35 research outputs found
On the instability of a nonlocal conservation law
We are interested in a nonlocal conservation law which describes the
morphodynamics of sand dunes sheared by a fluid flow, recently proposed by
Andrew C. Fowler. We prove that constant solutions of Fowler's equation are
non-linearly unstable. We also illustrate this fact using a finite difference
scheme
Multiphysics optimal transportation and image analysis
Benamou and Brenier formulation of Monge transportation problem has proven to be of great interest in image processing to compute warpings and distances between pair of images. In some applications, however, the built-in minimization of kinetic energy does not give satisfactory results. In particular cases where some specific regions represent physical objects, it does not make sense, as produces genuine optimal transport, to split, merge or arbitrarily deform these regions along the optimal path. The aim of this work is to introduce several extended energies to take care of physical properties of the image in the interpolation process. We present algorithms to compute approximations of the corresponding generalized optimal transportation plans
Analysis and approximation of a vorticity-velocity-pressure formulation for the Oseen equations
We introduce a family of mixed methods and discontinuous Galerkin
discretisations designed to numerically solve the Oseen equations written in
terms of velocity, vorticity, and Bernoulli pressure. The unique solvability of
the continuous problem is addressed by invoking a global inf-sup property in an
adequate abstract setting for non-symmetric systems. The proposed finite
element schemes, which produce exactly divergence-free discrete velocities, are
shown to be well-defined and optimal convergence rates are derived in suitable
norms. In addition, we establish optimal rates of convergence for a class of
discontinuous Galerkin schemes, which employ stabilisation. A set of numerical
examples serves to illustrate salient features of these methods
Simultaneous denoising and enhancement of signals by a fractal conservation law
In this paper, a new filtering method is presented for simultaneous noise
reduction and enhancement of signals using a fractal scalar conservation law
which is simply the forward heat equation modified by a fractional
anti-diffusive term of lower order. This kind of equation has been first
introduced by physicists to describe morphodynamics of sand dunes. To evaluate
the performance of this new filter, we perform a number of numerical tests on
various signals. Numerical simulations are based on finite difference schemes
or Fast and Fourier Transform. We used two well-known measuring metrics in
signal processing for the comparison. The results indicate that the proposed
method outperforms the well-known Savitzky-Golay filter in signal denoising.
Interesting multi-scale properties w.r.t. signal frequencies are exhibited
allowing to control both denoising and contrast enhancement
Analysis, simulation and optimization of nonlocal models for coastline morphodynamics.
Ce travail est motivé par une demande croissante d'informations quantitatives sur l'évolution du littoral. Nous avons étudié deux approches pour l'analyse de la dynamique sédimentaire. Les deux techniques aboutissent à la résolution de modèles non-locaux pour le fond. L'étude mathématique a porté sur l'analyse de l'existence et l'unicité de perturbations autour des ondes progressives solutions du modèle de Fowler. Nous avons montré que les solutions constantes de l'équation de Fowler sont instables. Pour la simulation numérique de ce modèle, nous avons dans un premier temps considéré des schémas aux différences finies explicites pour lesquels nous avons obtenu des critères de stabilité numérique. Dans un second temps, nous avons utilisé une approche par splitting de sorte à pouvoir résoudre la convection, puis la diffusion et l'anti-diffusion fractionnaire de façon exacte. Ensuite, il est apparu que nous pouvions utiliser les principes de minimisation pour décrire l'évolution d'un lit érodable sous l'action de l'eau où le fond est considéré comme une structure déformable de faible rigidité s'adaptant en minimisant une certaine fonctionnelle d'énergie. Il est intéressant de constater que cette seconde approche peut être liée à la première car elle débouche aussi sur une équation de type Exner avec un terme non-local. En nous inspirant du modèle morphodynamique non-local de Fowler, nous concluons cette thèse par une application exotique au traitement de signal où nous proposons une nouvelle méthode de filtrage.This work is motivated by a growing demand for quantitative information on the evolution of the coastline.We have studied two approaches for the analysis of sand morphodynamics.Both techniques lead to the resolution of nonlocal models for the seabottom.The mathematical study focused on the analysis of the existence and uniqueness of perturbations around the travelling-waves solutions of the Fowler model. We have shown that constant solutions of Fowler's equation are unstable.For the numerical simulation of this model, we have first considered explicit finite difference schemes for which we got numerical stability criteria. We have next used an approach by splitting method in order to solve first the convection, then the diffusion/fractional anti-diffusion exactly. We have also used minimization principles to describe the evolution of an erodible bed sheared by a fluid flow where the seabed is considered as a deformable structure with low stiffness whichadapts itself by minimizing a certain energy functional. It is interesting to note that this secondapproach can be linked to the first one because it also leads to a new Exner equation with a nonlocal term for the flux. Inspired by Fowler's morphodynamical model, we conclude this dissertation with an unexpected application to signal processing
Global Existence of Solutions to the Fowler Equation in a Neighbourhood of Travelling-Waves
We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation