Tahar BOULMEZAOUD - Academia.edu (original) (raw)

Papers by Tahar BOULMEZAOUD

Image Processing On Line, 2020

We describe a transparent method calculating an "effective reproduction number (ERN)" from the da... more We describe a transparent method calculating an "effective reproduction number (ERN)" from the daily count (incidence) of newly detected cases in each country, in the EU and in each US state. We aim at getting a result as faithful as possible to the observed data, which are very noisy. The noise, being specific of administrations, shows a seven days period. Hence the incidence curve is first filtered by a seven days mean or median filter. Then the ERN is computed by a classic reproduction formula due to Nishiura. To do so requires knowledge of the serial interval function Φ(s) which models the time between the onset of symptoms in a primary case and the onset of symptoms in secondary cases, or equivalently the probability that a person confirmed infected today was actually infected s days earlier by another confirmed infected person. We use and compare several recently proposed evaluations of Φ, and verify that their variation has moderate practical incidence on the evaluation of the ERN. The method we present derives from Nishiura's formula but we prove that for the adequate choice of parameters it is identical to one of the methods proposed by the classic EpiEstim (Estimate Time Varying Reproduction Numbers from Epidemic Curves) software. We find that the same method can be applied to compute an effective reproduction number from the daily death count, which yields therefore another prediction of the expansion of the pandemic. Although this application has no clear theoretical justification, we find good experimental fit of the ERN curves obtained from the incidence and from the death curve, up to a time shift. In most countries, both curves appear to be similar, with a time delay that depends on each country's detection and administrative processing delays. Both ERNs can be consulted daily online in the demo tag associated with this paper. We refer the readers to the online demo 1 to experiment by themselves. In the case of France, an ERN based on hospitalizations, new entries in ICU's and deaths at hospitals is also computed daily 2 .

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific ... more HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. An iterative method for solving non-linear hydromagnetic equations

Dans cette etude, on s'interesse a la dynamique d'une epidemie de type Covid-19 avec un d... more Dans cette etude, on s'interesse a la dynamique d'une epidemie de type Covid-19 avec un double objectif. Le premier objectif est de proposer un modele mathematique macroscopique fiable, s'appuyant sur les donnees reelles et integrant les differents delais cliniques inherents a la maladie, tout en restant tres peu consommateur de donnees et de temps de calcul. Il s'agit d'un modele temporel discret, journalier, a diffe-rents delais de retard, et dont les parametres sont calibres a partir des principaux indicateurs macroscopiques de l'epidemie. Le modele se decline en deux versions decouplees: une version mortalite-mortalite, utili-sable avec les donnees sur le nombre de deces, et une version infection-infection utilisable quand on dispose d'estimations fiables des taux d'infection de la population. Le modele permet de decrire de maniere rea-liste l'evolution des principaux marqueurs de l'epidemie: nombre de deces (ou de deces a l'hopital), ...

Mathematical Modelling of Natural Phenomena, 2020

This study looks at the dynamics of a Covid-19 type epidemic with a dual purpose. The first objec... more This study looks at the dynamics of a Covid-19 type epidemic with a dual purpose. The first objective is to propose a reliable temporal mathematical model, based on real data and integrating the course of illness. It is a daily discrete model with different delay times, and whose parameters are calibrated from the main indicators of the epidemic. The model can be broken down in two decoupled versions: a mortality-mortality version, which can be used with the data on the number of deaths, and an infection-infection version to be used when reliable estimates of infection rate are available. The model allows to describe realistically the evolution of the main markers of the epidemic. In addition, in terms of deaths and occupied ICU beds, the model is not very sensitive to the current uncertainties about IFR. The second objective is to study several original scenarios for the epidemic’s evolution, especially after the period of strict lockdown. A coherent strategy is therefore proposed ...

SIAM Journal on Optimization, 2018

We disclose an interesting connection between the gradient flow of a C 2smooth function ψ and str... more We disclose an interesting connection between the gradient flow of a C 2smooth function ψ and strongly evanescent orbits of the second order gradient system defined by the square-norm of ∇ψ, under adequate convexity assumption. As a consequence, we obtain the following surprising result for two C 2 , convex and bounded from below functions ψ 1 , ψ 2 : if ||∇ψ 1 || = ||∇ψ 2 ||, then ψ 1 = ψ 2 + k, for some k ∈ R.

Hiroshima Mathematical Journal, 2005

We deal with Oseen's equations in the whole space. A class of existence, uniqueness and regularit... more We deal with Oseen's equations in the whole space. A class of existence, uniqueness and regularity results for both the scalar and the vectorial equations are given. Isotropic weighted Sobolev spaces are used for describing the growth or the decay of functions at infinity.

We propose a new model for recovering an image from a blurry observation. This model is based on ... more We propose a new model for recovering an image from a blurry observation. This model is based on a new form of the blur operator and consists to solve a particular optimization problem which is intimately linked to the moment problem. We combine recent mathematical results on moments and semidefinite programming to provide a reliable skill for deblurring images. The methodology allows to handle isotropic and anisotropic blur.

ESAIM: Mathematical Modelling and Numerical Analysis, 2016

We propose a new spectral method for solving multi-dimensional second order elliptic equations wi... more We propose a new spectral method for solving multi-dimensional second order elliptic equations with varying coefficients in the whole space. This method employs an orthogonal family of quasi-rational functions recently discovered by Arar and Boulmezaoud. After proving an error estimate, we present some computational tests which demonstrate the efficiency of the method and the significance of its developmental potential.

2010 IEEE 11th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2010

We propose a new strategy for improving classical Blind Source Separation (BSS) methods. This str... more We propose a new strategy for improving classical Blind Source Separation (BSS) methods. This strategy consists in denoising both the observed signal and the estimated source signal, and is based on the minimization of regularized criterion which takes into account the Total Variation of the signal. We prove by the way that the method leads to a projection problem which is solved by means of projected gradient algorithm. The effectiveness and the robustness of the proposed separating process are shown on numerical examples.

Discrete and Continuous Dynamical Systems - Series S, 2011

In this paper we compare some families of weigthted Sobolev spaces which are commonly used for so... more In this paper we compare some families of weigthted Sobolev spaces which are commonly used for solving partial differential equations in unbounded domains. The first result is an identity between two particular spaces. The second result is another identity which generalises partially the first one.

We present some preliminary results on different mathematical problems encountered in attempts to... more We present some preliminary results on different mathematical problems encountered in attempts to reconstruct the coronal magnetic field, assumed to be in a force-free state, from its values in the photosphere. We discuss the formulations associated with these problems, and some new numerical methods that can be used to get their approximate solutions. Both the linear constant-and the nonlinear cases are considered. We also discuss the possible use of dynamical 3D MHD codes to construct approximate solutions of the equilibrium force-free equations, which are needed for testing numerical extrapolation schemes.

Lecture Notes in Computational Science and Engineering, 2015

Preface.- Part I: Space Discretisation Methods for PDEs.- Part II: Time Integration Schemes.- Par... more Preface.- Part I: Space Discretisation Methods for PDEs.- Part II: Time Integration Schemes.- Part III: A Posteriori Error Estimation and Adaptive Methods.- Part IV: Numerical Linear Algebra.- Part V: Multiscale Modeling and Simulation.- Part VI: Reduced Order Modeling.- Part VII: Optimal Control.- Part VIII: Uncertainty, Stochastic Modeling and Applications.- Part IX: Solvers, High Performance Computing and Software Libraries.- Part X: Computational Fluid and Structural Mechanics.- Part XI: Computational Electromagnetics.

In this paper we deal with time-harmonic Maxwell's equations in Lipschitz and multiply conne... more In this paper we deal with time-harmonic Maxwell's equations in Lipschitz and multiply connected bounded regions of IR3. We prove the wellposedness of the

SIAM Journal on Scientific Computing, 2009

For solving the non-linear equations governing force-free fields, an iterative methodology based ... more For solving the non-linear equations governing force-free fields, an iterative methodology based on the splitting of the problem is described. On the basis of this splitting, three families of subproblems have to be solved numerically. The first problem consists to find a potential field. A mixed hybrid method is used to solve it. The second problem, which is a curl-div system, is solved by means of a mixed method. The last problem is a transport equation which is approximated using a streamline diffusion technique. Numerical 3D experiences and results are given to illustrate the efficiency of the method.

Mathematical Methods in the Applied Sciences, 2005

The main objective of this paper is to propose two vorticity-vector potential formulations of the... more The main objective of this paper is to propose two vorticity-vector potential formulations of the Stokes problem in the half space of R 3. Weighted spaces are used for describing the behaviour at large distances. R Ã ESUM Ã E On propose deux formulations de type vorticitÃ e-potentiel vecteur du problÂ eme de Stokes dans le demiespace de R 3. Les espaces de Sobolev avec poids sont utilisÃ es pour dÃ ecrire le comportement des fonctions Â a l'inÿni. Copyright ? 2004 John Wiley & Sons, Ltd.

Journal of Scientific Computing, 2006

The behaviour of the eigenvalues of the spectral second-order differentiation operator is studied... more The behaviour of the eigenvalues of the spectral second-order differentiation operator is studied and the results are used to investigate the boundary observability of the one dimensional wave equation approximated with a spectral Galerkin method. New explicit estimates of the discrete eigenvalues are given. These estimates improve the previous results on the subject especially for the portion of eigenvalues converging exponentially to those of the continuous problem. Although the boundary observability property of the discretized wave equation is not uniform with respect to the discretization parameter, we show that a uniform observability estimate can be obtained by filtering out the highest eigenmodes.

Journal of Mathematical Analysis and Applications, 2013

We study the spectrum of the weighted Laplacian ϱ −1 ∆ in the whole space R n. We prove, under ad... more We study the spectrum of the weighted Laplacian ϱ −1 ∆ in the whole space R n. We prove, under adequate conditions on ϱ −1 , that this spectrum is discrete and we derive an explicit formula for eigenvalues and eigenfunctions when ϱ −1 = (|x| 2 + 1) 2. We get by the way a complete family of rational functions which are mutually orthogonal in a weighted L 2 space.

Journal de Mathématiques Pures et Appliquées, 2013

Abstract We show that, under adequate norms, the Fourier transform is an isometry over a chain of... more Abstract We show that, under adequate norms, the Fourier transform is an isometry over a chain of nested weighted Sobolev spaces. As a result, an infinite number of useful Plancherel-like identities are derived. Possible extensions are discussed, giving rise to some open questions.

IMA Journal of Numerical Analysis, 2005

In this paper we propose a mortar spectral element method for solving Maxwell's equations in 3D b... more In this paper we propose a mortar spectral element method for solving Maxwell's equations in 3D bounded cavities. The method is based on a non-conforming decomposition of the domain into the union of non-overlapping parallelepipeds. After proving an error estimate, we present some 3D computational results which confirm the performance of the method.

ESAIM: Mathematical Modelling and Numerical Analysis, 2005

In this paper, we propose a new numerical method for solving elliptic equations in unbounded regi... more In this paper, we propose a new numerical method for solving elliptic equations in unbounded regions of R n. The method is based on the mapping of a part of the domain into a bounded region. An appropriate family of weighted spaces is used for describing the growth or the decay of functions at large distances. After exposing the main ideas of the method, we analyse carefully its convergence. Some 3D computational results are displayed to demonstrate its efficiency and its high performance.