arXiv subjects

Florent Dewez

Publications and source records attributed to Florent Dewez.

6 recordsLinked to original sources

arXiv2021

Time-asymptotic propagation of approximate solutions of Schrödinger equations with both potential and initial condition in Fourier-frequency bands

In this paper, we consider the Schrödinger equation in one space-dimension with potential and we aim at exhibiting dynamic interaction phenomena produced by the potential. To this end, we focus our attention on the time-asymptotic behaviour of the two first terms of the Dyson-Phillips series, which gives a representation of the solution of the equation according to semigroup theory. The first term is actually the free wave packet while the second term corresponds to the wave packet resulting from a first interaction between the free solution and the potential. In order to follow a method developed in a series of papers and aiming at describing propagation features of wave packets, we suppose that both the potential and the initial datum are in bounded Fourier-frequency bands; in particular a family of potentials satisfying this hypothesis is constructed for illustration. We show then that the two terms are time-asymptotically localised in space-time cones which depend explicitly on the frequency bands. Since the inclination and the width of these cones indicate the time-asymptotic motion and dispersion of the two terms, our approach permits to highlight interaction phenomena produced by the potential.

math.AP↗

arXiv2021

An end-to-end data-driven optimisation framework for constrained trajectories

Many real-world problems require to optimise trajectories under constraints. Classical approaches are based on optimal control methods but require an exact knowledge of the underlying dynamics, which could be challenging or even out of reach. In this paper, we leverage data-driven approaches to design a new end-to-end framework which is dynamics-free for optimised and realistic trajectories. We first decompose the trajectories on function basis, trading the initial infinite dimension problem on a multivariate functional space for a parameter optimisation problem. A maximum \emph{a posteriori} approach which incorporates information from data is used to obtain a new optimisation problem which is regularised. The penalised term focuses the search on a region centered on data and includes estimated linear constraints in the problem. We apply our data-driven approach to two settings in aeronautics and sailing routes optimisation, yielding commanding results. The developed approach has been implemented in the Python library PyRotor.

stat.AP↗

arXiv2021

From industry-wide parameters to aircraft-centric on-flight inference: improving aeronautics performance prediction with machine learning

Aircraft performance models play a key role in airline operations, especially in planning a fuel-efficient flight. In practice, manufacturers provide guidelines which are slightly modified throughout the aircraft life cycle via the tuning of a single factor, enabling better fuel predictions. However this has limitations, in particular they do not reflect the evolution of each feature impacting the aircraft performance. Our goal here is to overcome this limitation. The key contribution of the present article is to foster the use of machine learning to leverage the massive amounts of data continuously recorded during flights performed by an aircraft and provide models reflecting its actual and individual performance. We illustrate our approach by focusing on the estimation of the drag and lift coefficients from recorded flight data. As these coefficients are not directly recorded, we resort to aerodynamics approximations. As a safety check, we provide bounds to assess the accuracy of both the aerodynamics approximation and the statistical performance of our approach. We provide numerical results on a collection of machine learning algorithms. We report excellent accuracy on real-life data and exhibit empirical evidence to support our modelling, in coherence with aerodynamics principles.

stat.AP↗

arXiv2020

Stability of propagation features under time-asymptotic approximations for a class of dispersive equations

We consider solutions in frequency bands of dispersive equations on the line defined by Fourier multipliers, these solutions being considered as wave packets. In this paper, a refinement of an existing method permitting to expand time-asymptotically the solution formulas is proposed, leading to a first term inheriting the mean position of the true solution together with a constant variance error. In particular, this first term is supported in a space-time cone whose origin position depends explicitly on the initial state, implying especially a shifted time-decay rate. This method, which takes into account both spatial and frequency information of the initial state, makes then stable some propagation features and permits a better description of the motion and the dispersion of the solutions of interest. The results are achieved firstly by making apparent the cone origin in the solution formula, secondly by applying precisely an adapted version of the stationary phase method with a new error bound, and finally by minimizing the error bound with respect to the cone origin.

math.AP↗

arXiv2017

Estimates of oscillatory integrals with stationary phase and singular amplitude: Applications to propagation features for dispersive equations

In this paper, we study time-asymptotic propagation phenomena for a class of dispersive equations on the line by exploiting precise estimates of oscillatory integrals. We propose first an extension of the van der Corput Lemma to the case of phases which may have a stationary point of real order and amplitudes allowed to have an integrable singular point. The resulting estimates provide optimal decay rates which show explicitly the influence of these two particular points. Then we apply these abstract results to solution formulas of a class of dispersive equations on the line defined by Fourier multipliers. Under the hypothesis that the Fourier transform of the initial data has a compact support or an integrable singular point, we derive uniform estimates of the solutions in space-time cones, describing their motions when the time tends to infinity. The method permits also to show that symbols having a restricted growth at infinity may influence the dispersion of the solutions: we prove the existence of a cone, depending only on the symbol, in which the solution is time-asymptotically localized. This corresponds to an asymptotic version of the notion of causality for initial data without compact support.

math.AP↗

arXiv2015

Asymptotic estimates of oscillatory integrals with general phase and singular amplitude: Applications to dispersive equations

In this paper, we furnish van der Corput types estimates for oscillatory integrals with respect to a large parameter, where the phase is allowed to have a stationary point of real order and the amplitude to have an integrable singularity. The resulting estimates show explicitly the influence of these two particular points on the decay. These results are then applied to the solutions of a family of dispersive equations whose generators are Fourier multipliers. We explore the effect of a limitation to compact frequency bands and of singular frequencies of the initial condition on the decay. Uniform estimates in space-time cones as well as L-infinity-norm estimates are furnished and the optimality of the decay rates is proved under certain hypotheses. Moreover the influence of a growth limitation at infinity of the symbols on the dispersion is exhibited.

math.AP↗