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Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.
ano.nymous@ccsd.cnrs.fr.invalid (Mokhtar Z. Alaya), Mokhtar Z. Alaya
We consider a repairable system modeled by a semi-Markov process (SMP), where we include a geometric renewal process for system degradation upon repair, and replacement strategies for non-repairable failure or upon N repairs. First Pérez-Ocón and Torres-Castro studied this system (Pérez-Ocón and Torres-Castro in Appl Stoch Model Bus Ind 18(2):157–170, 2002) and proposed availability calculation using the Laplace Transform. In our work, we consider an extended state space for up and down times separately. This allows us to leverage the standard theory for SMP to obtain all reliability related measurements such as reliability, availability (point and steady-state), mean times and rate of occurrence of failures of the system with general initial law. We proceed with a convolution algebra, which allows us to obtain final closed form formulas for the above measurements. Finally, numerical examples are given to illustrate the methodology.
ano.nymous@ccsd.cnrs.fr.invalid (Jingqi Zhang), Jingqi Zhang
Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation.
ano.nymous@ccsd.cnrs.fr.invalid (Mokhtar Z. Alaya), Mokhtar Z. Alaya
Dans ce manuscrit, nous développons une méthode numérique adaptée à la simulation d’écoulements de fluides compressibles non miscibles. Pour modéliser ces écoulements, nous analysons un système totalement conservatif original comptant six équations, ferme par une équation d’état stiffened-gas et une équation d’équilibre en pression. Nous introduisons également un schéma numérique d’ordre 2, en espace et en temps, spécialement conçu pour la capture des interfaces entre les fluides dans des configurations à plusieurs dimensions. Pour atteindre l’ordre 2, nous mettons au point une méthode de reconstruction de pente multidimensionnelle basée sur le critère de stabilité : local extremum diminishing (LED). Le schéma d’ordre 2 associé au modèle totalement conservatif entraine l’apparition d’oscillations dans les profils de pression. Pour éviter ces oscillations parasites, nous démontrons un ensemble de propriétés essentielles. Tout d’abord, nous trouvons des conditions de stabilité, de type CFL, imposées par les reconstructions de pente. Puis, nous démontrons un théorème garantissant la consistance entre l’équation d’énergie et le transport des fractions volumiques. Ensuite, nous proposons une reconstruction de la pression en deux temps pour assurer la positivité de l’énergie interne. Enfin, nous développons une méthode numérique à une seule étape adaptée à la simulation d’écoulements faisant intervenir plus de deux fluides. L’ensemble des résultats présentés dans ce document est illustré par des cas test, à une, deux ou trois dimensions d’espace.
ano.nymous@ccsd.cnrs.fr.invalid (Vincent Mahy), Vincent Mahy
We consider a Semi-Markov Process (SMP) to model the evolution of bladder cancer, which takes different states over time. A multi-state model has been constructed and applied to data collected from 847 patients during a period of fifteen years. Biomedicine databases usually contain censored data and this study shows that, despite this, a good fit of the main survival measures is achieved by using our specific model. This paper aims to present estimators for the semi-Markov kernel, the survival function and the mean time to disease progression. The strong consistency properties of the estimators are proved.
ano.nymous@ccsd.cnrs.fr.invalid (Alicia Perez A. P. das Neves Yedig), Alicia Perez A. P. das Neves Yedig
We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. This paper aims to provide theoretical justifications for the exchangeably weighted bootstrap consistency in the Markov setup. General structural conditions on the classes of functions (possibly unbounded) and the underlying distributions are required to establish our results. This paper provides the first general theoretical study of the bootstrap of the empirical U-processes in the Markov setting. Potential applications include the symmetry test, Kendall’s tau and the test of independence.
ano.nymous@ccsd.cnrs.fr.invalid (Inass Soukarieh), Inass Soukarieh
U-statistics are a fundamental class of statistics derived from modeling quantities of interest characterized by responses from multiple subjects. U-statistics make generalizations the empirical mean of a random variable X to the sum of all k-tuples of X observations. This paper examines a setting for nonparametric statistical curve estimation based on an infinite-dimensional covariate, including Stute’s estimator as a special case. In this functional context, the class of “delta sequence estimators” is defined and discussed. The orthogonal series method and the histogram method are both included in this class. We achieve almost complete uniform convergence with the rates of these estimators under certain broad conditions. Moreover, in the same context, we show the uniform almost-complete convergence for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Among the potential applications are discrimination problems, metric learning and the time series prediction from the continuous set of past values.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
Stute presented the so-called conditional U-statistics generalizing the Nadaraya–Watson estimates of the regression function. Stute demonstrated their pointwise consistency and the asymptotic normality. In this paper, we extend the results to a more abstract setting. We develop an asymptotic theory of conditional U-statistics for locally stationary random fields {Xs,An:sinRn} observed at irregularly spaced locations in Rn=[0,An]d as a subset of Rd. We employ a stochastic sampling scheme that may create irregularly spaced sampling sites in a flexible manner and includes both pure and mixed increasing domain frameworks. We specifically examine the rate of the strong uniform convergence and the weak convergence of conditional U-processes when the explicative variable is functional. We examine the weak convergence where the class of functions is either bounded or unbounded and satisfies specific moment conditions. These results are achieved under somewhat general structural conditions pertaining to the classes of functions and the underlying models. The theoretical results developed in this paper are (or will be) essential building blocks for several future breakthroughs in functional data analysis.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
In this study, we look at the wavelet basis for nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that it is ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination.
ano.nymous@ccsd.cnrs.fr.invalid (Sultana Didi), Sultana Didi
Three dimensional elliptic problems with variable coefficients and line Dirac sources arise in a number of fields. The lack of regularity on the solution prompts users to turn towards alternative variational formulations. Rather than using weighted Sobolev spaces, we prefer the dual variational formulation written in the Hilbertian Lebesgue space, the one used by G. Stampacchia [Séminaire Jean Leray, 1964]. The key work is to show a singular/regular expansion where the singularity of the potential is fully expressed by a convolution formula, based on the Green kernel of the Laplacian. The correction term restores the boundary condition and fits with the standard variational formulation of Poisson equation (in the Sobolev space H^1). We intend to develop a thorough analysis of the proposed expansion while avoiding stringent assumptions on the conductivities. Sharp technical tools, as those developed in [E. Di-Giorgi, Mem. Accad. Sci. Torino. 1957] and [N. G. Meyers Ann. Scuo. Norm. Sup. Pisa, 1963], are necessary in the proofs.
ano.nymous@ccsd.cnrs.fr.invalid (Eya Bejaoui), Eya Bejaoui
By constructing jointly a random graph and an associated exploration process, we define the dynamics of a “parking process” on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given–possibly unbounded–degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.
ano.nymous@ccsd.cnrs.fr.invalid (Paola Bermolen), Paola Bermolen
In this paper, we design a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the energy difference. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions.
ano.nymous@ccsd.cnrs.fr.invalid (André Harnist), André Harnist
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ano.nymous@ccsd.cnrs.fr.invalid (Hanna Bacave), Hanna Bacave
In industrial Computer-Assisted Engineering, it is common to deal with vector fields or multiple field variables. In this paper, different vector-valued extensions of the Empirical Interpolation Method (EIM) are considered. EIM has been shown to be a valuable tool for dimensionality reduction, reduced-order modeling for nonlinear problems and/or synthesis of families of solutions for parametric problems. Besides already existing vector-valued extensions, a new vector-valued EIM-the so-called VEIM approach-allowing interpolation on all the vector components is proposed and analyzed in this paper. This involves vector-valued basis functions, same magic points shared by all the components and linear combination matrices rather than scalar coefficients. Coefficient matrices are determined under constraints of point-wise interpolation properties for all the components and exact reconstruction property for the snapshots selected during the greedy iterative process. For numerical experiments, various vector-valued approaches including VEIM are tested and compared on various one, two and three-dimensional problems. All methods return robustness, stability and rather good convergence properties as soon as the Kolmogorov width of the dataset is not too big. Depending of the use case, a suitable and convenient method can be chosen among the different vector-valued EIM candidates.
ano.nymous@ccsd.cnrs.fr.invalid (Florian de Vuyst), Florian de Vuyst
We propose a way to account for inspection errors in a particular framework. We consider a situation where the lifetime of a system depends essentially of a particular part. A deterioration of this part is regarded as an unacceptable state for the safety of the system and a major renewal is deemed necessary. Thus the statistical analysis of the deterioration time distribution of this part is of primary interest for the preventive maintenance of the system. In this context, we faced the following problem. In the early life of the system, unwarranted renewals of the part are decided upon, caused by overly cautious behaviour. Such unnecessary renewals make the statistical analysis of deterioration time data difficult and can induce an underestimation of the mean life of the part. To overcome this difficulty, we propose to regard the problem as an incomplete data model. We present its estimation under the maximum likelihood methodology. Numerical experiments show that this approach eliminates the pessimistic bias in the estimation of the mean life of the part. We also present a Bayesian analysis of the problem which can be useful in a small sample setting.
ano.nymous@ccsd.cnrs.fr.invalid (Gilles Celeux), Gilles Celeux
The present thesis constructs an alternative framework to online matching algorithms on large graphs. Using the configuration model to mimic the degree distributions of large networks, we are able to build algorithms based on local matching policies for nodes. Thus, we are allowed to predict and approximate the performances of a class of matching policies given the degree distributions of the initial network. Towards this goal, we use a generalization of the differential equation method to measure valued processes. Throughout the text, we provide simulations and a comparison to the seminal work of Karp, Vazirani and Vazirani based on the prevailing viewpoint in online bipartite matching.
ano.nymous@ccsd.cnrs.fr.invalid (Mohamed Habib Aliou Diallo Aoudi), Mohamed Habib Aliou Diallo Aoudi
The main goal of this research is to develop a data-driven reduced order model (ROM) strategy from high-fidelity simulation result data of a full order model (FOM). The goal is to predict at lower computational cost the time evolution of solutions of Fluid-Structure Interaction (FSI) problems. For some FSI applications like tire/water interaction, the FOM solid model (often chosen as quasistatic) can take far more computational time than the HF fluid one. In this context, for the sake of performance one could only derive a reduced-order model for the structure and try to achieve a partitioned HF fluid solver coupled with a ROM solid one. In this paper, we present a datadriven partitioned ROM on a study case involving a simplified 1D-1D FSI problem representing an axisymmetric elastic model of an arterial vessel, coupled with an incompressible fluid flow. We derive a purely data-driven solid ROM for FOM fluid-ROM structure partitioned coupling and present early results.
ano.nymous@ccsd.cnrs.fr.invalid (Azzeddine Tiba), Azzeddine Tiba
The hidden Markov models (HMM) are used in many different fields, to study the dynamics of a process that cannot be directly observed. However, in some cases, the structure of dependencies of a HMM is too simple to describe the dynamics of the hidden process. In particular, in some applications in finance or in ecology, the transition probabilities of the hidden Markov chain can also depend on the current observation. In this work we are interested in extending the classical HMM to this situation. We define a new model, referred to as the Observation Driven-Hidden Markov Model (OD-HMM). We present a complete study of the general non-parametric OD-HMM with discrete and finite state spaces (hidden and observed variables). We study its identifiability. Then we study the consistency of the maximum likelihood estimators. We derive the associated forward-backward equations for the E-step of the EM algorithm. The quality of the procedure is tested on simulated data sets. Finally, we illustrate the use of the model on an application on the study of annual plants dynamics. This works sets theoretical and practical foundations for a new framework that could be further extended, on one hand to the non-parametric context to simplify estimation, and on the other hand to the hidden semi-Markov models for more realism.
ano.nymous@ccsd.cnrs.fr.invalid (Hanna Bacave), Hanna Bacave
We study the time evolution of an increasing stochastic process governed by a first-order stochastic differential system. This defines a particular piecewise deterministic Markov process (PDMP). We consider a Markov renewal process (MRP) associated to the PDMP and its Markov renewal equation (MRE) which is solved in order to obtain a closed-form solution of the transition function of the PDMP. It is then applied in the framework of survival analysis to evaluate the reliability function of a given system. We give a numerical illustration and we compare this analytical solution with the Monte Carlo estimator.
ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet
In this paper, we use a particular piecewise deterministic Markov process (PDMP) to model the evolution of a degradation mechanism that may arise in various structural components, namely, the fatigue crack growth. We first derive some probability results on the stochastic dynamics with the help of Markov renewal theory: a closed-form solution for the transition function of the PDMP is given. Then, we investigate some methods to estimate the parameters of the dynamical system, involving Bogolyubov's averaging principle and maximum likelihood estimation for the infinitesimal generator of the underlying jump Markov process. Numerical applications on a real crack data set are given.
ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet
An implicit Euler finite-volume scheme for a nonlocal cross-diffusion system on the one-dimensional torus, arising in population dynamics, is proposed and analyzed. The kernels are assumed to be in detailed balance and satisfy a weak cross-diffusion condition. The latter condition allows for negative off-diagonal coefficients and for kernels defined by an indicator function. The scheme preserves the nonnegativity of the densities, conservation of mass, and production of the Boltzmann and Rao entropies. The key idea is to "translate" the entropy calculations for the continuous equations to the finite-volume scheme, in particular to design discretizations of the mobilities, which guarantee a discrete chain rule even in the presence of nonlocal terms. Based on this idea, the existence of finite-volume solutions and the convergence of the scheme are proven. As a by-product, we deduce the existence of weak solutions to the continuous cross-diffusion system. Finally, we present some numerical experiments illustrating the behavior of the solutions to the nonlocal and associated local models.
ano.nymous@ccsd.cnrs.fr.invalid (Ansgar Jüngel), Ansgar Jüngel
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions.
ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. Sobolev spaces are the mathematical framework in which most weak formulations of partial derivative equations are stated, and where solutions are sought. These functional spaces are built on integration and measure theory. Hence, this chapter in functional analysis is a mandatory theoretical cornerstone for the definition of the finite element method. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs of the main results from integration and measure theory.
ano.nymous@ccsd.cnrs.fr.invalid (François Clément), François Clément
Statistical models with multiple change points in presence of censored data are used in many fields; however, the theoretical properties of M-estimators of such models have received relatively little attention. The main purpose of the present work is to investigate the asymptotic properties of M-estimators of the parameters of a multiple change-point model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments, in the setting of a known number of change points. Consistency of the M-estimators of the change points is established and the rate of convergence is determined. The asymptotic normality of the M-estimators of the parameters of the within-segment distributions is established. Since the approaches used in the complete data models are not easily extended to multiple change-point models in the presence of censoring, we have used some general results of Kaplan-Meier integrals. We investigate the performance of the methodology for small samples through a simulation study.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions.
ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of global weak solutions to the problem. For this, a semi-discrete in time minimizing movements scheme à la De Giorgi is introduced. First, the existence of solutions to the scheme is established and then, using a priori estimates, it is proved that as the time step goes to zero these solutions converge up to extraction towards a weak solution to the free boundary model.
ano.nymous@ccsd.cnrs.fr.invalid (Benoît Merlet), Benoît Merlet
The variational finite element solution of Cauchy's problem, expressed in the Steklov-Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in [Inverse Problems in Science and Engineering, 18, 1063-1086 (2011)]. The present work concentrates on the numerical analysis of the semi-discrete problem. We perform the mathematical study of the error to rigorously establish the convergence of the global bias-variance error.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
Motivation: Comparing single-stranded nucleic acids (ssNAs) secondary structures is fundamental when investigating their function and evolution and predicting the effect of mutations on their structures. Many comparison metrics exist, although they are either too elaborate or not sensitive enough to distinguish close ssNAs structures. Results: In this context, we developed AptaMat, a simple and sensitive algorithm for ssNAs secondary structures comparison based on matrices representing the ssNAs secondary structures and a metric built upon the Manhattan distance in the plane. We applied AptaMat to several examples and compared the results to those obtained by the most frequently used metrics, namely the Hamming distance and the RNAdistance, and by a recently developed image-based approach. We showed that AptaMat is able to discriminate between similar sequences, outperforming all the other here considered metrics. In addition, we showed that AptaMat was able to correctly classify 14 RFAM families within a clustering procedure.
ano.nymous@ccsd.cnrs.fr.invalid (Thomas Binet), Thomas Binet
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ano.nymous@ccsd.cnrs.fr.invalid (Mustapha Mohammedi), Mustapha Mohammedi
In this dissertation we are concerned with semiparametric models. These models have success and impact in mathematical statistics due to their excellent scientific utility and intriguing theoretical complexity. In the first part of the thesis, we consider the problem of the estimation of a parameter θ, in Banach spaces, maximizing some criterion function which depends on an unknown nuisance parameter h, possibly infinite-dimensional. We show that the m out of n bootstrap, in a general setting, is weakly consistent under conditions similar to those required for weak convergence of the non smooth M-estimators. In this framework, delicate mathematical derivations will be required to cope with estimators of the nuisance parameters inside non-smooth criterion functions. We then investigate an exchangeable weighted bootstrap for function-valued estimators defined as a zero point of a function-valued random criterion function. The main ingredient is the use of a differential identity that applies when the random criterion function is linear in terms of the empirical measure. A large number of bootstrap resampling schemes emerge as special cases of our settings. Examples of applications from the literature are given to illustrate the generality and the usefulness of our results. The second part of the thesis is devoted to the statistical models with multiple change-points. The main purpose of this part is to investigate the asymptotic properties of semiparametric M-estimators with non-smooth criterion functions of the parameters of multiple change-points model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the semiparametric M-estimators of the change-points is established and the rate of convergence is determined. The asymptotic normality of the semiparametric M-estimators of the parameters of the within-segment distributions is established under quite general conditions. We finally extend our study to the censored data framework. We investigate the performance of our methodologies for small samples through simulation studies.
ano.nymous@ccsd.cnrs.fr.invalid (Anouar Abdeldjaoued Ferfache), Anouar Abdeldjaoued Ferfache
We study the existence and uniqueness of a nonlinear system of eikonal equations in one space dimension for any BV initial data. We present two results. In the first one, we prove the existence of a discontinuous viscosity solution without any monotony conditions neither on the velocities nor on the initial data. In the second, we show the continuity of the constructed solution under continuous initial data, and continuous velocities verifying a certain monotony condition. We present an application to a system modeling the dynamics of dislocations densities.
ano.nymous@ccsd.cnrs.fr.invalid (Maryam Al Zohbi), Maryam Al Zohbi
In this thesis, we are mainly interested in the theoretical and numerical study of certain equations that describe the dynamics of dislocation densities. Dislocations are microscopic defects in materials, which move under the effect of an external stress. As a first work, we prove a global in time existence result of a discontinuous solution to a diagonal hyperbolic system, which is not necessarily strictly hyperbolic, in one space dimension. Then in another work, we broaden our scope by proving a similar result to a non-linear eikonal system, which is in fact a generalization of the hyperbolic system studied first. We also prove the existence and uniqueness of a continuous solution to the eikonal system. After that, we study this system numerically in a third work through proposing a finite difference scheme approximating it, of which we prove the convergence to the continuous problem, strengthening our outcomes with some numerical simulations. On a different direction, we were enthused by the theory of differential contraction to evolutionary equations. By introducing a new distance, we create a new family of contracting positive solutions to the evolutionary p-Laplacian equation.
ano.nymous@ccsd.cnrs.fr.invalid (Maryam Al Zohbi), Maryam Al Zohbi
Dans cette thèse, nous nous intéressons à l’analyse théorique et numérique de la dynamique des densités des dislocations. Les dislocations sont des défauts linéaires qui se déplacent dans les cristaux lorsque ceux-ci sont soumis à des contraintes extérieures. D’une manière générale, la dynamique des densités des dislocations est décrite par un système d’équations de transport, où les champs de vitesse dépendent de manière non-locale des densités des dislocations. Au départ, notre travail se focalise sur l’étude d’un système unidimensionnel (2 × 2) de type Hamilton-Jacobi dérivé d’un système bidimensionnel proposé par Groma et Balogh en 1999. Pour ce modèle, nous montrons un résultat d’existence globale et d’unicité. En addition, nous nous intéressons à l’étude numérique de ce problème, complété par des conditions initiales croissantes, en proposant un schéma aux différences finies implicite dont on prouve la convergence. Ensuite, en s’inspirant du travail effectué pour la résolution de la dynamique des densités des dislocations, nous mettons en œuvre une théorie plus générale permettant d’obtenir un résultat similaire d’existence et d’unicité d’une solution dans le cas des systèmes de type eikonal unidimensionnels. En considérant des conditions initiales croissantes, nous faisons une étude numérique pour ce système. Sous certaines conditions de monotonies sur la vitesse, nous proposons un schéma aux différences finies implicite permettant de calculer la solution discrète et simuler ainsi la dynamique des dislocations à travers ce modèle.
ano.nymous@ccsd.cnrs.fr.invalid (Aya Oussaily), Aya Oussaily
Adverse Outcome Pathways (AOPs) are increasingly used to support the integration of in vitro data in hazard assessment for chemicals. Quantitative AOPs (qAOPs) use mathematical models to describe the relationship between key events (KEs). In this paper, data obtained in three cell lines, LHUMES, HepG2 and RPTEC/TERT1, using similar experimental protocols, was used to calibrate a qAOP of mitochondrial toxicity for two chemicals, rotenone and deguelin. The objectives were to determine whether the same qAOP could be used for the three cell types, and to test chemical-independence by cross-validation with a dataset obtained on eight other chemicals in LHUMES cells. Repeating the calibration approach for both chemicals in three cell lines highlighted various practical difficulties. Even when the same readouts of KEs are measured, the mathematical functions used to describe the key event relationships may not be the same. Cross-validation in LHUMES cells was attempted by estimating chemical-specific potency at the molecular initiating events and using the rest of the calibrated qAOP to predict downstream KEs: toxicity of azoxystrobin, carboxine, mepronil and thifluzamide was underestimated.Selection of most relevant readouts and accurate characterization of the molecular initiating event for cross validation are critical when designing in vitro experiments targeted at calibrating qAOPs.
ano.nymous@ccsd.cnrs.fr.invalid (Cleo Tebby), Cleo Tebby
In this paper we analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. We prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, we investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities.
ano.nymous@ccsd.cnrs.fr.invalid (Maxime Herda), Maxime Herda
In this paper, we investigate the asymptotic properties of Le Cam's one-step estimator for weak Fractionally AutoRegressive Integrated Moving-Average (FARIMA) models. For these models, noises are uncorrelated but neither necessarily independent nor martingale differences errors. We show under some regularity assumptions that the onestep estimator is strongly consistent and asymptotically normal with the same asymptotic variance as the least squares estimator. We show through simulations that the proposed estimator reduces computational time compared with the least squares estimator. An application for providing remotely computed indicators for time series is proposed.
ano.nymous@ccsd.cnrs.fr.invalid (Samir Ben Hariz), Samir Ben Hariz
In this paper, we consider the problem of identifying a single moving point source for a three-dimensional wave equation from boundary measurements. Precisely, we show that the knowledge of the field generated by the source at six different points of the boundary over a finite time interval is sufficient to determine uniquely its trajectory. We also derive a Lipschitz stability estimate for the inversion.
ano.nymous@ccsd.cnrs.fr.invalid (Hanin Al Jebawy), Hanin Al Jebawy
The purpose of this note is to provide an approximation for the generalized bootstrapped empirical process achieving the rate in [38]. The proof is based on the same arguments used in [36]. As a conséquence, we establish an approximation of the bootstrapped kernel distribution estimation. Purthermore, our results are applied to two-sample testing procedures as well as to change-point problems. We end with establishing strong approximations of the bootstrapped empirical process when the parameters are estimated.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
Our thesis is dedicated in big part, to solving certain problems in biology (biologic sequences and lifespan using the competing risk framework) under semi-Markovian hypothesis. In recent years, computing the properties of words through random sequences has become a topic of interest in the intersection between mathematics and biology. In the literature, a vast number of methods have tackled this problem under the assumption that sequences of symbols are modeled by Markov processes. Nevertheless, the markovian hypothesis has some disadvantages. In Markov processes, the sojourn time is modeled by the exponential (geometric) distribution in continuous (discrete) time. By contrast, in semi-Markov processes the sojourn time in a state can be modeledby any probability law. Therefore, in order to propose a more general approach to compute the properties of words through a random sequence, in this PhD work weconsider that biological sequences are modeled by semi-Markovian discrete processes. We also compute the average number of times that the elements from a specific set of words appear through a sequence of letters. To achieve our goal, we use the strong law of large numbers and we provide the central limit theorem. To prove the applicationof our proposed model, we find a particular enzyme in a bacteriophage DNA sequence. Competing risk problems conform another interesting topic in the lifespan domain. In general, competing risk problems have been dealt with a statistic approach. In this thesis, we present competing risk models within a semi-Markov framework. We consider continuous and discrete time semi-Markov processes with a finite number of transient and absorbing states. Each absorbing state represents a failure mode (in reliability of a system) or a cause of death of an individual (in survival analysis). We express the probability that a failure occurs at a given time due to a unique cause. We give the joint distribution of the life time and the failure cause via the transition function of the semi-Markov process in continuous and discrete-time respectively. Some examples are given for illustration. We also present a method for solving continuous time Markovian renewal equations based on well-established algorithms in their discrete time corresponding counter parts. The great advantage drawn by this approach is that the in finite series of the renewal function, in continuous time, is replaced, in discrete time, by a finite series. Results for error estimation are also established. To illustrate this approach we propose a digital application concerning cyber-attacks where the functions of conditional transitions are of the Weibull type.
ano.nymous@ccsd.cnrs.fr.invalid (Brenda Ivette Garcia Maya), Brenda Ivette Garcia Maya
In this work, we design and analyze a Hybrid High-Order (HHO) discretization method for incompressible flows of non-Newtonian fluids with power-like convective behaviour. We work under general assumptions on the viscosity and convection laws, that are associated with possibly different Sobolev exponents r ∈ (1, ∞) and s ∈ (1, ∞). After providing a novel weak formulation of the continuous problem, we study its well-posedness highlighting how a subtle interplay between the exponents r and s determines the existence and uniqueness of a solution. We next design an HHO scheme based on this weak formulation and perform a comprehensive stability and convergence analysis, including convergence for general data and error estimates for shear-thinning fluids and small data. The HHO scheme is validated on a complete panel of model problems.
ano.nymous@ccsd.cnrs.fr.invalid (Daniel Castanon Quiroz), Daniel Castanon Quiroz
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $\sigma$-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou's lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $L^p$~spaces such as Banach spaces.
ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
Integration, just as much as differentiation, is a fundamental calculus tool that is widely used in many scientific domains. Formalizing the mathematical concept of integration and the associated results in a formal proof assistant helps in providing the highest confidence on the correctness of numerical programs involving the use of integration, directly or indirectly. By its capability to extend the (Riemann) integral to a wide class of irregular functions, and to functions defined on more general spaces than the real line, the Lebesgue integral is perfectly suited for use in mathematical fields such as probability theory, numerical mathematics, and real analysis. In this article, we present the Coq formalization of $\sigma$-algebras, measures, simple functions, and integration of nonnegative measurable functions, up to the full formal proofs of the Beppo Levi (monotone convergence) theorem and Fatou's lemma. More than a plain formalization of the known literature, we present several design choices made to balance the harmony between mathematical readability and usability of Coq theorems. These results are a first milestone toward the formalization of $L^p$~spaces such as Banach spaces.
ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo
An implicit Euler finite-volume scheme for a parabolic reaction-diffusion system modeling biofilm growth is analyzed and implemented. The system consists of a degenerate-singular diffusion equation for the biomass fraction, which is coupled to a diffusion equation for the nutrient concentration, and it is solved in a bounded domain with Dirichlet boundary conditions. By transforming the biomass fraction to an entropy-type variable, it is shown that the numerical scheme preserves the lower and upper bounds of the biomass fraction. The existence and uniqueness of a discrete solution and the convergence of the scheme are proved. Numerical experiments in one and two space dimensions illustrate, respectively, the rate of convergence in space of our scheme and the temporal evolution of the biomass fraction and the nutrient concentration.
ano.nymous@ccsd.cnrs.fr.invalid (Antoine Zurek), Antoine Zurek
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ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
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ano.nymous@ccsd.cnrs.fr.invalid (Elias Zgheib), Elias Zgheib
Nonlocal cross-diffusion systems on the torus, arising in population dynamics and neuroscience, are analyzed. The global existence of weak solutions, the weak-strong uniqueness, and the localization limit are proved. The kernels are assumed to be positive definite and in detailed balance. The proofs are based on entropy estimates coming from Shannon-type and Rao-type entropies, while the weak-strong uniqueness result follows from the relative entropy method. The existence and uniqueness theorems hold for nondifferentiable kernels. The associated local cross-diffusion system, derived in the localization limit, is also discussed.
ano.nymous@ccsd.cnrs.fr.invalid (Ansgar Jüngel), Ansgar Jüngel
An innovative data-driven model-order reduction technique is proposed to model dilute micrometric or nanometric suspensions of microcapsules, i.e., microdrops protected in a thin hyperelastic membrane, which are used in Healthcare as innovative drug vehicles. We consider a microcapsule flowing in a similar-size microfluidic channel and vary systematically the governing parameter, namely the capillary number, ratio of the viscous to elastic forces, and the confinement ratio, ratio of the capsule to tube size. The resulting space-time-parameter problem is solved using two global POD reduced bases, determined in the offline stage for the space and parameter variables, respectively. A suitable low-order spatial reduced basis is then computed in the online stage for any new parameter instance. The time evolution of the capsule dynamics is achieved by identifying the nonlinear low-order manifold of the reduced variables; for that, a point cloud of reduced data is computed and a diffuse approximation method is used. Numerical comparisons between the full-order fluid-structure interaction model and the reduced-order one confirm both accuracy and stability of the reduction technique over the whole admissible parameter domain. We believe that such an approach can be applied to a broad range of coupled problems especially involving quasistatic models of structural mechanics.
ano.nymous@ccsd.cnrs.fr.invalid (Toufik Boubehziz), Toufik Boubehziz
Concise formulae are given for the cumulant matrices of a real-valued (zero-mean) random vector up to order 6. In addition to usual matrix operations, they involve only the Kronecker product, the vec operator, and the commutation matrix. Orders 5 and 6 are provided here for the first time; the same method as provided in the paper can be applied to compute higher orders. An immediate consequence of these formulae is to return 1) upper bounds on the rank of the cumulant matrices and 2) the expression of the sixth-order moment matrix of a Gaussian vector. Due to their conciseness, the proposed formulae also have a computational advantage as compared to the repeated use of Leonov and Shiryaev formula.
ano.nymous@ccsd.cnrs.fr.invalid (Hanany Ould-Baba), Hanany Ould-Baba
We extend the general stochastic matching model on graphs introduced in [13], to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is 'First Come, First Matched'.
ano.nymous@ccsd.cnrs.fr.invalid (Jocelyn Begeot), Jocelyn Begeot
This thesis is dedicated to the estimation of two statistical models: the simultaneous regression quantiles model and the blind deconvolution model. It therefore consists of two parts. In the first part, we are interested in estimating several quantiles simultaneously in a regression context via the Bayesian approach. Assuming that the error term has an asymmetric Laplace distribution and using the relation between two distinct quantiles of this distribution, we propose a simple fully Bayesian method that satisfies the noncrossing property of quantiles. For implementation, we use Metropolis-Hastings within Gibbs algorithm to sample unknown parameters from their full conditional distribution. The performance and the competitiveness of the underlying method with other alternatives are shown in simulated examples. In the second part, we focus on recovering both the inverse filter and the noise level of a noisy blind deconvolution model in a parametric setting. After the characterization of both the true noise level and inverse filter, we provide a new estimation procedure that is simpler to implement compared with other existing methods. As well, we consider the estimation of the unknown discrete distribution of the input signal. We derive strong consistency and asymptotic normality for all our estimates. Including a comparison with another method, we perform a consistent simulation study that demonstrates empirically the computational performance of our estimation procedures.
ano.nymous@ccsd.cnrs.fr.invalid (Josephine Merhi Bleik), Josephine Merhi Bleik
We present here the results regarding the characterization of chemical composition and size distribution of aerosols released during laser cutting of two types of fuel debris simulants (Ex-Vessel and In-Vessel scenarios) in air and underwater conditions in the context of Fukushima Daiichi dismantling. The aerosols have systematically an aerodynamic mass median diameter below 1 μm, with particle sizes generally comprised between 60 nm and 160 nm for air cutting conditions, and larger diameters (300-400 nm) for underwater experiments. Regarding the chemical composition, iron, chromium and nickel are mainly found by more than 50 % in the samples whereas radioactive surrogates of Uranium (Hafnium) are undetectable. When compositions are transposed to radioactivity, taking into account radioisotope inventories 10 years after the accident, it is well evidenced that the radioactivity is carried out by small particles in air condition tests (median size around 100 nm) than underwater (median size around 400 nm): 50 % of the radioactivity is present in particles below 90 nm, and 99 % below 950 nm. Caesium carries the largest part of the radioactivity at all sizes below 1 μm in the case of an Ex- Vessel fuel debris simulant. For the In-Vessel, the aerosol median size for the radioactivity is situated around 100 nm, with 59 % of the radioactivity is carried by strontium, 17 % by barium and 16 % by minor actinides (modelled by cerium) and 7% by the caesium. For sizes above 1.6 μm, cerium representing alpha particles (surrogate of plutonium) is almost the only radioactivity-bearing element (96–97 % of the radioactivity). The data produced here could already be used for modelling or designing development of strategies to implement insitu the laser cutting for fuel debris retrieval and safety associated strategies.
ano.nymous@ccsd.cnrs.fr.invalid (Claire Dazon), Claire Dazon
We consider a two-component mixture model where one component distribution is known while the mixing proportion and the other component distribution are unknown. These kinds of models were first introduced in biology to study the differences in expression between genes. The various estimation methods proposed till now have all assumed that the unknown distribution belongs to a parametric family. In this paper, we show how this assumption can be relaxed. First, we note that generally the above model is not identifiable, but we show that under moment and symmetry conditions some ‘almost everywhere’ identifiability results can be obtained. Where such identifiability conditions are fulfilled we propose an estimation method for the unknown parameters which is shown to be strongly consistent under mild conditions. We discuss applications of our method to microarray data analysis and to the training data problem. We compare our method to the parametric approach using simulated data and, finally, we apply our method to real data from microarray experiments.
ano.nymous@ccsd.cnrs.fr.invalid (Laurent Bordes), Laurent Bordes
Dans le contexte du démantèlement des réacteurs de Fukushima Daiichi, plusieurs projets ont été subventionnés par le gouvernement japonais pour préparer les opérations de retrait du corium. Dans ce cadre, une étude conjointe menée entre ONET Technologies et les laboratoires du CEA et de l’IRSN a permis de démontrer la faisabilité de l’utilisation de la technique de découpe par laser et d’estimer le terme source aérosol ainsi généré. Deux simulants du corium synthétisés et caractérisés par le CEA-Cadarache ont fait l’objet d’essais de tirs laser sous air et sous eau au sein de l’installation DELIA du CEA Saclay, et les aérosols émis ont été caractérisés par l’IRSN. La caractérisation des particules émises en termes de concentration et de distribution granulométrique a permis d’apporter des informations pour prédire notamment le transport et le dépôt des particules, mais la connaissance de la composition chimique par classe de taille est une information nécessaire pour une meilleure gestion des risques professionnels et environnementaux. Cet article présente les résultats concernant la caractérisation de la composition chimique de l’aérosol d’un simulant du corium, en condition de découpe laser sous air, et la distribution granulométrique associée
ano.nymous@ccsd.cnrs.fr.invalid (Emmanuel Porcheron), Emmanuel Porcheron
The first aim of the present paper, is to establish strong approximations of the uniform non-overlapping k-spacings process extending the results of Aly et al. (1984). Our methods rely on the invariance principle in Mason and van Zwet (1987). The second goal, is to generalize the Dindar (1997) results for the increments of the spacings quantile process to the uniforme non-overlapping k-spacings quantile process. We apply the last result to characterize the limit laws of functionals of the increments k-spacings quantile process.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
We consider in this paper a model parabolic variational inequality. This problem is discretized with conforming Lagrange finite elements of order $p ≥ 1$ in space and with the backward Euler scheme in time. The nonlinearity coming from the complementarity constraints is treated with any semismooth Newton algorithm and we take into account in our analysis an arbitrary iterative algebraic solver. In the case $p = 1$, when the system of nonlinear algebraic equations is solved exactly, we derive an a posteriori error estimate on both the energy error norm and a norm approximating the time derivative error. When $p ≥ 1$, we provide a fully computable and guaranteed a posteriori estimate in the energy error norm which is valid at each step of the linearization and algebraic solvers. Our estimate, based on equilibrated flux reconstructions, also distinguishes the discretization, linearization, and algebraic error components. We build an adaptive inexact semismooth Newton algorithm based on stopping the iterations of both solvers when the estimators of the corresponding error components do not affect significantly the overall estimate. Numerical experiments are performed with the semismooth Newton-min algorithm and the semismooth Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative algebraic solver to illustrate the strengths of our approach.
ano.nymous@ccsd.cnrs.fr.invalid (Jad Dabaghi), Jad Dabaghi
We propose an adaptive inexact version of a class of semismooth Newton methods that is aware of the continuous (variational) level. As a model problem, we study the system of variational inequalities describing the contact between two membranes. This problem is discretized with conforming finite elements of order $p \geq 1$, yielding a nonlinear algebraic system of variational inequalities. We consider any iterative semismooth linearization algorithm like the Newton-min or the Newton--Fischer--Burmeister which we complementby any iterative linear algebraic solver. We then derive an a posteriori estimate on the error between the exact solution at the continuous level and the approximate solution which is valid at any step of the linearization and algebraic resolutions. Our estimate is based on flux reconstructions in discrete subspaces of $\mathbf{H}(\mathrm{div}, \Omega)$ and on potential reconstructions in discrete subspaces of $H^1(\Omega)$ satisfying the constraints. It distinguishes the discretization, linearization, and algebraic components of the error. Consequently, we can formulate adaptive stopping criteria for both solvers, giving rise to an adaptive version of the considered inexact semismooth Newton algorithm. Under these criteria, the efficiency of the leading estimates is also established, meaning that we prove them equivalent with the error up to a generic constant. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.
ano.nymous@ccsd.cnrs.fr.invalid (Jad Dabaghi), Jad Dabaghi
Let (S D-Omega) be the Stokes operator defined in a bounded domain Omega of R-3 with Dirichlet boundary conditions. We prove that, generically with respect to the domain Omega with C-5 boundary, the spectrum of (S D-Omega) satisfies a non-resonant property introduced by C. Foias and J.C. Saut in [17] to linearize the Navier-Stokes system in a bounded domain Omega of R-3 with Dirichlet boundary conditions. For that purpose, we first prove that, generically with respect to the domain Omega with C-5 boundary, all the eigenvalues of (SD Omega) are simple. That answers positively a question raised by J.H. Ortega and E. Zuazua in [27, Section 6]. The proofs of these results follow a standard strategy based on a contradiction argument requiring shape differentiation. One needs to shape differentiate at least twice the initial problem in the direction of carefully chosen domain variations. The main step of the contradiction argument amounts to study the evaluation of Dirichlet-to-Neumann operators associated to these domain variations. (C) 2014 Elsevier Masson SAS. All rights reserved.
ano.nymous@ccsd.cnrs.fr.invalid (Yacine Chitour), Yacine Chitour
In this thesis, we are interested in the theoretical and numerical analysis o the dynamics of dislocation densities, where dislocations are crystalline defects appearing at the microscopic scale in metallic alloys. Particularly, the study of the Groma-Czikor-Zaiser model (GCZ) and the study of the Groma-Balog model (GB) are considered. The first is actually a system of parabolic type equations, where as, the second is a system of non-linear Hamilton-Jacobi equations. Initially, we demonstrate an existence and uniqueness result of a regular solution using a comparison principle and a fixed point argument for the GCZ model. Next, we establish a time-based global existence result for the GB model, based on notions of discontinuous viscosity solutions and a new estimate of total solution variation, as well as finite velocity propagation of the governed equations. This result is extended also to the cas of general Hamilton-Jacobi equation systems. Finally, we propose a semi-explicit numerical scheme allowing the discretization of the GB model. Based on the theoretical study, we prove that the discrete solution converges toward the continuous solution, as well as an estimate of error between the continuous solution and the numerical solution has been established. Simulations showing the robustness of the numerical scheme are also presented.
ano.nymous@ccsd.cnrs.fr.invalid (Vivian Rizik), Vivian Rizik
This paper investigates an identifiability method for a class of systems of reaction diffusion equations in the L^2 framework. This class is composed of a master system of ordinary differential equations coupled with a slave system of diffusion equations. It can model two populations, the second one being diffusive contrary to the first one. The identifiability method is based on an elimination procedure providing relations called input-output polynomials and linking the unknown parameters , the inputs and the outputs of the model. These polynomials can also be used to estimate the parameters as shown in this article. To our best knowledge, such an identifiability method and a parameter estimation procedure have not yet been explored for such a system in the L^2 framework. This work is applied on an epidemiological model describing the propagation of the chikungunya in a local population.
ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière
We introduce the binacox, a prognostic method to deal with the problem of detect- ing multiple cut-points per features in a multivariate setting where a large number of continuous features are available. The method is based on the Cox model and com- bines one-hot encoding with the binarsity penalty, which uses total-variation regular- ization together with an extra linear constraint, and enables feature selection. Original nonasymptotic oracle inequalities for prediction (in terms of Kullback-Leibler diver- gence) and estimation with a fast rate of convergence are established. The statistical performance of the method is examined in an extensive Monte Carlo simulation study, and then illustrated on three publicly available genetic cancer datasets. On these high- dimensional datasets, our proposed method signi cantly outperforms state-of-the-art survival models regarding risk prediction in terms of the C-index, with a computing time orders of magnitude faster. In addition, it provides powerful interpretability from a clinical perspective by automatically pinpointing signi cant cut-points in relevant variables.
ano.nymous@ccsd.cnrs.fr.invalid (Simon Bussy), Simon Bussy
We discuss the use of a continuous-time jump Markov process as the driving process in stochastic differential systems. Results are given on the estimation of the infinitesimal generator of the jump Markov process, when considering sample paths on random time intervals. These results are then applied within the framework of stochastic dynamical systems modeling and estimation. Numerical examples are given to illustrate both consistency and asymptotic normality of the estimator of the infinitesimal generator of the driving process. We apply these results to fatigue crack growth modeling as an example of a complex dynamical system, with applications to reliability analysis.
ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet
This paper is devoted to the identification of a pollution source in a river. A simple mathematical model of such a problem is given by a one-dimensional linear advection–dispersion–reaction equation with a right hand side spatially supported in a point (the source) and a time varying intensity, both unknown. There exist some identifiability results about this distributed system. But the numerical estimation of the unknown quantities require the introduction of an approximated model, whose identifiability properties are not analyzed usually. This paper has a double purpose: – to do the identifiability analysis of the differential system considered for estimating the parameters, – to propose a new numerical global search of these parameters, based on the previous analysis. Another consequence of this approach is to give the unknown pollution intensity directly as the solution of a differential equation. Lastly, the numerical algorithm is described in detail, completed with some applications.
ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière
We consider Bienaymé-Galton-Watson and continuous-time Markov branching processes and prove diffusion approximation results in the near critical case, in fixed and random environment. In one hand, in the fixed environment case, we give new proofs and derive necessary and sufficient conditions for diffusion approximation to get hold of Feller-Jiřina and Jagers theorems. In the other hand, we propose a continuous-time Markov branching process with random environments and obtain diffusion approximation results. An averaging result is also presented. Proofs here are new, where weak convergence in the Skorohod space is proved via singular perturbation technique for convergence of generators and tightness of the distributions of the considered families of stochastic processes.
ano.nymous@ccsd.cnrs.fr.invalid (Nikolaos Limnios), Nikolaos Limnios
For each pair ε = (ε 1 , ε 2) of positive parameters, we define a perforated domain Ω ε by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n (n ≥ 3). The hole is situated at distance ε 1 from the outer boundary ∂Ω of the domain. Then, when ε → (0, 0) both the size of the hole and its distance from ∂Ω tend to zero, but the size shrinks faster than the distance. In such perforated domain Ω ε we consider a Dirichlet problem for the Laplace equation and we denote by u ε its solution. Our aim is to represent the map that takes ε to u ε in term of real analytic functions of ε defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of u ε to suitable subsets of Ω ε , we prove a global representation formula that holds on the whole of Ω ε. Such a formula allows to rigorously justify multi-scale expansions, which we subsequently construct.
ano.nymous@ccsd.cnrs.fr.invalid (Virginie Bonnaillie-Noël), Virginie Bonnaillie-Noël
L'analyse par microsonde électronique (EPMA) permet de quantifier, avec une grande précision, les concentrations élémentaires d'échantillons de compositions inconnues. Elle permet, par exemple, de quantifier les actinides présents dans les combustibles nucléaires neufs ou irradiés, d'aider à la gestion des déchets nucléaires ou encore de dater certaines roches. Malheureusement, ces analyses quantitatives ne sont pas toujours réalisables dû à l'indisponibilité des étalons de référence pour certains actinides. Afin de pallier cette difficulté, une méthode d'analyse dite « sans standard » peut-être employée au moyen d'étalons virtuels. Ces derniers sont obtenus à partir de formules empiriques ou à partir de calculs basés sur des modèles théoriques. Toutefois, ces calculs requièrent la connaissance de paramètres physiques généralement mal connus, comme c'est le cas pour les sections efficaces de production de rayons X. La connaissance précise de ces sections efficaces est requise dans de nombreuses applications telles que dans les codes de transport de particules et dans les simulations Monte-Carlo. Ces codes de calculs sont très utilisés en médecine et particulièrement en imagerie médicale et dans les traitements par faisceau d'électrons. Dans le domaine de l'astronomie, ces données sont utilisées pour effectuer des simulations servant à prédire les compositions des étoiles et des nuages galactiques ainsi que la formation des systèmes planétaires.Au cours de ce travail, les sections efficaces de production des raies L et M du plomb, du thorium et de l'uranium ont été mesurées par impact d'électrons sur des cibles minces autosupportées d'épaisseur variant de 0,2 à 8 nm. Les résultats expérimentaux ont été comparés avec les prédictions théoriques de sections efficaces d'ionisation calculées grâce à l'approximation de Born en ondes distordues (DWBA) et avec les prédictions de formules analytiques utilisées dans les applications pratiques. Les sections efficaces d'ionisation ont été converties en sections efficaces de productions de rayons X grâce aux paramètres de relaxation atomique extraits de la littérature. Les résultats théoriques du modèle DWBA sont en excellents accords avec les résultats expérimentaux. Ceci permet de confirmer les prédictions de ce modèle et de valider son utilisation pour le calcul de standards virtuels.Les prédictions de ce modèle ont été intégrées dans le code Monte-Carlo PENELOPE afin de calculer l'intensité de rayons X produite par des standards pur d'actinides. Les calculs ont été réalisés pour les éléments dont le numéro atomique est 89 ≤ Z ≤ 99 et pour des tensions d'accélération variant du seuil d'ionisation jusque 40 kV, par pas de 0,5 kV. Pour une utilisation pratique, les intensités calculées pour les raies L et M les plus intenses ont été regroupées dans une base de données.Les prédictions des standards virtuels ainsi obtenus ont été comparées avec des mesures effectuées sur des échantillons de composition connue (U, UO2, ThO2, ThF4, PuO2…) et avec les données acquises lors de précédentes campagnes de mesures. Le dosage des actinides à l'aide de ces standards virtuels a montré un bon accord avec les résultats attendus. Ceci confirme la fiabilité des standards virtuels développés et démontre que la quantification des actinides par microsonde électronique est réalisable sans standards d'actinides et avec un bon niveau de confiance.
ano.nymous@ccsd.cnrs.fr.invalid (Aurélien Moy), Aurélien Moy
This paper deals with optimal input design for parameter estimation in a bounded-error context. Uncertain controlled nonlinear dynamical models, when the input can be parametrized by a finite number of parameters, are considered. The main contribution of this paper concerns criteria for obtaining optimal inputs in this context. Two input design criteria are proposed and analysed. They involve sensitivity functions. The first criterion requires the inversion of the Gram matrix of sensitivity functions. The second one does not require this inversion and is then applied for parameter estimation of a model taken from the aeronautical domain. The estimation results obtained using an optimal input are compared with those obtained with an input optimized in a more classical context (Gaussian measurement noise and parameters a priori known to belong to some boxes). These results highlight the potential of optimal input design in a bounded-error context.
ano.nymous@ccsd.cnrs.fr.invalid (Carine Jauberthie), Carine Jauberthie
One of the important challenges for the decommissioning of the damaged reactors of the Fukushima Daiichi Nuclear Power Plant is the safe retrieval of the fuel debris or corium. It is especially primordial to investigate the cutting conditions for air configuration and for underwater configuration at different water levels. Concerning the cutting techniques, the laser technique is well adapted to the cutting of expected material such as corium that has an irregular shape and heterogeneous composition. A French consortium (ONET Technologies, CEA and IRSN) is being subsidized by the Japanese government to implement R&D related to the laser cutting of Fukushima Daiichi fuel debris and related to dust collection technology. Debris simulant have been manufactured in the PLINIUS platform to represent Molten Core Concrete Interaction as estimated from Fukushima Daiichi calculations. In this simulant, uranium is replaced by hafnium and the major fission products have been replaced by their natural isotopes. During laser cutting experiments in the DELIA facility, aerosols have been collected thanks to filters and impactors. The collected aerosols have been analyzed. Both chemical analysis (dissolution + ICP MS and ICP AES) and microscopic analyses (SEM EDS) will be presented and discussed. These data provide insights on the expected dust releases during cutting and can be converted to provide radioactivity estimates. They have also been successfully compared to thermodynamic calculations with the NUCLEA database.
ano.nymous@ccsd.cnrs.fr.invalid (Christophe Journeau), Christophe Journeau
In this work we present a novel discrete fracture model for single-phase Darcy flow in porous media with fractures of co-dimension one, which introduces an additional unknown at the fracture interface. Inspired by the fictitious domain method this Lagrange multiplier couples fracture and matrix domain and represents a local exchange of the fluid. The multipliers naturally impose the equality of the pressures at the fracture interface. The model is thus appropriate for domains with fractures of permeability higher than that in the surrounding bulk domain. In particular the novel approach allows for independent, regular meshing of fracture and matrix domain and therefore avoids the generation of small elements. We show existence and uniqueness of the weak solution of the continuous primal formulation. Moreover we discuss the discrete inf-sup condition of two different finite element formulations. Several numerical examples verify the accuracy and convergence of proposed method.
ano.nymous@ccsd.cnrs.fr.invalid (Markus Köppel), Markus Köppel
In this work we introduce a stabilized, numerical method for a multi-dimensional, discrete-fracture model (DFM) for single-phase Darcy flow in fractured porous media. In the model, introduced in an earlier work, flow in the (n − 1)-dimensional fracture domain is coupled with that in the n-dimensional bulk or matrix domain by the use of Lagrange multipliers. Thus the model permits a finite element discretization in which the meshes in the fracture and matrix domains are independent so that irregular meshing and in particular the generation of small elements can be avoided. In this paper we introduce in the numerical formulation, which is a saddle-point problem based on a primal, variational formulation for flow in the matrix domain and in the fracture system, a consistent stabilizing term which penalizes discontinuities in the Lagrange multipliers. For this penalized scheme we show stability and prove convergence. With numerical experiments we analyze the performance of the method for various choices of the penalization parameter and compare with other numerical DFM's.
ano.nymous@ccsd.cnrs.fr.invalid (Markus Köppel), Markus Köppel
The purpose is a finite element approximation of the heat diffusion problem in composite media, with non-linear contact resistance at the interfaces. As already explained in [Journal of Scientific Computing, {\bf 63}, 478-501(2015)], hybrid dual formulations are well fitted to complicated composite geometries and provide tractable approaches to variationally express the jumps of the temperature. The finite elements spaces are standard. Interface contributions are added to the variational problem to account for the contact resistance. This is an important advantage for computing codes developers. We undertake the analysis of the non-linear heat problem for a large range of contact resistance and we investigate its discretization by hybrid dual finite element methods. Numerical experiments are presented at the end to support the theoretical results.
ano.nymous@ccsd.cnrs.fr.invalid (F Ben Belgacem), F Ben Belgacem
We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation---in the mean parametric norm associated to the elliptic operator---of the error between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the proper orthogonal decomposition (POD) subspaces, except that in our case the norm is parameter-dependent. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the online step, and we prove that the partial sums converge to the continuous solution in the mean parametric elliptic norm. We show that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper. This opens the possibility of computing the PGD expansion by directly solving the optimization problems that yield the optimal subspaces.
ano.nymous@ccsd.cnrs.fr.invalid (M. Azaïez), M. Azaïez
Background and Objective: This paper deals with the improvement of parameter estimation in terms of precision and computational time for dynamical models in a bounded error context. Methods: To improve parameter estimation, an optimal initial state design is proposed combined with a contractor. This contractor is based on a volumetric criterion and an original condition initializing this contractor is given. Based on a sensitivity analysis, our optimal initial state design methodology consists in searching the minimum value of a proposed criterion for the interested parameters. In our framework, the uncertainty (on measurement noise and parameters) is supposed unknown but belongs to known bounded intervals. Thus guaranteed state and sensitivity estimation have been considered. An elementary effect analysis on the number of sampling times is also implemented to achieve the fast and guaranteed parameter estimation. Results: The whole procedure is applied to a pharmacokinetics model and simulation results are given. Conclusions: The good improvement of parameter estimation in terms of computational time and precision for the case study highlights the potential of the proposed methodology.
ano.nymous@ccsd.cnrs.fr.invalid (Qiaochu Li), Qiaochu Li
We consider a degenerate parabolic system modelling the flow of fresh and saltwater in an anisotropic porous medium in the context of seawater intrusion. We propose and analyze a nonlinear Control Volume Finite Element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the mesh and on the anisotropy tensor. Moreover It also provides a control on the entropy. Based on these nonlinear stability results, we show that the scheme converges towards a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme.
ano.nymous@ccsd.cnrs.fr.invalid (Ahmed Ait Hammou Oulhaj), Ahmed Ait Hammou Oulhaj
The present paper aims at the introduction of the semi-Markov model in continuous time as a candidate model for the description of seismicity patterns in time domain in the Northern Aegean Sea (Greece). Estimators of the semi-Markov kernels, Markov renewal functions and transition functions are calculated through a nonparametric method. Moreover , the hitting times for spatial occurrence of the strongest earthquakes as well as the confidence intervals of certain important indicators are estimated. Firstly, the classification of model states is based on earthquakes magnitude. The instantaneous earthquake occurrence rate between the states of the model as well as the total earthquake occurrence rate are calculated. In order to increase the consistency between the model and the process of earthquake generation, seismotectonic features have been incorporated as an important component in the model. Therefore, a new classification of states is proposed which combines both magnitude and fault orientation states. This model which takes into account seismotectonic features contributes significantly to the seismic hazard assessment in the region under study. The model is applied to earthquake catalogues for the Northern Aegean Sea, an area that accommodates high seismicity, being a key structure from the seismotec-tonic point of view.
ano.nymous@ccsd.cnrs.fr.invalid (Irene Votsi), Irene Votsi
We introduce in this paper a technique for the reduced order approximation of parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation in mean of the error with respect to the parameter in the quadratic norm associated to the elliptic operator between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the Proper Orthogonal Decomposition (POD) subspaces, excepting that in our case the norm is parameter-depending, and then the POD optimal sub-spaces cannot be characterized by means of a spectral problem. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the on-line step. We prove that the partial sums converge to the continuous solutions in mean quadratic elliptic norm.
ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaiez), Mejdi Azaiez
This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structure is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.
ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet
The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles.
ano.nymous@ccsd.cnrs.fr.invalid (Stéphanie Chaillat), Stéphanie Chaillat
In this paper, we study a Chikungunya epidemic transmission model which describes an epidemic disease transmitted by Aedes mosquitoes. This model includes the spatial mobility of humans which is probably a factor that has influenced the re-emergence of several diseases. Assuming that the spatial mobility of humans is random described as Brownian random motion, an original model including a reaction-diffusion system is proposed. Since the displacement of mosquitoes is limited to a few meters, compared with humans, one can ignore mosquitoes mobility. Therefore, the complete model is composed of a reaction-diffusion system coupled with ordinary differential equations (ODEs). In this paper, we prove the existence and uniqueness, the positivity and boundedness of the global solution for the model and give some numerical simulations.
ano.nymous@ccsd.cnrs.fr.invalid (Shousheng Zhu), Shousheng Zhu
The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on results of Koml\'os, Major and Tusn\'ady (1975). Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes.
ano.nymous@ccsd.cnrs.fr.invalid (Sergio Alvarez-Andrade), Sergio Alvarez-Andrade
Faults and geological barriers can drastically affect the flow patterns in porous media. Such fractures can be modeled as interfaces that interact with the surrounding matrix. We propose a new technique for the estimation of the location and hydrogeological properties of a small number of large fractures in a porous medium from given distributed pressure or flow data. At each iteration, the algorithm builds a short list of candidates by comparing fracture indicators. These indicators quantify at the first order the decrease of a data misfit function; they are cheap to compute. Then, the best candidate is picked up by minimization of the objective function for each candidate. Optimally driven by the fit to the data, the approach has the great advantage of not requiring remeshing, nor shape derivation. The stability of the algorithm is shown on a series of numerical examples representative of typical situations.
ano.nymous@ccsd.cnrs.fr.invalid (Hend Ben Ameur), Hend Ben Ameur
To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. The Lax-Milgram theorem may be seen as one of those theoretical cornerstones: under some completeness and coercivity assumptions, it states existence and uniqueness of the solution to the weak formulation of some boundary value problems. The purpose of this document is to provide the formal proof community with a very detailed pen-and-paper proof of the Lax-Milgram theorem.
ano.nymous@ccsd.cnrs.fr.invalid (François Clément), François Clément
A mathematical model for the forward problem in electroencephalographic (EEG) source localization in neonates is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy (H 1-)norm. Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model have been obtained and corroborate the theoretical results. A mathematical tool related to the concept of Gâteau derivatives is introduced which is able to measure the sensitivity of the electric potential with respect to small variations in the fontanel conductivity. Numerical simulations attest that the presence of fontanels in neonates does have an impact on EEG measurements. The present work is an essential preamble to the numerical analysis of the corresponding EEG source reconstruction.
ano.nymous@ccsd.cnrs.fr.invalid (M Darbas), M Darbas
Using a preconditioned Richardson iterative method as a regularization to the data completion problem is the aim of the contribution. The problem is known to be exponentially ill posed that makes its numerical treatment a hard task. The approach we present relies on the Steklov-Poincaré variational framework introduced in [Inverse Problems, vol. 21, 2005]. The resulting algorithm turns out to be equivalent to the Kozlov-Maz’ya-Fomin method in [Comp. Math. Phys., vol. 31, 1991]. We conduct a comprehensive analysis on the suitable stopping rules that provides some optimal estimates under the General Source Condition on the exact solution. Some numerical examples are finally discussed to highlight the performances of the method.
ano.nymous@ccsd.cnrs.fr.invalid (Duc Thang Du), Duc Thang Du
The magnetohydrodynamics laws govern the motion of a conducting fluid, such as blood, in an externally applied static magnetic field B 0. When an artery is exposed to a magnetic field, the blood charged particles are deviated by the Lorentz force thus inducing electrical currents and voltages along the vessel walls and in the neighboring tissues. Such a situation may occur in several bio-medical applications: magnetic resonance imaging (MRI), magnetic drug transport and targeting, tissue engineering… In this paper, we consider the steady unidirectional blood flow in a straight circular rigid vessel with non-conducting walls, in the presence of an exterior static magnetic field. The exact solution of Gold (1962) (with the induced fields not neglected) is revisited. It is shown that the integration over a cross section of the vessel of the longitudinal projection of the Lorentz force is zero, and that this result is related to the existence of current return paths, whose contributions compensate each other over the section. It is also demonstrated that the classical definition of the shear stresses cannot apply in this situation of magnetohydrodynamic flow, because, due to the existence of the Lorentz force, the axisymmetry is broken.
ano.nymous@ccsd.cnrs.fr.invalid (Agnès Drochon), Agnès Drochon
This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called Generalised Impedance Boundary Condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first order GIBCs characterizing the presence of an interior thin layer with either a constant or a variable thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non vanishing jumps across the exterior and the interior boundary of the thin layer. In the special case of thin layers with a constant thickness, we show that we can interchange the first order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlights the various theoretical results.
ano.nymous@ccsd.cnrs.fr.invalid (Djalil Kateb), Djalil Kateb
L’auteur s’intéresse au problème de l’analyse en composantes indépendantes multidimensionnelles (ACIM) qui est la généralisation naturelle du problème ordinaire de l’analyse en composantes indépendantes (ACI). Tout d’abord, afin de faciliter l’utilisation des cumulants des ordres supérieurs, nous présentons de nou- velles formules pour le calcul matriciel des matrices de cumulants d’un vecteur aléatoire réel à partir de ses matrices de moments. Outre les opérations matricielles usuelles, ces formules utilisent uniquement le produit de Kronecker, l’opérateur vec et des matrices de commutation. Nous pouvons immédiatement à partir de ces formules examiner de plus près les structures particulières des matrices de cumulants et ainsi donner des résultats sur les rangs de ces matrices qui caractérisent la dépendance entre les variables aléatoires constituant le vecteur aléatoire. L’intérêt pratique principal de nos formules matricielles réside certainement dans une évaluation des cumulants beaucoup plus aisée et rapide qu’avec la méthode usuelle basée sur une utilisation répétée des formules de Leonov et Shiryaev. Dans la deuxième partie de cette thèse, nous montrons que sous les hypothèses usuelles de l’analyse en composantes indépendantes mul- tidimensionnelles, les matrices de cumulants contractées à un ordre statistique quelconque sont toutes bloc-diagonalisables dans la même base. Nous en déduisons des algorithmes de résolution d’ACIM par bloc-diagonalisation conjointe et comparons les résultats obtenus aux ordres 3 à 6, entre eux et avec d’autres méthodes, sur quelques signaux synthétiques. Des exemples simples ont élaborés afin de justifier la nécessité de combiner des ordres différents pour garantir la meilleure séparation. Nous prouvons aussi que le cas le plus simple à traiter est celui de mélanges de sources qui ont différentes dimensions. Dans la dernière partie de cette thèse nous proposons une famille de méthodes qui exploitent uniquement les sta- tistiques d’ordres supérieurs à deux. Sous certaines hypothèses supplémentaires, ces méthodes permettent après un blanchiment d’ordre quatre des observations de résoudre complètement le problème ACIM bruité en bloc diagonalisant conjointement un ensemble de matrices de cumulants issues des statistiques d’ordres supérieurs strictement à quatre. Une comparaison avec les méthodes ACIM à blanchiment d’ordre deux pour la séparation des activités électriques foetale et maternelle (mesurées via trois électrodes placées sur l’abdomen de la mère) montre que cette nouvelle famille est mieux adaptée à cette application : elles permettent une séparation quasi parfaite de ces deux contributions.
ano.nymous@ccsd.cnrs.fr.invalid (Hanany Ould-Baba), Hanany Ould-Baba
Dans cet article, nous traitons un problème de Cauchy dans le cadre de la localisation des sources épileptiques en Electro-Encéphalo-Graphie (EEG). Plus particulièrement, il s'agit du problème de construction des données de Cauchy sur la surface du cerveau à partir des données du potentiel mesuré par l'EEG à la surface de la tête. Notre résolution est basée sur un algorithme itératif alternatif initialement proposé par Kozlov, Mazjya et Fomin. Nous présentons dans ce papier l'étude umérique de cette méthode que nous avons implémentée en trois dimensions. Nous donnons également des applications et des résultats numériques.
ano.nymous@ccsd.cnrs.fr.invalid (Abdellatif El-Badia), Abdellatif El-Badia
We derive rates of contraction of posterior distributions on non-parametric models resulting from sieve priors. The aim of the study was to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter is, for example, a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate.
ano.nymous@ccsd.cnrs.fr.invalid (Julyan Arbel), Julyan Arbel
It has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies.
ano.nymous@ccsd.cnrs.fr.invalid (Stéphanie Chaillat), Stéphanie Chaillat
The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to far-field operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß Newton method and show numerical experiments in the special case of star-shaped obstacles.
ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of the integrated empirical process.Reference: Koml\'os, J., Major, P. and Tusn\'ady, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, 111-131.
ano.nymous@ccsd.cnrs.fr.invalid (Sergio Alvarez-Andrade), Sergio Alvarez-Andrade
In this communication, we perform the sensitivity analysis of a building energy model. The aim is to assess the impact of the weather data on the performance of a model of a passive house, in order to better control it. The weather data are uncertain dynamic inputs to the model. To evaluate their impact, the problem of generating coherent weather data arises. To solve it, we carry out the Karhunen-Loève decomposition of the uncertain dynamic inputs. We then propose an approach for the sensitivity analysis of this kind of models. The originality for sensitivity analysis purpose is to separate the random variable of the dynamic inputs, propagated to the model response, from the deterministic spatio/temporal function. This analysis highlights the role of the solar gain on a high-insulated passive building, during winter time.
ano.nymous@ccsd.cnrs.fr.invalid (Floriane Anstett-Collin), Floriane Anstett-Collin
In this paper we describe a high order spectral algorithm for solving the time-harmonic Navier equations in the exterior of a bounded obstacle in three space dimensions, with Dirichlet or Neumann boundary conditions. Our approach is based on combined-field boundary integral equation (CFIE) reformulations of the Navier equations. We extend the spectral method developped by Ganesh and Hawkins - for solving second kind boundary integral equations in electromagnetism - to linear elasticity for solving CFIEs that commonly involve integral operators with a strongly singular or hypersingular kernel. The numerical scheme applies to boundaries which are globally parameterised by spherical coordinates. The algorithm has the interesting feature that it leads to solve linear systems with substantially fewer unknowns than with other existing fast methods. The computational performances of the proposed spectral algorithm are demonstrated on numerical examples for a variety of three-dimensional convex and non-convex smooth obstacles.
ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.
ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine
In this paper, we address the issue of performing sensitivity analysis of complex models presenting uncertain static and dynamic inputs. The dynamic inputs are viewed as random processes which can be represented by a linear combination of the deterministic functions depending on time whose coefficients are uncorrelated random variables. To achieve this, the Karhunen-Loève decomposition of the dynamic inputs is performed. For sensitivity analysis purposes, the influence of the dynamic inputs onto the model response is then given by the one of the uncorrelated random coefficients of the Karhunen-Loève decomposition, which is the originality here. The approach is applied to a building energy model, in order to assess the impact of the uncertainties of the material properties and the weather data on the energy performance of a real low energy consumption house.
ano.nymous@ccsd.cnrs.fr.invalid (Floriane Anstett-Collin), Floriane Anstett-Collin
Karhunen-Loève's decompositions (KLD) or the proper orthogonal decompositions (POD) of bivariate functions are revisited in this work. We investigate the truncation error first for regular functions and try to improve and sharpen bounds found in the literature. However it happens that (KL)-series expansions are in fact more sensitive to the liability of fields to approximate to be well represented by a small sum of products of separated variables functions. We consider this very issue for some interesting fields solutions of partial differential equations such as the transient heat problem and Poisson's equation. The main tool to state approximation bounds is linear algebra. We show how the singular value decomposition underlying the (KL)-expansion is connected to the spectrum of some Gram matrices. Deriving estimates on the truncation error is thus strongly tied to the spectral properties of these Gram matrices which are structured matrices with low displacement ranks.
ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaïez), Mejdi Azaïez
Ill-posedness and/or Ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
Observability Gramians of diffusion equations have been recently connected to infinite Pick and Cauchy matrices. In fact, inverse or observability inequalities can be obtained after estimating the extreme eigenvalues of these structured matrices, with respect to the diffusion semi-group matrix. The purpose is hence to conduct a spectral study of a subclass of symmetric Cauchy matrices and present an algebraic way to show the desired observability results. We revisit observability inequalities for three different observation problems of the diffusion equation and show how they can be (re)stated through simple proofs.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional transport model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the outfall point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequences of these non-standard boundary conditions is that dispersive transport equations on both oxygen tracers are strongly coupled and the resulting system becomes ill-posed. The main purpose is a finite element space-discretization of the variational problem put under a non-symmetric mixed form. Combining analytical calculations, numerical computations and theoretical justifications, we try to elucidate the characteristics related to the ill-posedness of this data completion dynamical problem and understand its mathematical structure.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
We construct and analyze a family of well-conditioned boundary integral equations for the Krylov iterative solution of three-dimensional elastic scattering problems by a bounded rigid obstacle. We develop a new potential theory using a rewriting of the Somigliana integral representation formula. From these results, we generalize to linear elasticity the well-known Brakhage-Werner and Combined Field Integral Equation formulations. We use a suitable approximation of the Dirichlet-to-Neumann (DtN) map as a regularizing operator in the proposed boundary integral equations. The construction of the approximate DtN map is inspired by the On-Surface Radiation Conditions method. We prove that the associated integral equations are uniquely solvable and possess very interesting spectral properties. Promising analytical and numerical investigations, in terms of spherical harmonics, with the elastic sphere are provided.
ano.nymous@ccsd.cnrs.fr.invalid (Marion Darbas), Marion Darbas
Uncertainty Analysis and Sensitivity Analysis of complex models: Coping with dynamic and static inputs
ano.nymous@ccsd.cnrs.fr.invalid (Floriane Anstett-Collin), Floriane Anstett-Collin
We consider an inverse problem that arises in the management of water resources and pertains to the analysis of the surface waters pollution by organic matter. Most of physical models used by engineers derive from various additions and corrections to enhance the earlier deoxygenation-reaeration model proposed by Streeter and Phelps in 1925, the unknowns being the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) concentrations. The one we deal with includes Taylor's dispersion to account for the heterogeneity of the contamination in all space directions. The system we obtain is then composed of two reaction-dispersion equations. The particularity is that both Neumann and Dirichlet boundary conditions are available on the DO tracer while the BOD density is free of any condition. In fact, for real-life concerns, measurements on the dissolved oxygen are easy to obtain and to save. In the contrary, collecting data on the biochemical oxygen demand is a sensitive task and turns out to be a long-time process. The global model pursues the reconstruction of the BOD density, and especially of its flux along the boundary. Not only this problem is plainly worth studying for its own interest but it can be also a mandatory step in other applications such as the identification of the pollution sources location. The non-standard boundary conditions generate two difficulties in mathematical and computational grounds. They set up a severe coupling between both equations and they are cause of ill-posedness for the data reconstruction problem. Existence and stability fail. Identifiability is therefore the only positive result one can seek after ; it is the central purpose of the paper. We end by some computational experiences to assess the capability of the mixed finite element capability in the missing data recovery (on the biochemical oxygen demand).
ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaïez), Mejdi Azaïez
We are interested in an inverse problem of recovering the position of a pollutant or contaminant source in a stream water. Advection, dispersive transport and the reaction of the solute is commonly modeled by a linear or non-linear parabolic equation. In former works, it is established that a point-wise source is fully identifiable from measurements recorded by a couple of sensors placed, one up-stream and the other down-stream of the pollution source. The observability question we try to solve here is related to the redundancy of sensors when additional information is available on the point-wise source. It may occur, in hydrological engineering, that the intensity of the pollutant is known in advance. In this case, we pursue an identifiability result of a moving source location using a single observation. The chief mathematical tools to prove identifiability are the unique continuation theorem together with an appropriate maximum principle for the parabolic equation under investigation.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
The direct electrochemical reduction of UO2 solid pellets was carried out in LiF-CaF2 (+ 2 mass. % Li2O) at 850°C. An inert gold anode was used instead of the usual reactive sacrificial carbon anode. In this case, oxidation of oxide ions present in the melt yields O2 gas evolution on the anode. Electrochemical characterisations of UO2 pellets were performed by linear sweep voltammetry at 10mV/s and reduction waves associated to oxide direct reduction were observed at a potential 150mV more positive in comparison to the solvent reduction. Subsequent, galvanostatic electrolyses runs were carried out and products were characterised by SEM-EDX, EPMA/WDS and XRD. In one of the runs, uranium oxide was partially reduced and three phases were observed: non reduced UO2 in the centre, pure metallic uranium on the external layer and an intermediate phase representing the initial stage of reduction taking place at the grain boundaries. In another run, the UO2 sample was fully reduced. Due to oxygen removal, the U matrix had a typical coral-like structure which is characteristic of the pattern observed after the electroreduction of solid oxides.
ano.nymous@ccsd.cnrs.fr.invalid (Mathieu Gibilaro), Mathieu Gibilaro
We consider a mixed reaction diffusion system describing the organic pollution in stream-waters. It may be viewed as the static version of Streeter-Phelps equations relating the Biochemical Oxygen Demand and Dissolved Oxygen to which dispersion terms are added. In this work, we propose a mixed variational formulation and prove its well-posedness. Next, we develop two finite element discretizations of this problem and establish optimal a priori error estimates for the second discrete problem.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain Ω via boundary measurements on ∂Ω. The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied thanks to the tools of shape optimization by minimizing a least squares type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerated for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.
ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet
We study the stability of some critical (or equilibrium) shapes in the minimization problem of the energy dissipated by a fluid (i.e. the drag minimization problem) governed by the Stokes equations. We first compute the shape derivative up to the second order, then provide a sufficient condition for the shape Hessian of the energy functional to be coercive at a critical shape. Under this condition, the existence of such a local strict minimum is then proved using a precise upper bound for the variations of the second order shape derivative of the functional with respect to the coercivity and differentiability norms. Finally, for smooth domains, a lower bound of the variations of the drag is obtained in terms of the measure of the symmetric difference of domains.
ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet
We are concerned with fast methods for the numerical implementation of the direct and inverse scattering problems for a perfectly conducting obstacle. The scattering problem is usually reduced to a single uniquely solvable modified combined-field integral equation (M-CFIE). For the numerical solution of the M-CFIE we propose a new high-order spectral algorithm by transporting this equation on the unit sphere via the Piola transform. The inverse problem is formulated as a nonlinear least squares problem for which the iteratively regularized Gauss-Newton method is applied to recover an approximate solution. Numerical experiments are presented in the special case of star-shaped obstacles.
ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
This article concerns maximum-likelihood estimation for discrete time homogeneous nonparametric semi-Markov models with finite state space. In particular, we present the exact maximum-likelihood estimator of the semi-Markov kernel which governs the evolution of the semi-Markov chain (SMC). We study its asymptotic properties in the following cases: (i) for one observed trajectory, when the length of the observation tends to infinity, and (ii) for parallel observations of independent copies of an SMC censored at a fixed time, when the number of copies tends to infinity. In both cases, we obtain strong consistency, asymptotic normality, and asymptotic efficiency for every finite dimensional vector of this estimator. Finally, we obtain explicit forms for the covariance matrices of the asymptotic distributions.
ano.nymous@ccsd.cnrs.fr.invalid (Samis Trevezas), Samis Trevezas
The aim of this article is to explore the possibility of using a family of fixed finite elements shape functions to solve a Dirichlet boundary value problem with an alternative variational formulation. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for example shape optimization or moving interfaces. This is not a new idea, but we analyze here a special approach. The main difficulty of the approach is that the associated quadratic form is not coercive and an inf-sup condition has to be checked. In dimension one, we prove that this formulation is well posed and we provide error estimates. Nevertheless, our proof relying on explicit computations is limited to that case and we give numerical evidence in dimension two that the formulation does not provide a reliable method. We first add a regularization through a Nitscheterm and we observe that some instabilities still remain. We then introduce and justify a geometrical regularization. A reliable method is obtained using both regularizations.
ano.nymous@ccsd.cnrs.fr.invalid (Gaël Dupire), Gaël Dupire
The aim of this article is to explore the possibility of using a family of fixed finite element shape functions that does not match the domain to solve a boundary value problem with Dirichlet boundary condition. The domain is embedded in a bounding box and the finite element approximation is associated to a regular structured mesh of the box. The shape of the domain is independent of the discretization mesh. In these conditions, a meshing tool is never required. This may be especially useful in the case of evolving domains, for example shape optimization or moving interfaces. Nitsche method has been intensively applied. However, Nitsche is weighted with the mesh size h and therefore is a purely discrete point of view with no interpretation in terms of a continuous variational approach associated with a boundary value problem. In this paper, we introduce an alternative to Nitsche method which is associated with a continuous bilinear form. This extension has strong restrictions: it needs more regularity on the data than the usual method. We prove the well-posedness of our formulation and error estimates. We provide numerical comparisons with Nitsche method.
ano.nymous@ccsd.cnrs.fr.invalid (Jean-Paul Boufflet), Jean-Paul Boufflet
In the last years, several epidemics have been reported in particular the chikungunya epidemic on the R eunion Island. For predicting its possible evolution, new models describing the transmission of the chikungunya to the human population have been proposed and studied in the literature. In such models, some parameters are not directely accessible from experiments and for estimating them, iterative algorithms can be used. However, before searching for their values, it is essential to verify the identi ability of models parameters to assess wether the set of unknown parameters can be uniquely determined from the data. Thus, identi ability is particularly important in modeling. Indeed, if the model is not identi able, numerical procedures can fail and in that case, some supplementary data have to be added or the set of admissible data has to be reduced. Thus, this paper proposes to study the identi ability of the proposed models by (Moulay, Aziz-Alaoui & Cadivel 2011).
ano.nymous@ccsd.cnrs.fr.invalid (Djamila Moulay), Djamila Moulay
In order to optimize the performance of a diesel engine subject to legislative constraints on pollutant emissions, it is necessary to improve their design, and to identify how design parameters a ect engine behaviours. One speci city of this work is that it does not exist a physical model of engine behaviour under all possible operational conditions. A powerful strategy for engine modeling is to build a fast emulator based on carefully chosen observations, made according to an experimental design. In this paper, two Kriging models are considered. One is based on a geostatistical approach and the other corresponds to a Gaussian process metamodel approach. Our aim is to show that the use of each of these methods does not lead to the same results, particularly when "atypical" points are present in our database. In a more precise way, the statistical approach allows us to obtain a good quality modeling even if atypical data are present, while this situation leads to a bad quality of the modeling by the geostatistical approach. This behaviour takes a fundamental importance for the problem of the pollutant emissions, because the analysis of these atypical data, which are rarely erroneous data, can supply precious information for the engine tuning in the design stage.
ano.nymous@ccsd.cnrs.fr.invalid (Sébastien Castric), Sébastien Castric
The aim of this paper is to identify the localization and the flow rate of a pollution source in a river by measuring the concentration of a substrate giving significant information. This concentration is assumed to be measured in two points of the river. The simplest model of such a problem consists in a parabolic partial derivative equation. We propose to discretize this P.D.E. in space, which leads to a system of differential equations in time. Then, the analysis of identifiability is carried out using an approach based on differential algebra. A numerical parameter estimation is inferred from this procedure, which gives a first parameter estimate without a priori knowledge about unknown parameters.
ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdiere), Nathalie Verdiere
This paper is concerned by the analysis of nonlinear controlled or uncontrolled dynamical model identifiability. The proposed approach is based on the construction of an input-output ideal. The aim is to develop an algorithm which gives identifiability results from this approach. Differential algebra theory allows realization of such a project. In order to state the algorithm, new results of differential algebra must be proved. Then the implementation is done in a symbolic computing language
ano.nymous@ccsd.cnrs.fr.invalid (Lilianne Denis-Vidal), Lilianne Denis-Vidal
This paper considers two different methods in the analysis of nonlinear controlled dynamical system identifiability. The corresponding identifiability definitions are not equivalent. Moreover one is based on the construction of an input-output ideal and the other on the similarity transformation theorem. Our aim is to develop algorithms which give identifiability results from both approaches. Differential algebra theory allows realization of such a project. In order to state these algorithms, new results of differential algebra must be proved. Then the implementation of these algorithms is done in a symbolic computation language.
ano.nymous@ccsd.cnrs.fr.invalid (Lilianne Denis-Vidal), Lilianne Denis-Vidal
In many applications, the estimation of derivatives has to be done from noisy measured signal. In this paper, an original method based on a distribution approach is presented. Its interest is to report the derivatives on infinitely differentiable functions. Thus, the estimation of the derivatives is done only from the signal. Besides, this method gives some explicit formulae leading to fast calculus. For all these reasons, it is an efficient method in the case of noisy signals as it will be confirmed in several examples.
ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière
In this paper, we investigate the existence and characterizations of the Fréchet derivative of solutions to time-harmonic elastic scattering problems with respect to the boundary of the obstacle. Our analysis is based on a technique - the factorization of the difference of the far-field pattern for two different scatterers - introduced by Kress and Päivärinta to establish Fréchet differentiability in acoustic scattering. For the Dirichlet boundary condition an alternative proof of a differentiability result due to Charalambopoulos is provided and new results are proven for the Neumann and impedance exterior boundary value problems.
ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
Ventcel boundary conditions are second order di erential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coe cient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surface are sought for numerical purposes. The well-posedness of such problems was recently investigated : up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation.
ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine
The aim of our work is to reconstruct an inclusion immersed in a fluid flowing in a larger bounded domain via a boundary measurement. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method.
ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet
We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity $TH\sp{-\frac{1}{2}}(\Div_{\Gamma},\Gamma)$. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem.
ano.nymous@ccsd.cnrs.fr.invalid (Martin Costabel), Martin Costabel
In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators.
ano.nymous@ccsd.cnrs.fr.invalid (Martin Costabel), Martin Costabel
We are concerned with the inverse problem of detecting sources in a coupled diffusion-reaction system. This problem arises from the Biochemical Oxygen Demand-Dissolved Oxygen model(\footnote{The acronym BOD-DO model is currently used.}) governing the interaction between organic pollutants and the oxygen available in stream waters. The sources we consider are point-wise and simulate stationary or moving pollution sources. The ultimate objective is to obtain their discharge location and recover their output rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain. It is, as a matter of fact, the most realistic configuration. The subject to address here is the identifiability of these sources, in other words to determine if the observations uniquely determine the sources. The key tool is the study of coupled parabolic systems derived after restricting the global model to regions at the exterior of the observations. The absence of any prescribed condition on the BOD density is compensated by data recorded on the DO which provide over-determined Cauchy boundary conditions. Now, the first step toward the identifiability of the sources is precisely to recover the BOD at the observation points (of DO). This may be achieved by handling and solving the coupled systems. Unsurprisingly, they turn out to be ill-posed. That issue is investigated first. Then, we state a uniqueness result owing to a suitable saddle-point variational framework and to Pazy's uniqueness Theorem. This uniqueness complemented by former identifiability results proved in [2011, Inverse problems] for scalar reaction-diffusion equations yields the desired identifiability for the global model.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
This article concerns the study of the asymptotic properties of the maximum likelihood estimator (MLE) for the general hidden semi-Markov model (HSMM) with backward recurrence time dependence. By transforming the general HSMM into a general hidden Markov model, we prove that under some regularity conditions, the MLE is strongly consistent and asymptotically normal. We also provide useful expressions for the asymptotic covariance matrices, involving the MLE of the conditional sojourn times and the embedded Markov chain of the hidden semi-Markov chain.
ano.nymous@ccsd.cnrs.fr.invalid (Samis Trevezas), Samis Trevezas
The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface Γ. In this paper, following an idea developed by Kleinman and Martin for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over Γ. for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the well-posedness of the integral equations in the space of finite energy on smooth and non-smooth boundaries.
ano.nymous@ccsd.cnrs.fr.invalid (Martin Costabel), Martin Costabel
Nowdays, one of the greatest problems that earth has to face up is pollution, and that is what leads European Union to make stricter laws about pollution constraints. Moreover, the European laws lead to the increase of emission constraints. In order to take into account these constraints, automotive constructors are obliged to create more and more complex systems. The use of model to predict systems behavior in order to make technical choices or to understand its functioning, has become very important during the last decade. This paper presents two stage approaches for the prediction of NOx (nitrogen oxide) emissions, which are based on an ordinary Kriging method. In the first stage, a reduction of data will take place by selecting signals with correlations studies and by using a fast Fourier transformation. In the second stage, the Kriging method is used to solve the problem of the estimation of NOx emissions under given conditions. Numerical results are presented and compared to highlight the effectiveness of the proposed methods
ano.nymous@ccsd.cnrs.fr.invalid (El Hassane Brahmi), El Hassane Brahmi
In this paper, we study the uniqueness of solutions for diagonal hyperbolic systems in one space dimension. We present two uniqueness results. The first one is a global existence and uniqueness result of a continuous solution for strictly hyperbolic systems. The second one is a global existence and uniqueness result of a Lipschitz solution for hyperbolic systems not necessarily strictly hyperbolic. An application of these two results is shown in the case of the one-dimensional isentropic gas dynamics.
ano.nymous@ccsd.cnrs.fr.invalid (Ahmad El Hajj), Ahmad El Hajj
This work is devoted to the numerical simulation of an incompressible fluid through a porous interface, modeled as a macroscopic resistive interface term in the Stokes equations. We improve the results reported in [M2AN, 42(6):961-990, 2008], by showing that the standard Pressure Stabilized Petrov-Galerkin (PSPG) finite element method is stable, and optimally convergent, without the need for controlling the pressure jump across the interface.
ano.nymous@ccsd.cnrs.fr.invalid (Alfonso Caiazzo), Alfonso Caiazzo
Dans ce travail je présente une étude unifiée basée sur l'estimation du maximum de vraisemblance pour des modèles markoviens, semi-markoviens et semi-markoviens cachés. Il s'agit d'une étude théorique des propriétés asymptotiques de l'EMV des modèles mentionnés ainsi que une étude algorithmique. D'abord, nous construisons l'estimateur du maximum de vraisemblance (EMV) de la loi stationnaire et de la variance asymptotique du théorème de la limite centrale (TLC) pour des fonctionnelles additives des chaînes de Markov ergodiques et nous démontrons sa convergence forte et sa normalité asymptotique. Ensuite, nous considérons un modèle semi-markovien non paramétrique. Nous présentons l'EMV exact du noyau semi-markovien qui gouverne l'évolution de la chaîne semi-markovienne (CSM) et démontrons la convergence forte, ainsi que la normalité asymptotique de chaque sous-vecteur fini de cet estimateur en obtenant des formes explicites pour les matrices de covariance asymptotiques. Ceci a été appliqué pour une observation de longue durée d'une seule trajectoire d'une CSM, ainsi que pour une suite des trajectoires i.i.d. d'une CSM censurée à un instant fixe. Nous introduisons un modèle semi-markovien caché (MSMC) général avec dépendance des temps de récurrence en arrière. Nous donnons des propriétés asymptotiques de l'EMV qui correspond à ce modèle. Nous déduisons également des expressions explicites pour les matrices de covariance asymptotiques qui apparaissent dans le TLC pour l'EMV des principales caractéristiques des CSM. Enfin, nous proposons une version améliorée de l'algorithme EM (Estimation-Maximisation) et une version stochastique de cet algorithme (SAEM) afin de trouver l'EMV pour les MSMC non paramétriques. Des exemples numériques sont présentés pour ces deux algorithmes.
ano.nymous@ccsd.cnrs.fr.invalid (Samis Trevezas), Samis Trevezas
This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the MLE estimator for the stationary distribution.
ano.nymous@ccsd.cnrs.fr.invalid (Samis Trevezas), Samis Trevezas
We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem.
ano.nymous@ccsd.cnrs.fr.invalid (Martin Costabel), Martin Costabel
Les antennes lentilles sont des dispositifs ayant pour support les ondes électromagnétiques et sont constituées d'une source primaire et d'un système focalisant diélectrique. La montée en importance récente d'applications en ondes millimétriques (exemple : radars d'assistance et d'aide à la conduite), nécessite la construction d'antennes lentilles de quelques centimètres qui répondent à des cahiers des charges spécifiques à chaque cas. L'une des problématiques à résoudre consiste à déterminer la forme optimale de la lentille étant données : (i) les caractéristiques de la source primaire, (ii) les caractéristiques en rayonnement fixées. Ce projet de thèse vise à développer de nouveaux outils pour l'optimisation de forme en utilisant une formulation intégrale du problème. Cette thèse s'articule en deux parties. Dans la première nous avons construit plusieurs formulations intégrales pour le problème de diffraction diélectrique en utilisant une approche par équation intégrale surfacique. Dans la seconde nous avons étudié les dérivées de forme des opérateurs intégraux standard en électromagnétisme dans le but de les incorporer dans un algorithme d'optimisation de forme.
ano.nymous@ccsd.cnrs.fr.invalid (Frédérique Le Louër), Frédérique Le Louër
We consider the flow of a viscous incompressible fluid in a rigid homogeneous porous medium provided with boundary conditions on the pressure around a circular well. When the boundary pressure presents high variations, the permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a spectral discretization of the resulting system of equations which takes into account the axisymmetry of the domain and of the flow. We prove optimal error estimates and present some numerical experiments which confirm the interest of the discretization.
ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaïez), Mejdi Azaïez
Ce travail porte sur un problème inverse de sources dipolaires et son application à l'identification des sources de l'activité cérébrale telle qu'elle peut être mesurée par l'Electro-Encéphalo-Graphie (EEG). Des résultats d'identifiabilité et de stabilité ont été établis. Par ailleurs, une étude du problème de Cauchy en 3D, motivée par l'application de la méthode d'identification dite "algébrique", a été faite à l'aide de la méthode itérative introduite par Kozlov, Maz'ya et Fomin et au moyen des équations intégrales de frontières. En outre, une autre méthode basée sur une fonctionnelle coût de type Kohn et Vogelius a été considérée pour l'identification des sources et dont les résultats numériques sont avérés plus performants que ceux donnés par la méthode des moindres carrés.
ano.nymous@ccsd.cnrs.fr.invalid (Maha Farah), Maha Farah
The level set method has become widely used in shape optimization where it allows a popular implementation of the steepest descent method. Once coupled with a weak material approximation, a single mesh is only used leading to very efficient and cheap numerical schemes in optimization of structures. However, it has some limitations and cannot be applied in every situation. This work aims at exploring such a limitation. We estimate the systematic error committed by using the weak material approximation and, on a model case, explain that they amplifies instabilities by a second order analysis of the objective function.
ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine
A new theorem is provided to test the identifiability of discrete-time systems with polynomial nonlinearities. That extends to discrete-time systems the local state isomorphism approach for continuous-time systems. Two examples are provided to illustrate the approach.
ano.nymous@ccsd.cnrs.fr.invalid (Floriane Anstett), Floriane Anstett
[...]
ano.nymous@ccsd.cnrs.fr.invalid (Chérif Amrouche), Chérif Amrouche
Enhancing the safety of high-temperature reactors (HTRs) is based on the quality of the fuel particles, requiring good knowledge of the microstructure of the four-layer particles designed to retain the fission products during irradiation and under accidental conditions. This paper focuses on the intensive research work performed to characterize the micro- and nanostructure of each unirradiated layer (silicon carbide and pyrocarbon coatings). The analytic expertise developed in the 1970s has been recovered and innovative advanced characterization methods have been developed to improve the process parameters and to ensure the production reproducibility of coatings.
ano.nymous@ccsd.cnrs.fr.invalid (D. Helary), D. Helary
Nous décrivons le niveau de dégradation caractéristique d'une structure à l'aide d'un processus stochastique appelé processus de dégradation. La dynamique de ce processus est modélisée par un système différentiel à environnement markovien. Nous étudions la fiabilité du système en considérant la défaillance de la structure lorsque le processus de dégradation dépasse un seuil fixe. Nous obtenons la fiabilité théorique à l'aide de la théorie du renouvellement markovien. Puis, nous proposons une procédure d'estimation des paramètres des processus aléatoires du système différentiel. Les méthodes d'estimation et les résultats théoriques de la fiabilité, ainsi que les algorithmes de calcul associés, sont validés sur des données simulés. Notre méthode est appliquée à la modélisation d'un mécanisme réel de dégradation, la propagation des fissures, pour lequel nous disposons d'un jeu de données expérimental.
ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet
Electron back-scattering diffraction (EBSD) can be successfully performed on SiC coatings for HTR fuel particles. EBSD grain maps obtained from thick and thin unirradiated samples are presented, along with pole figures showing textures and a chart showing the distribution of grain aspect ratios. This information is of great interest, and contributes to improving the process parameters and ensuring the reproducibility of coatings
ano.nymous@ccsd.cnrs.fr.invalid (D. Helary), D. Helary
We present a first version of a software dedicated to an application of a classical nonlinear control theory problem to the study of compartmental models in biology. The software is being developed over a new free computer algebra library dedicated to differential and algebraic elimination.
ano.nymous@ccsd.cnrs.fr.invalid (François Boulier), François Boulier
This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partial\omega$, then we choose a cost function in order to recover the geometry of $\partial \omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.
ano.nymous@ccsd.cnrs.fr.invalid (Lekbir Afraites), Lekbir Afraites
We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging.
ano.nymous@ccsd.cnrs.fr.invalid (Miguel Angel Fernández), Miguel Angel Fernández
The mortar spectral element method is a domain decomposition technique that allows for discretizing second- or fourth-order elliptic equations when set in standard Sobolev spaces.he aim of this paper is to extend this method to problems formulated in the space of square-integrable vector fields with square-integrable curl.We consider the problem of computing the vector potential associated with a divergence- free function in dimension 3 and propose a discretization of it. The numerical analysis of the discrete problem is performed and numerical experiments are presented, they turn out to be in good coherency with the theoretical results.
ano.nymous@ccsd.cnrs.fr.invalid (Mjedi Azaïez), Mjedi Azaïez
In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uε based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We conclude with numerical results.
ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine
We consider the inverse conductivity problem with one measurement for the equation $div((\sigma_1+(\sigma_2-\sigma_1)\chi_D)\nabla{u})=0$ determining the unknown inclusion $D$ included in $\Omega$. We suppose that $\Omega$ is the unit disk of $\mathbb{R}^2$. With the tools of the conformal mappings, of elementary Fourier analysis and also the action of some quasi-conformal mapping on the Sobolev space $\sH^{1/2}(S^1)$, we show how to approximate the Dirichlet-to-Neumann map when the original inclusion $D$ is a $\varepsilon-$ approximation of a disk. This enables us to give some uniqueness and stability results.
ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine
Avant d'estimer les paramètres intervenant dans des systèmes dynamiques, linéaires ou non-linéaires, contrôlés ou non contrôlés, il est important d'effectuer une étude d'identifiabilité, c'est à dire si, à partir des données expérimentales, les paramètres étudiés sont uniques ou non. Plusieurs méthodes ont été développées ces dernières années, en particulier une qui est basée sur l'algèbre différentielle. Celle-ci a conduit à un algorithme utilisant le package Diffalg implémenté sous Maple et permettant de tester l'identifiabilité de systèmes d'équations différentielles. Les résultats obtenus à partir de cette étude permettent de mettre en place des méthodes numériques pour obtenir une première estimation des paramètres, ceci sans aucune connaissance à priori de leur valeur. Cette première estimation peut alors être utilisée comme point de départ d'algorithmes itératifs spécialisés dans l'étude des problèmes mal posés : la régularisation de Tikhonov. Dans cette thèse, deux modèles non linéaires en pharmacocinétique de type Michaelis-Menten ont tout d'abord été étudiés. Ensuite, nous nous sommes intéressés à un modèle de pollution décrit par une équation aux dérivées partielles parabolique. Le terme source à identifier était modélisé par le produit de la fonction débit avec la masse de Dirac, de support la position de la source polluante. Le but du travail était de fournir une première estimation de la source polluante. Après avoir obtenu l'identifiabilité du problème continu, nous avons étudié l'identifiabilité d'un problème approché en nous appuyant sur les méthodes d'algèbre différentielle. Celui-ci a été obtenu en approchant la masse de Dirac par une fonction gaussienne et en discrétisant ensuite le système en espace. Les résultats d'identifiabilité ont été obtenus quel que soit le nombre de points de discrétisation en espace. De cette étude théorique, nous en avons déduit des algorithmes numériques donnant une première estimation des paramètres à identifier.
ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière
In the present work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the shape derivative of the state function and of shape functionals. We consider both Least Squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parametrization of shapes coupled with a boundary element method. Several numerical exemples indicate the superiority of the Kohn and Vogelius functional over Least Squares fitting.
ano.nymous@ccsd.cnrs.fr.invalid (Lekbir Afraites), Lekbir Afraites
Recently several authors considered finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. In this paper we propose a generalization of the EM algorithm to semiparametric mixture models. Our approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the EM type estimators we propose is studied numerically through several Monte Carlo experiments but also by comparison with alternative methods existing in the literature. In addition to these numerical experiments we provide applications to real data showing that our estimation methods behaves well, that it is fast and easy to be implemented.
ano.nymous@ccsd.cnrs.fr.invalid (Laurent Bordes), Laurent Bordes
La première partie concerne l'estimation séquentielle du paramètre de régression pour le modèle de Cox pour des données censurées à droite. Il est ainsi possible de définir des règles d'arrêt garantissant une bonne estimation. Celles-ci conduisent alors à des estimateurs dépendant de tailles d'échantillons aléatoires pour lesquels le comportement asymptotique est le même que celui des estimateurs non séquentiels. Les propriétés démontrées sont étendues au cadre multidimensionnel et illustrées par des simulations. Cette première partie s'achève par l'étude théorique du comportement de la variable d'arrêt dans le cadre d'intervalles de confiance séquentiels. La règle d'arrêt normalisée est alors asymptotiquement normale. La seconde partie porte sur la construction de tests d'homogénéité dans le cadre d'un modèle de durées de vie non paramétrique incluant des covariables ainsi que la censure à droite. Une statistique de test est proposée et son comportement asymptotique est établi.
ano.nymous@ccsd.cnrs.fr.invalid (Christelle Breuils), Christelle Breuils
L'utilisation des méthodes intégrales dans le milieu pétrolier est récente et reste limitée à des problèmes 2D, le puits étant modélisé comme un terme source. Dans ce travail, nous proposons une nouvelle méthode intégrale pour évaluer la performance des puits dans un réservoir stratifié à géométrie quelconque en 3D. Ici, l'écoulement dans le puits est pris en compte par deux types de conditions aux limites, la première linéaire, la seconde non-linéaire et non-locale. Nous avons démontré que chacun des deux modèles (linéaire et non-linéaire) est bien posé. Du point de vue numérique, nous avons développé une nouvelle formulation intégrale, équivalente au modèle linéaire. Les équations intégrales ont été discrétisées par une méthode de Galerkin. D'autre part, nous avons pu tirer profit du problème d'échelle pour faire une approximation filaire du puits. Les tests numériques montrent que cette nouvelle méthode intégrale permet de calculer l'indice de productivité du puits à 1% près.
ano.nymous@ccsd.cnrs.fr.invalid (Valérie Moumas), Valérie Moumas