighbor graphs defined on the merged sequence of all agents’ event times, the algorithms cause all agents’ estimates to converge exponentially fast to the same solution to $Ax=b$ . The first algorithm requires a specific initialization step at each agent, and the second algorithm works for arbitrary initializations. Explicit expressions for convergence rates are provided, and a relation between local initializations and limiting consensus solutions is established, which is used to solve the least 2-norm solution. [ABSTRACT FROM PUBLISHER]