1,194 research outputs found
Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems
Development of robust dynamical systems and networks such as autonomous
aircraft systems capable of accomplishing complex missions faces challenges due
to the dynamically evolving uncertainties coming from model uncertainties,
necessity to operate in a hostile cluttered urban environment, and the
distributed and dynamic nature of the communication and computation resources.
Model-based robust design is difficult because of the complexity of the hybrid
dynamic models including continuous vehicle dynamics, the discrete models of
computations and communications, and the size of the problem. We will overview
recent advances in methodology and tools to model, analyze, and design robust
autonomous aerospace systems operating in uncertain environment, with stress on
efficient uncertainty quantification and robust design using the case studies
of the mission including model-based target tracking and search, and trajectory
planning in uncertain urban environment. To show that the methodology is
generally applicable to uncertain dynamical systems, we will also show examples
of application of the new methods to efficient uncertainty quantification of
energy usage in buildings, and stability assessment of interconnected power
networks
Lecture notes: Semidefinite programs and harmonic analysis
Lecture notes for the tutorial at the workshop HPOPT 2008 - 10th
International Workshop on High Performance Optimization Techniques (Algebraic
Structure in Semidefinite Programming), June 11th to 13th, 2008, Tilburg
University, The Netherlands.Comment: 31 page
Stein's Method, Jack Measure, and the Metropolis Algorithm
The one parameter family of Jack(alpha) measures on partitions is an
important discrete analog of Dyson's beta ensembles of random matrix theory.
Except for special values of alpha=1/2,1,2 which have group theoretic
interpretations, the Jack(alpha) measure has been difficult if not intractable
to analyze. This paper proves a central limit theorem (with an error term) for
Jack(alpha) measure which works for arbitrary values of alpha. For alpha=1 we
recover a known central limit theorem on the distribution of character ratios
of random representations of the symmetric group on transpositions. The case
alpha=2 gives a new central limit theorem for random spherical functions of a
Gelfand pair. The proof uses Stein's method and has interesting ingredients: an
intruiging construction of an exchangeable pair, properties of Jack
polynomials, and work of Hanlon relating Jack polynomials to the Metropolis
algorithm.Comment: very minor revisions; fix a few misprints and update bibliograph
The Hidden Convexity of Spectral Clustering
In recent years, spectral clustering has become a standard method for data
analysis used in a broad range of applications. In this paper we propose a new
class of algorithms for multiway spectral clustering based on optimization of a
certain "contrast function" over the unit sphere. These algorithms, partly
inspired by certain Independent Component Analysis techniques, are simple, easy
to implement and efficient.
Geometrically, the proposed algorithms can be interpreted as hidden basis
recovery by means of function optimization. We give a complete characterization
of the contrast functions admissible for provable basis recovery. We show how
these conditions can be interpreted as a "hidden convexity" of our optimization
problem on the sphere; interestingly, we use efficient convex maximization
rather than the more common convex minimization. We also show encouraging
experimental results on real and simulated data.Comment: 22 page
Quantum Gravity from Causal Dynamical Triangulations: A Review
This topical review gives a comprehensive overview and assessment of recent
results in Causal Dynamical Triangulations (CDT), a modern formulation of
lattice gravity, whose aim is to obtain a theory of quantum gravity
nonperturbatively from a scaling limit of the lattice-regularized theory. In
this manifestly diffeomorphism-invariant approach one has direct, computational
access to a Planckian spacetime regime, which is explored with the help of
invariant quantum observables. During the last few years, there have been
numerous new and important developments and insights concerning the theory's
phase structure, the roles of time, causality, diffeomorphisms and global
topology, the application of renormalization group methods and new observables.
We will focus on these new results, primarily in four spacetime dimensions, and
discuss some of their geometric and physical implications.Comment: 64 pages, 28 figure
"Zero" temperature stochastic 3D Ising model and dimer covering fluctuations: a first step towards interface mean curvature motion
We consider the Glauber dynamics for the Ising model with "+" boundary
conditions, at zero temperature or at temperature which goes to zero with the
system size (hence the quotation marks in the title). In dimension d=3 we prove
that an initial domain of linear size L of "-" spins disappears within a time
\tau_+ which is at most L^2(\log L)^c and at least L^2/(c\log L), for some c>0.
The proof of the upper bound proceeds via comparison with an auxiliary dynamics
which mimics the motion by mean curvature that is expected to describe, on
large time-scales, the evolution of the interface between "+" and "-" domains.
The analysis of the auxiliary dynamics requires recent results on the
fluctuations of the height function associated to dimer coverings of the
infinite honeycomb lattice. Our result, apart from the spurious logarithmic
factor, is the first rigorous confirmation of the expected behavior
\tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension
d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections:
the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here
we provide the lower bound. For d=2, we also prove that the spectral gap of the
generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli,
2002].Comment: 44 pages, 7 figures. v2: Theorem 1 improved to include a matching
lower bound on tau_
Harmonic analysis of finite lamplighter random walks
Recently, several papers have been devoted to the analysis of lamplighter
random walks, in particular when the underlying graph is the infinite path
. In the present paper, we develop a spectral analysis for
lamplighter random walks on finite graphs. In the general case, we use the
-symmetry to reduce the spectral computations to a series of eigenvalue
problems on the underlying graph. In the case the graph has a transitive
isometry group , we also describe the spectral analysis in terms of the
representation theory of the wreath product . We apply our theory to
the lamplighter random walks on the complete graph and on the discrete circle.
These examples were already studied by Haggstrom and Jonasson by probabilistic
methods.Comment: 29 page
The Type-problem on the Average for random walks on graphs
When averages over all starting points are considered, the Type Problem for
the recurrence or transience of a simple random walk on an inhomogeneous
network in general differs from the usual "local" Type Problem. This difference
leads to a new classification of inhomogeneous discrete structures in terms of
{\it recurrence} and {\it transience} {\it on the average}, describing their
large scale topology from a "statistical" point of view. In this paper we
analyze this classification and the properties connected to it, showing how the
average behavior affects the thermodynamic properties of statistical models on
graphs.Comment: 10 pages, 3 figures. to appear on EPJ
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