Supremum-Norm Convergence for Step-Asynchronous Successive Overrelaxation on M-matrices

Step-asynchronous successive overrelaxation updates the values contained in a single vector using the usual Gau\ss-Seidel-like weighted rule, but arbitrarily mixing old and new values, the only constraint being temporal coherence: you cannot use a value before it has been computed. We show that given a nonnegative real matrix $A$, a $\sigma\geq\rho(A)$ and a vector $\boldsymbol w>0$ such that $A\boldsymbol w\leq\sigma\boldsymbol w$, every iteration of step-asynchronous successive overrelaxation for the problem $(sI- A)\boldsymbol x=\boldsymbol b$, with $s >\sigma$, reduces geometrically the $\boldsymbol w$-norm of the current error by a factor that we can compute explicitly. Then, we show that given a $\sigma>\rho(A)$ it is in principle always possible to compute such a $\boldsymbol w$. This property makes it possible to estimate the supremum norm of the absolute error at each iteration without any additional hypothesis on $A$, even when $A$ is so large that computing the product $A\boldsymbol x$ is feasible, but estimating the supremum norm of $(sI-A)^{-1}$ is not

Wiener-Hopf factorization and distribution of extrema for a family of L\'{e}vy processes

In this paper we introduce a ten-parameter family of L\'{e}vy processes for which we obtain Wiener-Hopf factors and distribution of the supremum process in semi-explicit form. This family allows an arbitrary behavior of small jumps and includes processes similar to the generalized tempered stable, KoBoL and CGMY processes. Analytically it is characterized by the property that the characteristic exponent is a meromorphic function, expressed in terms of beta and digamma functions. We prove that the Wiener-Hopf factors can be expressed as infinite products over roots of a certain transcendental equation, and the density of the supremum process can be computed as an exponentially converging infinite series. In several special cases when the roots can be found analytically, we are able to identify the Wiener-Hopf factors and distribution of the supremum in closed form. In the general case we prove that all the roots are real and simple, and we provide localization results and asymptotic formulas which allow an efficient numerical evaluation. We also derive a convergence acceleration algorithm for infinite products and a simple and efficient procedure to compute the Wiener-Hopf factors for complex values of parameters. As a numerical example we discuss computation of the density of the supremum process.Comment: Published in at http://dx.doi.org/10.1214/09-AAP673 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Efficient Algorithms for the Consensus Decision Problem

We address the problem of determining if a discrete time switched consensus system converges for any switching sequence and that of determining if it converges for at least one switching sequence. For these two problems, we provide necessary and sufficient conditions that can be checked in singly exponential time. As a side result, we prove the existence of a polynomial time algorithm for the first problem when the system switches between only two subsystems whose corresponding graphs are undirected, a problem that had been suggested to be NP-hard by Blondel and Olshevsky.Comment: Small modifications after comments from reviewer

Energy-Aware Competitive Power Allocation for Heterogeneous Networks Under QoS Constraints

This work proposes a distributed power allocation scheme for maximizing energy efficiency in the uplink of orthogonal frequency-division multiple access (OFDMA)-based heterogeneous networks (HetNets). The user equipment (UEs) in the network are modeled as rational agents that engage in a non-cooperative game where each UE allocates its available transmit power over the set of assigned subcarriers so as to maximize its individual utility (defined as the user's throughput per Watt of transmit power) subject to minimum-rate constraints. In this framework, the relevant solution concept is that of Debreu equilibrium, a generalization of Nash equilibrium which accounts for the case where an agent's set of possible actions depends on the actions of its opponents. Since the problem at hand might not be feasible, Debreu equilibria do not always exist. However, using techniques from fractional programming, we provide a characterization of equilibrial power allocation profiles when they do exist. In particular, Debreu equilibria are found to be the fixed points of a water-filling best response operator whose water level is a function of minimum rate constraints and circuit power. Moreover, we also describe a set of sufficient conditions for the existence and uniqueness of Debreu equilibria exploiting the contraction properties of the best response operator. This analysis provides the necessary tools to derive a power allocation scheme that steers the network to equilibrium in an iterative and distributed manner without the need for any centralized processing. Numerical simulations are then used to validate the analysis and assess the performance of the proposed algorithm as a function of the system parameters.Comment: 37 pages, 12 figures, to appear IEEE Trans. Wireless Commu