Measuring the irreversibility of numerical schemes for reversible stochastic differential equations

Abstract. For a Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE’s) time discretization numerical schemes usually destroy the property of time-reversibility. Despite an extensive literature on the numerical analysis for SDE’s, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of the discrete-time approximation process. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein’s for reversible SDEs with additive or multiplicative noise. Additionally, we analyze the entropy production for th

Stable sets and mean Li-Yorke chaos in positive entropy systems

It is shown that in a topological dynamical system with positive entropy, there is a measure-theoretically "rather big" set such that a multivariant version of mean Li-Yorke chaos happens on the closure of the stable or unstable set of any point from the set. It is also proved that the intersections of the sets of asymptotic tuples and mean Li-Yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively.Comment: The final version, reference updated, to appear in Journal of Functional Analysi

Entropy-difference based stereo error detection

Stereo depth estimation is error-prone; hence, effective error detection methods are desirable. Most such existing methods depend on characteristics of the stereo matching cost curve, making them unduly dependent on functional details of the matching algorithm. As a remedy, we propose a novel error detection approach based solely on the input image and its depth map. Our assumption is that, entropy of any point on an image will be significantly higher than the entropy of its corresponding point on the image's depth map. In this paper, we propose a confidence measure, Entropy-Difference (ED) for stereo depth estimates and a binary classification method to identify incorrect depths. Experiments on the Middlebury dataset show the effectiveness of our method. Our proposed stereo confidence measure outperforms 17 existing measures in all aspects except occlusion detection. Established metrics such as precision, accuracy, recall, and area-under-curve are used to demonstrate the effectiveness of our method

Long-range interactions, doubling measures and Tsallis entropy

We present a path toward determining the statistical origin of the thermodynamic limit for systems with long-range interactions. We assume throughout that the systems under consideration have thermodynamic properties given by the Tsallis entropy. We rely on the composition property of the Tsallis entropy for determining effective metrics and measures on their configuration/phase spaces. We point out the significance of Muckenhoupt weights, of doubling measures and of doubling measure-induced metric deformations of the metric. We comment on the volume deformations induced by the Tsallis entropy composition and on the significance of functional spaces for these constructions.Comment: 26 pages, No figures, Standard LaTeX. Revised version: addition of a paragraph on a contentious issue (Sect. 3). To be published by Eur. Phys. J.