Minimax estimation of smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure

Application of water cumulative charges as a water spouts for intensive flame extinguishing

Shape cumulative charge is a set of explosive components that uses directional energy accumulation. The water cumulative charges are filled with water, which forms a water-directed beam that has the ability to effectively counteract the intense flame that is induced by gaseous flammable gas or liquid from the damaged gas duct and extinguishes it. Study contains description of the experimentally constructed cumulative charge as well as the analysis of results of experiments carried out in real conditions. Based on the facts gained from the experiments we can conclude that the cumulative water charge has a significant potential and possibilities to extinguish an intense flame.Web of Science68326426

Experimental Verification of the Mechanical Resistance of Forensic Marking by Means Synthetic DNA

This article deals with experimental verification of resistance of forensic identification marks (microdots in combination with artificial DNA) to property. It is considered mechanical abrasion from potential offender to remove or damage readability of marking and following identification. The aim of this work is to test the hypothesis that forensic marking can be completely removed by the process of mechanical abrasion without causing damages to a protected object. To fulfi ll this purpose it was designed and built a test equipment, where experiments were carried out to confirm or refute the above mentioned hypothesis

Volume Change and Non-Local Driving Force in Crystallization

A simplified crystallization model is developed with emphasis on situations of disparate specific volumes of the solid and liquid phases. Using the general equation for the nonequilibrium reversible-irreversible coupling (GENERIC), the model is formulated in terms of the average momentum density, the degree of crystallinity, a single temperature, and a single pressure, where in particular the latter two are appealing for comparison with experiments. In order to describe the volume expansion upon crystallization, a dissipative mass current density is introduced, for which a constitutive relation is derived. One finds that by way of the Onsager-Casimir symmetry, the introduction of this irreversible current also leads to a modification of the driving force for phase change. Rather than depending only on the local chemical potential difference, it also contains a non-local term, namely the Laplacian of the ratio of pressure p to temperature T, multiplied by the square of a screening length. The model is studied for the specific case of aluminum, for which a perturbation analysis is performed. The results show that the type and rate of relaxation of a perturbation depend strongly on its wavelength and on the screening lengt