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Time series anomaly detection (TSAD) focuses on identifying whether observations in streaming data deviate significantly from normal patterns. With the prevalence of connected devices, anomaly detection on time series has become paramount, as it enables real-time monitoring and early detection of irregular behaviors across various application domains. In this work, we introduce PatchTrAD, a Patch-based Transformer model for time series anomaly detection. Our approach leverages a Transformer encoder along with the use of patches under a reconstructionbased framework for anomaly detection. Empirical evaluations on multiple benchmark datasets show that PatchTrAD is on par, in terms of detection performance, with state-of-the-art deep learning models for anomaly detection while being time efficient during inference.
ano.nymous@ccsd.cnrs.fr.invalid (Samy-Melwan Vilhes), Samy-Melwan Vilhes
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 free 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 weak solutions to the problem which are maximal in time. For this, a time semidiscrete minimizing movements scheme based on a Wasserstein-like distance is introduced. The existence of solutions to the scheme is proved. Then, using a priori estimates, it is shown that as the time step goes to zero these solutions converge up to extraction towards a maximal weak solution to the free boundary model.
ano.nymous@ccsd.cnrs.fr.invalid (Benoît Merlet), Benoît Merlet
This paper investigates statistical inference for weak FARIMA models in the frequency domain. We estimate the asymptotic covariance matrix of the classical Whittle estimator to achieve full inference, thereby addressing an open question posed by Shao, X. (2010). However, computing this matrix numerically is costly. To mitigate this issue, we propose an alternative approach that circumvents trispectrum estimation at the cost of a slower convergence rate. Additionally, we introduce a fast alternative to the Whittle estimator based on a one-step procedure. This method refines an initial Whittle estimator computed on a subsample using a single Fisher scoring step. The resulting estimator retains the same asymptotic properties as the Whittle estimator computed on the full sample while significantly reducing computational time.
ano.nymous@ccsd.cnrs.fr.invalid (Samir Ben-Hariz), Samir Ben-Hariz
Locally stationary processes (LSPs) provide a robust framework for modeling time-varying phenomena, allowing for smooth variations in statistical properties such as mean and variance over time. In this paper, we address the estimation of the conditional probability distribution of LSPs using Nadaraya-Watson (NW) type estimators. The NW estimator approximates the conditional distribution of a target variable given covariates through kernel smoothing techniques. We establish the convergence rate of the NW conditional probability estimator for LSPs in the univariate setting under the Wasserstein distance and extend this analysis to the multivariate case using the sliced Wasserstein distance. Theoretical results are supported by numerical experiments on both synthetic and real-world datasets, demonstrating the practical usefulness of the proposed estimators.
ano.nymous@ccsd.cnrs.fr.invalid (Jan Nino G. Tinio), Jan Nino G. Tinio
In this paper, we derive a periodic model from a one dimensional nonlocal eikonal equation set on the full space modeling dislocation dynamics. Thanks to a gradient entropy estimate, we show that this periodic model converges toward the initial one when the period goes to infinity. Moreover, we design a semi-explicit numerical scheme for the periodic model that we introduce. We show the well-posedness of the scheme and a discrete gradient entropy inequality. We also prove the convergence of the scheme and we present some numerical experiments.
ano.nymous@ccsd.cnrs.fr.invalid (Diana Al Zareef), Diana Al Zareef
We introduce binacox, a prognostic method to deal with the problem of detecting multiple cut-points per feature in a multivariate setting where a large number of continuous features are available. The method is based on the Cox model and combines one-hot encoding with the binarsity penalty, which uses total-variation regularization together with an extra linear constraint, and enables feature selection. Original nonasymptotic oracle inequalities for prediction (in terms of Kullback-Leibler divergence) 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 data sets. On these high-dimensional data sets, our proposed method 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 significant cut-points in relevant variables.
ano.nymous@ccsd.cnrs.fr.invalid (Simon Bussy), Simon Bussy
In this paper we design, analyze and simulate a finite volume scheme for a cross-diffusion system which models chemotaxis with local sensing. This system has the same gradient flow structure as the celebrated minimal Keller-Segel system, but unlike the latter, its solutions are known to exist globally in 2D. The long-time behavior of solutions is only partially understood which motivates numerical exploration with a reliable numerical method. We propose a linearly implicit, two-point flux finite volume approximation of the system. We show that the scheme preserves, at the discrete level, the main features of the continuous system, namely mass, non-negativity of solution, entropy, and duality estimates. These properties allow us to prove the well-posedness, unconditional stability and convergence of the scheme. We also show rigorously that the scheme possesses an asymptotic preserving (AP) property in the quasi-stationary limit. We complement our analysis with thorough numerical experiments investigating convergence and AP properties of the scheme as well as its reliability with respect to stability properties of steady solutions.
ano.nymous@ccsd.cnrs.fr.invalid (Maxime Herda), Maxime Herda
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
In this paper, a novel surrogate model for shape-parametrized vehicle drag force prediction is proposed. It is assumed that only a limited dataset of high-fidelity CFD results is available, typically less than ten high-fidelity CFD solutions for different shape samples. The idea is to take advantage not only of the drag coefficients, but also physical fields such as velocity, pressure and kinetic energy evaluated on a cutting plane in the wake of the vehicle and perpendicular to the road. This additional 'augmented' information provides a more accurate and robust prediction of the drag force, compared to a standard surface response methodology. As a first step, an original reparametrization of the shape based on combination coefficients of shape principal components is proposed, leading to a low-dimensional representation of the shape space. The second step consists in determining principal components of the x-direction momentum flux through a cutting plane behind the car. The final step is to find the mapping between the reduced shape description and the momentum flux formula to achieve an accurate drag estimation. The resulting surrogate model is a space-parameter separated representation with shape principal component coefficients and spatial modes dedicated to drag-force evaluation. The algorithm can deal with shapes of variable mesh, by using an optimal transport procedure that interpolates the fields on a shared reference mesh. The Machine Learning algorithm is challenged on a car concept with a shape design space of dimensional three. With only two wellchosen samples, the numerical algorithm is able to return a drag surrogate model with reasonable uniform error over the validation dataset. An incremental learning approach involving additional high-fidelity computations is also proposed. The leading algorithm is shown to improve the model accuracy. The study also shows the sensitivity of the results with respect to the initial experimental design. As a feedback, we discuss and suggest what appear to be the correct choices of experimental designs for best results.
ano.nymous@ccsd.cnrs.fr.invalid (Kalinja Naffer-Chevassier), Kalinja Naffer-Chevassier
This work examines the asymptotic characteristics of a conditional set-indexed empirical process composed of functional ergodic random variables with missing at random (MAR). This paper’s findings enlarge the previous advancements in functional data analysis through the use of empirical process methodologies. These results are shown under specific structural hypotheses regarding entropy and under appealing situations regarding the model. The regression operator’s asymptotic (1−α)-confidence interval is provided for 0<α<1 as an application. Additionally, we offer a classification example to demonstrate the practical importance of the methodology.
ano.nymous@ccsd.cnrs.fr.invalid (Salim Bouzebda), Salim Bouzebda
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
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
In the last years, the automotive engineering industry has been deeply influenced by the use of «machine learning» techniques for new design and innovation purposes. However, some specific engineering aspects like numerical optimization study still require the development of suitable high-performance machine learning approaches involving parametrized Finite Elements (FE) structural dynamics simulation data. Weight reduction on a car body is a crucial matter that improves the environmental impact and the cost of the product. The actual optimization process at Renault SA uses numerical Design of Experiments (DOE) to find the right thicknesses and materials for each part of the vehicle that guarantees a reduced weight while keeping a good behavior of the car body, identified by criteria or sensors on the body (maximum displacements, upper bounds of instantaneous acceleration …). The usual DOE methodology generally uses between 3 and 10 times the numbers of parameters of the study (which means, for a 30-parameters study, at least 90 simulations, with typically 10 h per run on a 140-core computer). During the last 2 years, Renault’s teams strived to develop a disruptive methodology to conduct optimization study. By ‘disruptive’, we mean to find a methodology that cuts the cost of computational effort by several orders of magnitude. It is acknowledged that standard DoEs need a number of simulations which is at least proportional to the dimension of the parameter space, leading generally to hundreds of fine simulations for real applications. Comparatively, a disruptive method should require about 10 fine evaluations only. This can be achieved by means of a combination of massive data knowledge extraction of FE crash simulation results and the help of parallel high-performance computing (HPC). For instance, in the recent study presented by Assou et al. (A car crash reduced order model with random forest. In: 4th International workshop on reduced basis, POD and PGD Model Reduction Techniques—MORTech 2017. 2017), it took 10 runs to find a solution of a 34-parameter problem that fulfils the specifications. In order to improve this method, we must extract more knowledge from the simulation results (correlations, spatio-temporal features, explanatory variables) and process them in order to find efficient ways to describe the car crash dynamics and link criteria/quantities of interest with some explanatory variables. One of the improvements made in the last months is the use of the so-called Empirical Interpolation Method (EIM, [Barrault et al.]) to identify the few time instants and spatial nodes of the FE-mesh (referred to as magic points) that “explain” the behavior of the body during the crash, within a dimensionality reduction approach. The EIM method replaces a former K -Means algorithm (Davies et al. in IEEE Trans Pattern Anal Mach Intell, 1(2):224–227, 1979) which was processed online, for each ROM. Instead, the computation of EIM method is done offline, once for all, for each simulation. This new method allows us to compute a ROM quite faster, and to reduce the number of features that we use for the regression step (~ 100). The nonlinear regression step is achieved by a standard Random Forest (RF, [Breiman. Mach Learn 45:5–32, 2001]) algorithm. Another improvement of the method is the characterization of numerical features describing the shape of the body, at a nodal scale. The characteristics of orientation of the elements surrounding a mesh node must be taken into account to describe the behavior of the node during the crash. The actual method integrates some numerical features, computed from the orientation of the elements around each node, to explain the node behavior. The paper is organized as follows: The introduction states the scientific and industrial context of the research. Then, the ReCUR Method is detailed, and the recent improvements are highlighted. Results are presented and discussed before having some concluding remarks on this piece of work.
ano.nymous@ccsd.cnrs.fr.invalid (Etienne Gstalter), Etienne Gstalter
This paper focuses on the low-dimensional representation of multivariate functions. We study a recursive POD representation, based upon the use of the power iterate algorithm to recursively expand the modes retained in the previous step. We obtain general error estimates for the truncated expansion, and prove that the recursive POD representation provides a quasi-optimal approximation in $$L^2$$ L 2 norm. We also prove an exponential rate of convergence, when applied to the solution of the reaction-diffusion partial differential equation. Some relevant numerical experiments show that the recursive POD is computationally more accurate than the Proper Generalized Decomposition for multivariate functions. We also recover the theoretical exponential convergence rate for the solution of the reaction-diffusion equation.
ano.nymous@ccsd.cnrs.fr.invalid (M. Azaïez), M. Azaïez
The problem of estimating the spatio-functional expectile regression for a given spatial mixing structure Xi,Yi∈F×R, when i∈ZN,N≥1 and F is a metric space, is investigated. We have proposed the M-estimation procedure to construct the Spatial Local Linear (SLL) estimator of the expectile regression function. The main contribution of this study is the establishment of the asymptotic properties of the SLL expectile regression estimator. Precisely, we establish the almost-complete convergence with rate. This result is proven under some mild conditions on the model in the mixing framework. The implementation of the SLL estimator is evaluated using an empirical investigation. A COVID-19 data application is performed, allowing this work to highlight the substantial superiority of the SLL-expectile over SLL-quantile in risk exploration.
ano.nymous@ccsd.cnrs.fr.invalid (Ali Laksaci), Ali Laksaci
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 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 deploy artificial neural networks to unfold neutron spectra from measured energy-integrated quantities. These neutron spectra represent an important parameter allowing to compute the absorbed dose and the kerma to serve radiation protection in addition to nuclear safety. The built architectures are inspired from convolutional neural networks. The first architecture is made up of residual transposed convolution's blocks while the second is a modified version of the U-net architecture. A large and balanced dataset is simulated following "realistic" physical constraints to train the architectures in an efficient way. Results show a high accuracy prediction of neutron spectra ranging from thermal up to fast spectrum. The dataset processing, the attention paid to performances' metrics and the hyperoptimization are behind the architectures' robustness.
ano.nymous@ccsd.cnrs.fr.invalid (Maha Bouhadida), Maha Bouhadida
The world is producing 2.5 quintillion bytes daily, known as big data. Volume, value, variety, velocity, and veracity define the five characteristics of big data that represent a fundamental complexity for many machine learning algorithms, such as clustering, image recognition, and other modern learning techniques. With this large data, hyperparameter estimations do not take the form of the sample mean (not linear). Instead, they takethe form of average over m-tuples, known as the U-statistic estimator in probabilityand statistics. In this work, we treat the collection of U-statistics, known as the Uprocess,for two types of dependent variables, the Markovian data, and locally stationary random variables. Thus, we have divided our work into two parts to address each type independently.In the first part, we deal with Markovian data. The approach relies on regenerative methods, which essentially involve dividing the sample into independent and identically distributed (i.i.d.) blocks of data, where each block corresponds to the path segments between two visits of an atom called A, forming a renewal sequence. We derive the limiting theory for Harris recurrent Markov chain over uniformly bounded and unbounded classes of functions. We show that the results can be generalized also to the bootstrappe dU statistics. The bootstrap approach bypasses the problems faced with the asymptotic behavior due to the unknown parameters of limiting distribution. Furthermore, the bootstrap technique we use in this thesis is the renewal bootstrap, where the bootstrap samplevis formed by resampling the blocks. Since the non-bootstrapped blocks are independent, most proofs reduce to the i.i.d. case. The main difficulties are related to the randomsize of the resampled blocks, which creates a problem with random stopping times. This problem is degraded by replacing the random stopping time with their expectation. Also, since we resample from a random number of blocks, and the bootstrap equicontinuity can be verified by comparing with the initial process, the weak convergence of the bootstrap U-process must be treated very carefully. We successfully derive the results in the case of the k-Harris Markov chain. We extend all the above results to the case where the degreeof U-statistic grows with the sample size n, with the kernel varying in a class of functions. We provide the uniform limit theory for the renewal bootstrap for the infinite-degree U-process with the help of the decoupling technique combined with symmetrization techniques in addition to the chaining inequality. Remaining in the Markovian setting, we extend the weighted bootstrap empirical processes to a high-dimensional estimation. We consider an exchangeably weighted bootstrap of the general function-indexed empirical U-processes. In the second part of this thesis, dependent data are represented by locally stationary random variables. Propelled by the increasing representation of the data by functionalor curves time series and the non-stationary behavior of the latter, we are interested in the conditional U-process of locally stationary functional time series. More precisely, we investigate the weak convergence of the conditional U-processes in the locally stationary functional mixing data framework. We treat the weak convergence in both caseswhen the class of functions is bounded or unbounded, satisfying some moment conditions. Finally, we extend the asymptotic theory of conditional U-process to the locallystationary functional random field {Xs,An : s ∈ Rn} observed at irregular spaced locations in Rn = [0,An]d ∈ Rd, and include both pure increasing domain and mixed increasing domain. We treat the weak convergence in both cases when the class of functions is boundedor unbounded, satisfying some moment conditions. These results are established underfairly general structural conditions on the classes of functions and the underlying models.
ano.nymous@ccsd.cnrs.fr.invalid (Inass Soukarieh), Inass Soukarieh
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
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
Principal component analysis is a recognized powerful and practical method in statistics and data science. It can also be used in modeling as a dimensionality reduction tool to achieve low-order models of complex multiphysics or engineering systems. Model-order reduction (MOR) methodologies today are an important topic for engineering design and analysis. Design space exploration or accelerated numerical optimization for example are made easier by the use of reduced-order models. In this chapter, we will talk about the use of higher-order singular value decompositions (HOSVD) applied to spatiotemporal problems that are parameterized by a set of design variables or physical parameters. Here we consider a data-driven reduced order modeling based on a design of computer experiment: from high-dimensional computational results returned by high-fidelity solvers (e.g. finite element ones), the HOSVD allows us to determine spatial, time and parameters principal components. The dynamics of the system can then be retrieved by identifying the low-order discrete dynamical system. As application, we will consider the dynamics of deformable capsules flowing into microchannels. The study of such fluid-structure interaction problems is motivated by the use of microcapsules as innovative drug delivery carriers through blood vessels.
ano.nymous@ccsd.cnrs.fr.invalid (Florian de Vuyst), Florian de Vuyst
We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn-Vogelius artifice and the condensation Steklov-Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the discrete and exact problems behave in diametrically opposite ways. Indeed, existence of the discrete solution is always guaranteed while its uniqueness may be lost. In contrast, the solution of the exact problem may not exist, but it is unique. We show how existence of the so called "weak spurious modes", of the exact variational formulation, is source of instability and the reason why existence may fail. For the discrete problem, we find that the cause of non uniqueness is actually the occurrence of "spurious modes". We track their fading effect asymptotically when the mesh size tends to zero. In order to restore uniqueness, we recall the discrete version of the Holmgren principle, introduced in [Azaïez et al, IPSE, 18, 2011], and we discuss the effect on uniqueness of the finite element mesh, using some graph theory basic material.
ano.nymous@ccsd.cnrs.fr.invalid (F Ben Belgacem), F Ben Belgacem
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ano.nymous@ccsd.cnrs.fr.invalid (Mustapha Mohammedi), Mustapha Mohammedi
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
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
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ano.nymous@ccsd.cnrs.fr.invalid (Florian de Vuyst), Florian de Vuyst
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ano.nymous@ccsd.cnrs.fr.invalid (Tarik Fahlaoui), Tarik Fahlaoui
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
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ano.nymous@ccsd.cnrs.fr.invalid (Elias Zgheib), Elias Zgheib
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
For a system, a priori identifiability is a theoretical property depending only on the model and guarantees that its parameters can be uniquely determined from observations. This paper provides a survey of the various and numerous definitions of a priori identifiability given in the literature, for both deterministic continuous and discrete-time models. A classification is done by distinguishing analytical and algebraic definitions as well as local and global ones. Moreover, this paper provides an overview on the distinct methods to test the parameter identifiability. They are classified into the so-called output equality approaches, local state isomorphism approaches and differential algebra approaches. A few examples are detailed to illustrate the methods and complete this survey.
ano.nymous@ccsd.cnrs.fr.invalid (Floriane Anstett-Collin), Floriane Anstett-Collin
Internet of Things (IoT) applications using sensors and actuators raise new privacy related threats such as drivers and vehicles tracking and profiling. These threats can be addressed by developing adaptive and context-aware privacy protection solutions to face the environmental constraints (memory, energy, communication channel, etc.), which cause a number of limitations of applying cryptographic schemes. This paper proposes a privacy preserving solution in ITS context relying on a game theory model between two actors (data holder and data requester) using an incentive motivation against a privacy concession, or leading an active attack. We describe the game elements (actors, roles, states, strategies, and transitions), and find an equilibrium point reaching a compromise between privacy concessions and incentive motivation. Finally, we present numerical results to analyze and evaluate the game theory-based theoretical formulation.
ano.nymous@ccsd.cnrs.fr.invalid (Arbia Riahi Sfar), Arbia Riahi Sfar
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
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
Dans le cadre d’un programme pluriannuel, des campagnes de sondages ont été réalisées sur les deux versants du col du Petit-Saint-Bernard(2188 m, Alpes occidentales), entre 750 et 3000 m d’altitude. La méthode de travail néglige les prospections au sol, au profit de la multiplication des sondages manuels, implantés dans des contextes topographiques sélectionnés et menés jusqu’à la base des remplissages holocènes. Les résultats obtenus documentent dans la longue durée l’évolution de la dynamique pédo-sédimentaire et la fréquentation des différents étages d’altitude. La signification des données archéologiques collectées est discutée par rapport à l’état des connaissances dans une zone de comparaison groupant les vallées voisines des Alpes occidentales, par rapport aux modèles de peuplement existants et par rapport aux indications taphonomiques apportées par l’étude pédo-sédimentaire. Un programme d’analyses complémentaires destiné à préciser le contexte, la taphonomie et le statut fonctionnel
ano.nymous@ccsd.cnrs.fr.invalid (Pierre-Jérôme Rey), Pierre-Jérôme Rey
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
Many biological networks include cyclic structures. In such cases, Bayesian networks (BNs), which must be acyclic, are not sound models for structure learning. Dynamic BNs can be used but require relatively large time series data. We discuss an alternative model that embeds cyclic structures within acyclic BNs, allowing us to still use the fac-torization property and informative priors on network structure. We present an implementation in the linear Gaussian case, where cyclic structures are treated as multivariate nodes. We use a Markov Chain Monte Carlo algorithm for inference, allowing us to work with posterior distribution on the space of graphs.
ano.nymous@ccsd.cnrs.fr.invalid (Witold Wiecek), Witold Wiecek
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
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
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
Because of the huge number of graphs possible even with a small number of nodes, inference on network structure is known to be a challenging problem. Generating large random directed graphs with prescribed probabilities of occurrences of some meaningful patterns (motifs) is also difficult. We show how to generate such random graphs according to a formal probabilistic representation, using fast Markov chain Monte Carlo methods to sample them. As an illustration, we generate realistic graphs with several hundred nodes mimicking a gene transcription interaction network in Escherichia coli.
ano.nymous@ccsd.cnrs.fr.invalid (Frédéric Y. Bois), Frédéric Y. Bois
In this chapter, we present the empirical estimation of some reliability measures, such as the rate of occurrence of failures and the steady-state availability, for a discrete-time semi-Markov system. The probability of first occurred failure is introduced and estimated. A numerical application is given to illustrate the strong consistency of these estimators.
ano.nymous@ccsd.cnrs.fr.invalid (Stylianos Georgiadis), Stylianos Georgiadis
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 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
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
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
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
This paper deals with parameter and state estimation in a bounded-error context for uncertain dynamical aerospace models when the input is considered optimized or not. In a bounded-error context, perturbations are assumed bounded but otherwise unknown. The parameters to be estimated are also considered bounded. The tools of the presented work are based on a guaranteed numerical set integration solver of ordinary differential equations combined with adapted set inversion computation. The main contribution of this work consists in developing procedures for parameter estimation whose performance is highly related with the input of system. In this paper, a comparison with a classical non-optimized input is proposed.
ano.nymous@ccsd.cnrs.fr.invalid (Qiaochu Li), Qiaochu Li
A real time algorithm for cardiac and respiratory gating, which only requires an ECG sensor, is proposed here. Three ECG electrodes are placed in such a manner that the modulation of the recorded ECG by the respiratory signal would be maximal; hence, given only one signal we can achieve both cardiac and respiratory MRI gating. First, an off-line learning phase based on wavelet decomposition is run to compute an optimal QRS filter. Afterwards, on one hand the QRS filter is used to accomplish R peak detection, and on the other, a low pass filtering process allows the retrieval of the respiration cycle so that the image acquisition sequences would be triggered by the R peaks only during the expiration phase.
ano.nymous@ccsd.cnrs.fr.invalid (D Abi-Abdallah), D Abi-Abdallah
Blood flow in high static magnetic fields induces elevated voltages that disrupt the ECG signal recorded simultaneously during MRI scans for synchronization purposes. This is known as the magnetohydrodynamic (MHD) effect, it increases the amplitude of the T wave, thus hindering correct R peak detection. In this paper, we present an algorithm for extracting an efficient reference signal from an ECG contaminated by the Nuclear Magnetic Resonance (NMR) environment, that performs a good separation of the R-wave and the MHD artifacts. The proposed signal processing method is based on sub-band decomposition using the wavelet transform, and has been tested on human and small rodents ECG signals acquired during MRI scans in various magnetic field intensities. The results showed an almost flawless trigger generation in fields up to 4.7 Tesla during the three tested imaging sequences (GE, FSE and IRSE)
ano.nymous@ccsd.cnrs.fr.invalid (D Abi-Abdallah), D Abi-Abdallah
Blood flow in high static magnetic fields induces elevated voltages that contaminate the ECG signal which is recorded simultaneously during MRI scans for synchronization purposes. This is known as the magnetohydrodynamic (MHD) effect, it increases the amplitude of the T wave, thus hindering correct R peak detection. In this paper, we inspect the MHD induced alterations of human ECG signals recorded in a 1.5 Tesla steady magnetic field and establish a primary characterization of the induced changes using time and frequency domain analysis. We also reexamine our previously developed real time algorithm for MRI cardiac gating and determine that, with a minor modification, this algorithm is capable of achieving perfect detection even in the presence of strong MHD artifacts.
ano.nymous@ccsd.cnrs.fr.invalid (Dima Abi Abdallah), Dima Abi Abdallah
Cardiac Magnetic Resonance Imaging (MRI) requires synchronization to overcome motion related artifacts caused by the heart’s contractions and the chest wall movements during respiration. Achieving good image quality necessitates combining cardiac and respiratory gating to produce, in real time, a trigger signal that sets off the consecutive image acquisitions. This guarantees that the data collection always starts at the same point of the cardiac cycle during the exhalation phase. In this paper, we present a real time algorithm for extracting a cardiac-respiratory trigger signal using only one, adequately placed, ECG sensor. First, an off-line calculation phase, based on wavelet decomposition, is run to compute an optimal QRS filter. This filter is used, afterwards, to accomplish R peak detection, while a low pass filtering process allows the retrieval of the respiration cycle. The algorithm’s synchronization capabilities were assessed during mice cardiac MRI sessions employing three different imaging sequences, and three specific wavelet functions. The prominent image enhancement gave a good proof of correct triggering. QRS detection was almost flawless for all signals. As for the respiration cycle retrieval it was evaluated on contaminated simulated signals, which were artificially modulated to imitate respiration. The results were quite satisfactory.
ano.nymous@ccsd.cnrs.fr.invalid (Dima Abi-Abdallah), Dima Abi-Abdallah
Blood flow in a steady magnetic field has been of great interest over the past years.Many researchers have examined the effects of magnetic fields on velocity profiles and arterial pressure, and major studies focused on steady or sinusoidal flows. In this paper we present a solution for pulsed magnetohydrodynamic blood flow with a somewhat realistic physiological pressure wave obtained using a windkessel lumped model. A pressure gradient is derived along a rigid vessel placed at the output of a compliant module which receives the ventricle outflow. Then, velocity profile and flow rate expressions are derived in the rigid vessel in the presence of a steady transverse magnetic field. As expected, results showed flow retardation and flattening. The adaptability of our solution approach allowed a comparison with previously addressed flow cases and calculations presented a good coherence with those well established solutions.
ano.nymous@ccsd.cnrs.fr.invalid (Dima Abi Abdallah), Dima Abi Abdallah
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
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
Nous considérons une ́equation qui modélise la diffusion de la température dans une mousse de graphite contenant des capsules de sel. Les conditions de transition de la température entre le graphite et le sel doivent être traitées correctement. Nous effectuons l'analyse de ce modèle et prouvons qu'il est bien posé. Puis nous en proposons une discrétisation par éléments finis et effectuons l'analyse a priori du problème discret. Quelques expériences numériques confirment l'intérêt de cette approche.
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
Dans cet article nous allons estimer la fiabilité d'un système binaire et obtenir son intervalle de confiance par l'approximation normale asymptotique. Cette méthode peut s'appliquer aux systèmes complexes et de grande taille réduisant la largeur de l'intervalle de confiance.
ano.nymous@ccsd.cnrs.fr.invalid (Yunhui Hou), Yunhui Hou
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
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
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
This paper addresses a complex multi-physical phenomemon involving cardiac electrophysiology and hemodynamics. The purpose is to model and simulate a phenomenon that has been observed in MRI machines: in the presence of a strong magnetic field, the T-wave of the electrocardiogram (ECG) gets bigger, which may perturb ECG-gated imaging. This is due a magnetohydrodynamic (MHD) eff ect occurring in the aorta. We reproduce this experimental observation through computer simulations on a realistic anatomy, and with a three-compartment model: inductionless magnetohydrodynamic equations in the aorta, bidomain equations in the heart and electrical di ffusion in the rest of the body. These compartments are strongly coupled and solved using fi nite elements. Several benchmark tests are proposed to assess the numerical solutions and the validity of some modeling assumptions. Then, ECGs are simulated for a wide range of magnetic field intensities (from 0 to 20 Tesla).
ano.nymous@ccsd.cnrs.fr.invalid (Vincent Martin), Vincent Martin
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 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
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
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
We consider a model for fluid flow in a porous medium with a fracture. In this model, the fracture is represented as an interface between subdomains, where specific equations have to be solved. In this article we analyse the discrete problem, assuming that the fracture mesh and the subdomain meshes are completely independent, but that the geometry of the fracture is respected. We show that despite this non-conformity, first order convergence is preserved with the lowest order Raviart-Thomas(-Nedelec) mixed finite elements. Numerical simulations confirm this result.
ano.nymous@ccsd.cnrs.fr.invalid (Najla Frih), Najla Frih
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
The contact between two membranes can be described by a system of variational inequalities, where the unknowns are the displacements of the membranes and the action of a membrane on the other one. A discretization of this system is proposed in Part 1 of this work, where the displacements are approximated by standard finite elements and the action by a local postprocessing which admits an equivalent mixed reformulation.Here, we perform the a posteriori analysis of this discretization and prove optimal error estimates. Next, we present numerical experiments that confirm the efficiency of the error indicators.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
We are interested in the optimal control problem of the heat equation where the quadratic cost functional involves a final observation and the control variable is a Dirichlet boundary condition. We first prove that this problem is well-posed. Next, we check its equivalence with a fixed point problem for a space-time mixed system of parabolic equations. Finally, we introduce a Robin penalization on the Dirichlet boundary control for the mixed problem and analyze the convergence when the penalty parameter tends to zero.
ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem
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
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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
Nous présentons un environnement de génération automatique de simulations entièrement basé sur les technologies XML. Le langage de description proposé permet de décrire des objets mathématiques tels que des systèmes d'équations différentielles, des systèmes d'équations non-linéaires, des équations aux dérivées partielles en dimension 2, ou bien de simples courbes et surfaces. Il permet aussi de décrire les paramètres dont dépendent ces objets. Ce langage est indépendant du logiciel et permet donc de garantir la pérennité du travail des auteurs ainsi que leur mutualisation et leur réutilisation. Nous décrivons aussi l'architecture d'une «chaîne de compilation» permettant de transformer ces fichiers XML sous forme de scripts et de les faire fonctionner dans le logiciel Scilab.
ano.nymous@ccsd.cnrs.fr.invalid (Stéphane Mottelet), Stéphane Mottelet