  In the past few years, the problem of distributed consensus has received a
lot of attention, particularly in the framework of ad hoc sensor networks. Most
methods proposed in the literature address the consensus averaging problem by
distributed linear iterative algorithms, with asymptotic convergence of the
consensus solution. The convergence rate of such distributed algorithms
typically depends on the network topology and the weights given to the edges
between neighboring sensors, as described by the network matrix. In this paper,
we propose to accelerate the convergence rate for given network matrices by the
use of polynomial filtering algorithms. The main idea of the proposed
methodology is to apply a polynomial filter on the network matrix that will
shape its spectrum in order to increase the convergence rate. Such an algorithm
is equivalent to periodic updates in each of the sensors by aggregating a few
of its previous estimates. We formulate the computation of the coefficients of
the optimal polynomial as a semi-definite program that can be efficiently and
globally solved for both static and dynamic network topologies. We finally
provide simulation results that demonstrate the effectiveness of the proposed
solutions in accelerating the convergence of distributed consensus averaging
problems.
