An algorithm for optimal transport between a simplex soup and a point cloud

We propose a numerical method to find the optimal transport map between a measure supported on a lower-dimensional subset of R^d and a finitely supported measure. More precisely, the source measure is assumed to be supported on a simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we recast this optimal transport problem as the resolution of a non-linear system where one wants to prescribe the quantity of mass in each cell of the so-called Laguerre diagram. We prove the convergence with linear speed of a damped Newton's algorithm to solve this non-linear system. The convergence relies on two conditions: (i) a genericity condition on the point cloud with respect to the simplex soup and (ii) a (strong) connectedness condition on the support of the source measure defined on the simplex soup. Finally, we apply our algorithm in R^3 to compute optimal transport plans between a measure supported on a triangulation and a discrete measure. We also detail some applications such as optimal quantization of a probability density over a surface, remeshing or rigid point set registration on a mesh

Notions of optimal transport theory and how to implement them on a computer

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.Comment: 32 pages, 17 figure

Intersection of paraboloids and application to Minkowski-type problems

In this article, we study the intersection (or union) of the convex hull of N confocal paraboloids (or ellipsoids) of revolution. This study is motivated by a Minkowski-type problem arising in geometric optics. We show that in each of the four cases, the combinatorics is given by the intersection of a power diagram with the unit sphere. We prove the complexity is O(N) for the intersection of paraboloids and Omega(N^2) for the intersection and the union of ellipsoids. We provide an algorithm to compute these intersections using the exact geometric computation paradigm. This algorithm is optimal in the case of the intersection of ellipsoids and is used to solve numerically the far-field reflector problem

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

NEW APPROACHES OF THE FORECAST OF THE AGEING OF PLASMA JET NOZZLE IN INDUSTRIAL SETTINGS OF THERMAL SPRAYING

p.19The optimization of the extension of the lifetime of the torches without hazard of drop of the product is an unceasing industrial difficulty. It needs a reliable criterion to characterize the wear, and thus indicate when the nozzle used should be switched by a new nozzle. These criteria are still quite subjective and hold the operator's experience. In 2003, David RIGOT, in collaboration with the Volvo Aero Corporation, has defined a number of criteria in his thesis dissertation [ ]. This study led to the realization of an electronic module alert detecting the hazardous time of process. The focus method was based on research and tracking characteristic frequencies in arc voltage fluctuations and in acoustic noise that is the image. It seems that a rough statistical study of signals is able to help define and visualize simple criteria ruthless variation is usable in objective phenomena wear alert. This is ad hoc measures for smooth power spectrum [ ] [ ] and its variability. Filtered signal is used to distinguish typically one or two epochs of slow wear followed by an epoch of fast wear. This difference is clear after a suitable choice of used frequency bands. Graphical representations are very expressive; they are supplemented by statistical tests. Signals & method: results, by David RIGOT on campaign made shots in September 2003 at the aircraft manufacturer Volvo Aero. Files led to collect a series of measures standardized on the time life of the nozzles. For one, whose use was particularly long, a few shots "vacuum" were prosecuted beyond of the usual threshold used in order to study the ageing. Sound torch and tension thus collected at various times of operation nozzles, with several frequencies of sampling, feed and no feed of powder. They allow first approach directly extract global statistical parameters which it represents dissimilarity based on time: moments, skewness, kurtosis, entropy [ ]. To each power spectra measurement, time is calculated from each signal multiple sampling. Each of the frequencies spectra is smooth inside of a convolution window by a kernel assessor [ ]. A simultaneous representation of the power of signals and their erraticism are used to visually choose the most discriminating frequency bands. Two sets of settings are then assessed: mean and variance of spectrum in the band selected; global parameters newly evaluated from filtered in the chosen band signals. Results: most selective statistics evolve in a similar manner and allows periods wear, while the overall results are most often efficient. For example templates to slope failure (piecewise linear models) are used. The spectra of tension frequencies and the spectra of sound frequencies give similar results. The sound is usable with the feed or without feed of powder. Sequential simple tests may be expected to assess the time of rapid wear

Dynamique de la rivière Poulvar et morphogenèse de la plaine de Tang-i Bulaghi (Fars, Iran) à l'Holocène. Premiers résultats

International audienceThe geomorphologic study of a part of the Poulvar River basin (Fars, Iran) has brought to light a series of Holocene deposits. The main deposit is 17 m thick, consisting mainly of silty sediments deposited during the early and middle Holocene. These deposits are unusual in such a mountainous environment. The study shows that structural and climate factors could explain this sedimentation. On the one hand, it occurred during the relatively humid Holocene Climatic Optimum, which may have contributed to soil erosion. On the other hand, the plains of structural origin located downstream from the Poulvar basin would explain the high base level of the river, the low incline of the longitudinal profile and the tendency towards sedimentation. This major episode is followed by a long period of erosion with only two phases of sedimentation. Data are still lacking to explain this phenomenon, which probably had two major causes: the return of aridity at about 6000 BP and the Zagros uplift

L'évolution ralentie du milieu naturel dans la steppe aride du nord de la Syrie à l'Holocène

International audienceThis geomorphological study of the Jabbûl lake region shows that landscape change was very slow during middle and late Holocene times. An intense erosional episode occurred at the beginning of the Holocene associated with the climatic phase known as "Holocene climatic optimum" (between 9000 yr BP and 7000 yr BP). It was followed by a decrease in morphogenic activity. Brief erosional episodes followed during the Middle Bronze (2100 to 1600 BC), Roman (64 BC to 395 AD) and Byzantine (395 to 636 AD) periods. These lower intensity episodes generated pebble terraces that are sometimes thick but in each case occupy a very narrow belt in river channels. Furthermore, the general phenomenon of soil erosion observed in the Mediterranean basin, from the Roman period to the early Middle Ages, is unknown here. Reasons given for this are the pattern of mountain topography in northern Syria, which explains the limited rainfall, and the widespread occurrence of resistant bedrock. However, extensive agricultural slope terracing during the Roman and Byzantine periods also assisted in mitigating soil erosion

Parental Height and the Sex Ratio

This paper tests the generalized Trivers Willard hypothesis, which predicts that parents with heritable traits that increase the relative reproductive success of males compared to females will have relatively more males than females. As in Kanazawa (2005) we test if taller mothers have relatively more sons in a pooled sample of Demographic Health Surveys(DHS) from 46 developing countries. Despite using a rich dataset and an array of statistical models that address some of the concerns raised by Gelman (2007), we provide further evidence against the hypothesis.Evolutionary psychology; sex ratio; Generalized Trivers Willard hypothesis (gTWH); height

Analysing the heavy goods vehicle “écotaxe” in France: Why did a promising idea fail in implementation?

In October 2014, France abandoned the implementation of the écotaxe, a major country-wide Electronic Tolling System (ETS) designed to charge for the use of national and local roads, which were not covered by the traditional toll system. The écotaxe originated from a political consensus and was designed with the collaboration of business stakeholders. However, unforeseen implementation difficulties resulted in a renouncement at a late stage, when the infrastructure was already deployed. According to a recent report from the French body in charge of auditing public expenses, it generated a cost of 953 million euros. In analysing what happened during the policy delivery stage, this paper provides insights to policymakers in countries where ETS is envisaged. The ETS system was piloted in summer 2013, and the “go live” date was 1st September 2014. Data was collected ‘live’, in March 2014, featuring 21 interviews, with the aim of better understanding the expected challenges and impacts on the business stakeholders. At the time, stakeholders accepted that the project would go ahead: while further implementation challenges remained, few anticipated them to fully derail the project within 6 months. Retrospective analysis of the data collected in March 2014 can help to deepen policymakers’ insights into why the project ultimately failed and better understand the main lessons to be learnt. The findings show that the écotaxe, presented as environmental taxation by policymakers, had been in reality perceived and accepted by business stakeholders as more of an infrastructure tax. Interestingly, this in itself does not explain the failure. The main explanation is to be found in the perceived inequities associated with the charging approach, excluding privately-operated motorways, and the failure of the ad valorem surcharging system designed as a means of passing the tax costs to shippers. These findings should be valuable for policymakers anywhere the introduction of a similar ETS system is being contemplated

Convergence of a Newton algorithm for semi-discrete optimal transport

Many problems in geometric optics or convex geometry can be recast as optimal transport problems and a popular way to solve these problems numerically is to assume that the source probability measure is absolutely continuous while the target measure is finitely supported. We introduce a damped Newton's algorithm for this type of problems, which is experimentally efficient, and we establish its global linear convergence for cost functions satisfying an assumption that appears in the regularity theory for optimal transport