In this paper, we review recent results on the role of entropy in the numerical integration of conservation laws. It is well known that weak solutions of systems of conservation laws may not be unique. Physically relevant weak solutions possess a viscous profile and satisfy entropy inequalities. In the discrete case entropy inequalities are used as a tool to prove convergence to entropy dissipa...

The measure-theoretic definition of Kullback-Leibler relative-entropy (or simply KLentropy) plays a basic role in defining various classical information measures on general spaces. Entropy, mutual information and conditional forms of entropy can be expressed in terms of KL-entropy and hence properties of their measure-theoretic analogs will follow from those of measure-theoretic KL-entropy. The...

We consider the problem of discrete distribution estimation under l1 loss. We provide tight upper and lower bounds on the maximum risk of the empirical distribution (the maximum likelihood estimator), and the minimax risk in regimes where the support size S may grow with the number of observations n. We show that among distributions with bounded entropy H , the asymptotic maximum risk for the e...

The heat transfer problem of a zero-mean oscillatory flow of a Maxwell fluid between infinite parallel plates with boundary conditions of the third kind is considered. The local and global time-averaged entropy production are computed, and the consequences of convective cooling of the plates are also assessed. It is found that the global entropy production is a minimum for certain suitable comb...

We consider semi-discrete and fully discrete approximations of nonlinear parabolic equations in the limit of unbounded domains, which by a scaling argument is equivalent to the limit of small viscosity. We define the spatial density of "–entropy, topological entropy and dimension for the attractors and show that these quantities are bounded. We also provide practical means of computing lower bo...

In this research paper, weighted / unweighted, directed / undirected graphs are associated with interesting Discrete Time Markov Chains (DTMCs) as well as Continuous Time Markov Chains ( CTMCs ). The equilibrium / transient behavior of such Markov chains is studied. Also entropy dynamics ( Shannon entropy ) of certain structured Markov chains is investigated. Finally certain structured graphs a...

The representation of discrete-time time-varying systems using infinite dimensional operators has shown to be suitable for extending the entropy as it is defined for time-invariant systems. In this paper we considerminimum entropy control for the disturbance feedforward problem, which is known to be the main step in solving the output feedback case.

We introduce a class of discrete velocity BGK type approximations to multidimensional scalar nonlinearly diiusive conservation laws. We prove the well-posedness of these models, a priori bounds and kinetic entropy inequalities that allow to pass into the limit towards the unique entropy solution recently obtained by Carrillo. Examples of such BGK models are provided.

Here, we consider the following inverse problem: Determination of an increasing continuous function U(x) on an interval [a, b] from the knowledge of the integrals ∫ U(x)dFXi (x) = πi where the Xi are random variables taking values on [a, b] and πi are given numbers. This is a linear integral equation with discrete data, which can be transformed into a generalized moment problem when U(x) is sup...