10,091 research outputs found
Joint Spectral Radius and Path-Complete Graph Lyapunov Functions
We introduce the framework of path-complete graph Lyapunov functions for
approximation of the joint spectral radius. The approach is based on the
analysis of the underlying switched system via inequalities imposed among
multiple Lyapunov functions associated to a labeled directed graph. Inspired by
concepts in automata theory and symbolic dynamics, we define a class of graphs
called path-complete graphs, and show that any such graph gives rise to a
method for proving stability of the switched system. This enables us to derive
several asymptotically tight hierarchies of semidefinite programming
relaxations that unify and generalize many existing techniques such as common
quadratic, common sum of squares, and maximum/minimum-of-quadratics Lyapunov
functions. We compare the quality of approximation obtained by certain classes
of path-complete graphs including a family of dual graphs and all path-complete
graphs with two nodes on an alphabet of two matrices. We provide approximation
guarantees for several families of path-complete graphs, such as the De Bruijn
graphs, establishing as a byproduct a constructive converse Lyapunov theorem
for maximum/minimum-of-quadratics Lyapunov functions.Comment: To appear in SIAM Journal on Control and Optimization. Version 2 has
gone through two major rounds of revision. In particular, a section on the
performance of our algorithm on application-motivated problems has been added
and a more comprehensive literature review is presente
Numerical Studies of the Gauss Lattice Problem
The difference between the number of lattice points N(R) that lie in x^2 + y^2 ≤ R^2 and the area of that circle, d(R) = N(R) - πR^2, can be bounded by |d(R)| ≤ KR^θ.
Gauss showed that this holds for θ = 1, but the least value for which it holds is an open problem in number
theory. We have sought numerical evidence by tabulating N(R) up to R ≈ 55,000. From the convex hull bounding log |d(R)| versus log R we obtain the bound θ ≤ 0.575, which is significantly better than the best analytical result θ ≤ 0.6301 ... due to Huxley. The behavior of d(R) is of interest to those studying quantum chaos
Graphical Models for Optimal Power Flow
Optimal power flow (OPF) is the central optimization problem in electric
power grids. Although solved routinely in the course of power grid operations,
it is known to be strongly NP-hard in general, and weakly NP-hard over tree
networks. In this paper, we formulate the optimal power flow problem over tree
networks as an inference problem over a tree-structured graphical model where
the nodal variables are low-dimensional vectors. We adapt the standard dynamic
programming algorithm for inference over a tree-structured graphical model to
the OPF problem. Combining this with an interval discretization of the nodal
variables, we develop an approximation algorithm for the OPF problem. Further,
we use techniques from constraint programming (CP) to perform interval
computations and adaptive bound propagation to obtain practically efficient
algorithms. Compared to previous algorithms that solve OPF with optimality
guarantees using convex relaxations, our approach is able to work for arbitrary
distribution networks and handle mixed-integer optimization problems. Further,
it can be implemented in a distributed message-passing fashion that is scalable
and is suitable for "smart grid" applications like control of distributed
energy resources. We evaluate our technique numerically on several benchmark
networks and show that practical OPF problems can be solved effectively using
this approach.Comment: To appear in Proceedings of the 22nd International Conference on
Principles and Practice of Constraint Programming (CP 2016
Dynamical Phase Transitions for Fluxes of Mass on Finite Graphs
We study the time-averaged flux in a model of particles that randomly hop on
a finite directed graph. In the limit as the number of particles and the time
window go to infinity but the graph remains finite, the large-deviation rate
functional of the average flux is given by a variational formulation involving
paths of the density and flux. We give sufficient conditions under which the
large deviations of a given time averaged flux is determined by paths that are
constant in time. We then consider a class of models on a discrete ring for
which it is possible to show that a better strategy is obtained producing a
time-dependent path. This phenomenon, called a dynamical phase transition, is
known to occur for some particle systems in the hydrodynamic scaling limit,
which is thus extended to the setting of a finite graph
Euler-Poincare' Characteristic and Phase Transition in the Potts Model
Recent results concerning the topological properties of random geometrical
sets have been successfully applied to the study of the morphology of clusters
in percolation theory. This approach provides an alternative way of inspecting
the critical behaviour of random systems in statistical mechanics. For the 2d
q-states Potts model with q <= 6, intensive and accurate numerics indicates
that the average of the Euler characteristic (taken with respect to the
Fortuin-Kasteleyn random cluster measure) is an order parameter of the phase
transition.Comment: 17 pages, 8 figures, 1 tabl
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