5,136 research outputs found
Regularized Optimal Transport and the Rot Mover's Distance
This paper presents a unified framework for smooth convex regularization of
discrete optimal transport problems. In this context, the regularized optimal
transport turns out to be equivalent to a matrix nearness problem with respect
to Bregman divergences. Our framework thus naturally generalizes a previously
proposed regularization based on the Boltzmann-Shannon entropy related to the
Kullback-Leibler divergence, and solved with the Sinkhorn-Knopp algorithm. We
call the regularized optimal transport distance the rot mover's distance in
reference to the classical earth mover's distance. We develop two generic
schemes that we respectively call the alternate scaling algorithm and the
non-negative alternate scaling algorithm, to compute efficiently the
regularized optimal plans depending on whether the domain of the regularizer
lies within the non-negative orthant or not. These schemes are based on
Dykstra's algorithm with alternate Bregman projections, and further exploit the
Newton-Raphson method when applied to separable divergences. We enhance the
separable case with a sparse extension to deal with high data dimensions. We
also instantiate our proposed framework and discuss the inherent specificities
for well-known regularizers and statistical divergences in the machine learning
and information geometry communities. Finally, we demonstrate the merits of our
methods with experiments using synthetic data to illustrate the effect of
different regularizers and penalties on the solutions, as well as real-world
data for a pattern recognition application to audio scene classification
A Smoothed Dual Approach for Variational Wasserstein Problems
Variational problems that involve Wasserstein distances have been recently
proposed to summarize and learn from probability measures. Despite being
conceptually simple, such problems are computationally challenging because they
involve minimizing over quantities (Wasserstein distances) that are themselves
hard to compute. We show that the dual formulation of Wasserstein variational
problems introduced recently by Carlier et al. (2014) can be regularized using
an entropic smoothing, which leads to smooth, differentiable, convex
optimization problems that are simpler to implement and numerically more
stable. We illustrate the versatility of this approach by applying it to the
computation of Wasserstein barycenters and gradient flows of spacial
regularization functionals
Hamiltonian dynamics, nanosystems, and nonequilibrium statistical mechanics
An overview is given of recent advances in nonequilibrium statistical
mechanics on the basis of the theory of Hamiltonian dynamical systems and in
the perspective provided by the nanosciences. It is shown how the properties of
relaxation toward a state of equilibrium can be derived from Liouville's
equation for Hamiltonian dynamical systems. The relaxation rates can be
conceived in terms of the so-called Pollicott-Ruelle resonances. In spatially
extended systems, the transport coefficients can also be obtained from the
Pollicott-Ruelle resonances. The Liouvillian eigenstates associated with these
resonances are in general singular and present fractal properties. The singular
character of the nonequilibrium states is shown to be at the origin of the
positive entropy production of nonequilibrium thermodynamics. Furthermore,
large-deviation dynamical relationships are obtained which relate the transport
properties to the characteristic quantities of the microscopic dynamics such as
the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, and the
fractal dimensions. We show that these large-deviation dynamical relationships
belong to the same family of formulas as the fluctuation theorem, as well as a
new formula relating the entropy production to the difference between an
entropy per unit time of Kolmogorov-Sinai type and a time-reversed entropy per
unit time. The connections to the nonequilibrium work theorem and the transient
fluctuation theorem are also discussed. Applications to nanosystems are
described.Comment: Lecture notes for the International Summer School Fundamental
Problems in Statistical Physics XI (Leuven, Belgium, September 4-17, 2005
Predictability: a way to characterize Complexity
Different aspects of the predictability problem in dynamical systems are
reviewed. The deep relation among Lyapunov exponents, Kolmogorov-Sinai entropy,
Shannon entropy and algorithmic complexity is discussed. In particular, we
emphasize how a characterization of the unpredictability of a system gives a
measure of its complexity. Adopting this point of view, we review some
developments in the characterization of the predictability of systems showing
different kind of complexity: from low-dimensional systems to high-dimensional
ones with spatio-temporal chaos and to fully developed turbulence. A special
attention is devoted to finite-time and finite-resolution effects on
predictability, which can be accounted with suitable generalization of the
standard indicators. The problems involved in systems with intrinsic randomness
is discussed, with emphasis on the important problems of distinguishing chaos
from noise and of modeling the system. The characterization of irregular
behavior in systems with discrete phase space is also considered.Comment: 142 Latex pgs. 41 included eps figures, submitted to Physics Reports.
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Large Scale Stochastic Dynamics
In focus are interacting stochastic systems with many components, ranging from stochastic partial differential equations to discrete systems as interacting particles on a lattice moving through random jumps.
More specifically one wants to understand the large scale behavior, large in spatial extent but also over long time spans, as entailed by the characterization of stationary measures, effective macroscopic evolution laws, transport of conserved fields, homogenization, self-similar structure and scaling, critical dynamics, aging, dynamical phase transitions, large deviations, to mention only a few key items
The automatic solution of partial differential equations using a global spectral method
A spectral method for solving linear partial differential equations (PDEs)
with variable coefficients and general boundary conditions defined on
rectangular domains is described, based on separable representations of partial
differential operators and the one-dimensional ultraspherical spectral method.
If a partial differential operator is of splitting rank , such as the
operator associated with Poisson or Helmholtz, the corresponding PDE is solved
via a generalized Sylvester matrix equation, and a bivariate polynomial
approximation of the solution of degree is computed in
operations. Partial differential operators of
splitting rank are solved via a linear system involving a block-banded
matrix in operations. Numerical
examples demonstrate the applicability of our 2D spectral method to a broad
class of PDEs, which includes elliptic and dispersive time-evolution equations.
The resulting PDE solver is written in MATLAB and is publicly available as part
of CHEBFUN. It can resolve solutions requiring over a million degrees of
freedom in under seconds. An experimental implementation in the Julia
language can currently perform the same solve in seconds.Comment: 22 page
An Optimization of Thermodynamic Efficiency vs. Capacity for Communications Systems
This work provides a fundamental view of the mechanisms which affect the power efficiency of communications processes along with a method for efficiency enhancement. Shannon\u27s work is the definitive source for analyzing information capacity of a communications system but his formulation does not predict an efficiency relationship suitable for calculating the power consumption of a system, particularly for practical signals which may only approach the capacity limit. This work leverages Shannon\u27s while providing additional insight through physical models which enable the calculation and improvement of efficiency for the encoding of signals. The proliferation of Mobile Communications platforms is challenging capacity of networks largely because of the ever increasing data rate at each node. This places significant power management demands on personal computing devices as well as cellular and WLAN terminals. The increased data throughput translates to shorter meantime between battery charging cycles and increased thermal footprint. Solutions are developed herein to counter this trend. Hardware was constructed to measure the efficiency of a prototypical Gaussian signal prior to efficiency enhancement. After an optimization was performed, the efficiency of the encoding apparatus increased from 3.125% to greater than 86% for a manageable investment of resources. Likewise several telecommunications standards based waveforms were also tested on the same hardware. The results reveal that the developed physical theories extrapolate in a very accurate manner to an electronics application, predicting the efficiency of single ended and differential encoding circuits before and after optimization
Dynamical Systems
Complex systems are pervasive in many areas of science integrated in our daily lives. Examples include financial markets, highway transportation networks, telecommunication networks, world and country economies, social networks, immunological systems, living organisms, computational systems and electrical and mechanical structures. Complex systems are often composed of a large number of interconnected and interacting entities, exhibiting much richer global scale dynamics than the properties and behavior of individual entities. Complex systems are studied in many areas of natural sciences, social sciences, engineering and mathematical sciences. This special issue therefore intends to contribute towards the dissemination of the multifaceted concepts in accepted use by the scientific community. We hope readers enjoy this pertinent selection of papers which represents relevant examples of the state of the art in present day research. [...
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