Consensus in non-commutative spaces

Convergence analysis of consensus algorithms is revisited in the light of the Hilbert distance. The Lyapunov function used in the early analysis by Tsitsiklis is shown to be the Hilbert distance to consensus in log coordinates. Birkhoff theorem, which proves contraction of the Hilbert metric for any positive homogeneous monotone map, provides an early yet general convergence result for consensus algorithms. Because Birkhoff theorem holds in arbitrary cones, we extend consensus algorithms to the cone of positive definite matrices. The proposed generalization finds applications in the convergence analysis of quantum stochastic maps, which are a generalization of stochastic maps to non-commutative probability spaces. ©2010 IEEE.

Quantitative convergence analysis of multi-agent systems

We introduce a characterization of contraction for bounded convex sets. For discrete-time multi-agent systems we provide an explicit upperbound on the rate of convergence to a consensus under the assumptions of contractiveness and (weak) connectedness (across an interval.) Convergence is shown to be exponential when either the system or the function characterizing the contraction is linear. Copyright © 2007 IFAC.

The stochastic implicit Euler method - A stable coupling scheme for Monte Carlo burnup calculations

Existing Monte Carlo burnup codes use various schemes to solve the coupled criticality and burnup equations. Previous studies have shown that the coupling schemes of the existing Monte Carlo burnup codes can be numerically unstable. Here we develop the Stochastic Implicit Euler method - a stable and efficient new coupling scheme. The implicit solution is obtained by the stochastic approximation at each time step. Our test calculations demonstrate that the Stochastic Implicit Euler method can provide an accurate solution to problems where the methods in the existing Monte Carlo burnup codes fail. © 2013 Elsevier Ltd. All rights reserved.

Numerically stable Monte Carlo-burnup-thermal hydraulic coupling schemes

This paper presents stochastic implicit coupling method intended for use in Monte-Carlo (MC) based reactor analysis systems that include burnup and thermal hydraulic (TH) feedbacks. Both feedbacks are essential for accurate modeling of advanced reactor designs and analyses of associated fuel cycles. In particular, we investigate the effect of different burnup-TH coupling schemes on the numerical stability and accuracy of coupled MC calculations. First, we present the beginning of time step method which is the most commonly used. The accuracy of this method depends on the time step length and it is only conditionally stable. This work demonstrates that even for relatively short time steps, this method can be numerically unstable. Namely, the spatial distribution of neutronic and thermal hydraulic parameters, such as nuclide densities and temperatures, exhibit oscillatory behavior. To address the numerical stability issue, new implicit stochastic methods are proposed. The methods solve the depletion and TH problems simultaneously and use under-relaxation to speed up convergence. These methods are numerically stable and accurate even for relatively large time steps and require less computation time than the existing methods. © 2013 Elsevier Ltd. All rights reserved.