Fonction de Heaviside

Résumé

En mathématiques, la fonction de Heaviside (également fonction échelon unité, fonction marche d'escalier), du nom d’Oliver Heaviside, est la fonction indicatrice de {\R}^+. C'est donc la fonction H (discontinue en 0) prenant la valeur 1 pour tous les réels strictement positifs et la valeur 0 pour les réels strictement négatifs. En 0, sa valeur n'a généralement pas d'importance, même si souvent elle vaut 1/2. Présentation et propriétés C'est une primitive de la distribution de Dirac en théorie des distributions. La valeur de H(0) a très peu d'importance, puisque la fonction est le plus souvent utilisée dans une intégrale. Certains auteurs donnent H(0) = 0, d'autres H(0) = 1. La valeur H(0) = 0,5 est souvent utilisée, parce que la fonction obtenue est ainsi symétrique. La définition est alors : :\forall x \in \R,\ H(x)=\left{\begin{matrix} 0 & \mathrm{si} & x < 0 \ \frac{1}{2} & \mathrm{si} & x = 0 \ 1 & \mathrm{si} & x > 0. \end{matrix}\right.

Source officielle

https://fr.wikipedia.org/wiki/Fonction_de_Heaviside

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Let f be an integrable function on RN, a a point in RN and B a complex number. If the mean value of f on the sphere of centre a and radius r tends to B when r tends to 0, we show that the Fourier integral at a of f is summable to B in Cesàro means of order λ > (N-1)/2. Let now U be a bounded open subset of RN whose boundary ∂U is a real analytic submanifold of RN with dimension N-1. We deduce from the preceding result that the Fourier integral at a of the indicator function of U is summable in Cesàro means of order λ > (N-1)/2 to 1 if a ∈ U, to 1/2 if a ∈ ∂U and to 0 if a ∉ U. We then show that if the function defined on ∂U by y → ‖ y - a ‖ has only a finite number of critical points, then we can take λ less or equal to (N-1)/2 ; more precisely, it suffices to have λ > (N-3)/2 + σ(a|∂U), where σ (a|∂U) < 0 is the maximum of the oscillatory indices associated to the critical points of y → ‖ y - a ‖ ; this generalizes results obtained by Pinsky, Taylor and Popov in 1997. Finally, writing μ∂U for the natural measure supported by ∂U, P(D) for a differential operator with constant coefficients of order m and b for a C∞ function on RN, we show that, if a is a point outside ∂U such that ‖ y - a ‖ has only a finite number of critical points on ∂U, the Fourier integral at a of the distribution P(D) bμ∂U is summable to 0 in Cesàro means of order λ > (N-1)/2 + m + σ (a|∂U) ; this generalizes a result obtained by Gonzàlez Vieli in 2002.

EPFL2007

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Numerical simulation of 3D free surface flows, with multiple incompressible immiscible phases. Applications to impulse waves

Alexandre Caboussat, Marco Picasso

A numerical method for the solution to the density-dependent incompressible Navier-Stokes equations modeling the flow of N immiscible incompressible liquid phases with a free surface is proposed. It allows to model the flow of an arbitrary number of liquid phases together with an additional vacuum phase separated with a free surface. It is based on a volume-of-fluid approach involving N indicator functions (one per phase, identified by its density) that guarantees mass conservation within each phase. An additional indicator function for the whole liquid domain allows to treat boundary conditions at the interface between the liquid domain and a vacuum. The system of partial differential equations is solved by implicit operator splitting at each time step: first, transport equations are solved by a forward characteristics method on a fine Cartesian grid to predict the new location of each liquid phase; second, a generalized Stokes problem with a density-dependent viscosity is solved with a FEM on a coarser mesh of the liquid domain. A novel algorithm ensuring the maximum principle and limiting the numerical diffusion for the transport of the N phases is validated on benchmark flows. Then, we focus on a novel application and compare the numerical and physical simulations of impulse waves, that is, waves generated at the free surface of a water basin initially at rest after the impact of a denser phase. A particularly useful application in hydraulic engineering is to predict the effects of a landslide-generated impulse wave in a reservoir. Copyright (c) 2014 John Wiley & Sons, Ltd.

Wiley-Blackwell2014

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Robust and Efficient Data Clustering with Signal Processing on Graphs

Lionel Jérémie Martin

Data is pervasive in today's world and has actually been for quite some time. With the increasing volume of data to process, there is a need for faster and at least as accurate techniques than what we already have. In particular, the last decade recorded the effervescence of social networks and ubiquitous sensing (through smartphones and the Internet of Things). These phenomena, including also the progresses in bioinformatics and traffic monitoring, pushed forward the research on graph analysis and called for more efficient techniques. Clustering is an important field of machine learning because it belongs to the unsupervised techniques (i.e., one does not need to possess a ground truth about the data to start learning). With it, one can extract meaningful patterns from large data sources without requiring an expert to annotate a portion of the data, which can be very costly. However, the techniques of clustering designed so far all tend to be computationally demanding and have trouble scaling with the size of today's problems. The emergence of Graph Signal Processing, attempting to apply traditional signal processing techniques on graphs instead of time, provided additional tools for efficient graph analysis. By considering the clustering assignment as a signal lying on the nodes of the graph, one may now apply the tools of GSP to the improvement of graph clustering and more generally data clustering at large. In this thesis, we present several techniques using some of the latest developments of GSP in order to improve the scalability of clustering, while aiming for an accuracy resembling that of Spectral Clustering, a famous graph clustering technique that possess a solid mathematical intuition. On the one hand, we explore the benefits of random signal filtering on a practical and theoretical aspect for the determination of the eigenvectors of the graph Laplacian. In practice, this attempt requires the design of polynomial approximations of the step function for which we provided an accelerated heuristic. We used this series of work in order to reduce the complexity of dynamic graphs clustering, the problem of defining a partition to a graph which is evolving in time at each snapshot. We also used them to propose a fast method for the determination of the subspace generated by the first eigenvectors of any symmetrical matrix. This element is useful for clustering as it serves in Spectral Clustering but it goes beyond that since it also serves in graph visualization (with Laplacian Eigenmaps) and data mining (with Principal Components Projection). On the other hand, we were inspired by the latest works on graph filter localization in order to propose an extremely fast clustering technique. We tried to perform clustering by only using graph filtering and combining the results in order to obtain a partition of the nodes. These different contributions are completed by experiments using both synthetic datasets and real-world problems. Since we think that research should be shared in order to progress, all the experiments made in this thesis are publicly available on my personal Github account.

EPFL2018

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Robust and Efficient Data Clustering with Signal Processing on Graphs

Lionel Jérémie Martin

EPFL2018