Publications sur H.A.L.

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[hal-04427506] Weak Convergence of the Conditional Set-Indexed Empirical Process for Missing at Random Functional Ergodic Data

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. (Salim Bouzebda), Salim Bouzebda

[hal-04282819] Gaussian-Smoothed Sliced Probability Divergences

Gaussian smoothed sliced Wasserstein distance has been recently introduced for comparing probability distributions, while preserving privacy on the data. It has been shown that it provides performances similar to its non-smoothed (non-private) counterpart. However, the computational and statistical properties of such a metric have not yet been well-established. This work investigates the theoretical properties of this distance as well as those of generalized versions denoted as Gaussian-smoothed sliced divergences. We first show that smoothing and slicing preserve the metric property and the weak topology. To study the sample complexity of such divergences, we then introduce $\hat{\hat\mu}_{n}$ the double empirical distribution for the smoothed-projected $\mu$. The distribution $\hat{\hat\mu}_{n}$ is a result of a double sampling process: one from sampling according to the origin distribution $\mu$ and the second according to the convolution of the projection of $\mu$ on the unit sphere and the Gaussian smoothing. We particularly focus on the Gaussian smoothed sliced Wasserstein distance and prove that it converges with a rate $O(n^{-1/2})$. We also derive other properties, including continuity, of different divergences with respect to the smoothing parameter. We support our theoretical findings with empirical studies in the context of privacy-preserving domain adaptation. (Mokhtar Z. Alaya), Mokhtar Z. Alaya

[hal-04543367] A Semi-Markov Model with Geometric Renewal Processes

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. (Jingqi Zhang), Jingqi Zhang

[hal-04458367] Recursive POD expansion for reaction-diffusion equation

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. (M. Azaïez), M. Azaïez

[hal-03812864] Exchangeably Weighted Bootstraps of General Markov U-Process

We explore an exchangeably weighted bootstrap of the general function-indexed empirical U-processes in the Markov setting, which is a natural higher-order generalization of the weighted bootstrap empirical processes. As a result of our findings, a considerable variety of bootstrap resampling strategies arise. This paper aims to provide theoretical justifications for the exchangeably weighted bootstrap consistency in the Markov setup. General structural conditions on the classes of functions (possibly unbounded) and the underlying distributions are required to establish our results. This paper provides the first general theoretical study of the bootstrap of the empirical U-processes in the Markov setting. Potential applications include the symmetry test, Kendall’s tau and the test of independence. (Inass Soukarieh), Inass Soukarieh

[hal-03915451] Uniform Consistency for Functional Conditional U-Statistics Using Delta-Sequences

U-statistics are a fundamental class of statistics derived from modeling quantities of interest characterized by responses from multiple subjects. U-statistics make generalizations the empirical mean of a random variable X to the sum of all k-tuples of X observations. This paper examines a setting for nonparametric statistical curve estimation based on an infinite-dimensional covariate, including Stute’s estimator as a special case. In this functional context, the class of “delta sequence estimators” is defined and discussed. The orthogonal series method and the histogram method are both included in this class. We achieve almost complete uniform convergence with the rates of these estimators under certain broad conditions. Moreover, in the same context, we show the uniform almost-complete convergence for the nonparametric inverse probability of censoring weighted (I.P.C.W.) estimators of the regression function under random censorship, which is of its own interest. Among the potential applications are discrimination problems, metric learning and the time series prediction from the continuous set of past values. (Salim Bouzebda), Salim Bouzebda

[hal-03909074] Non-Parametric Conditional U-Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design

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. (Salim Bouzebda), Salim Bouzebda

[hal-03864942] Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes

In this study, we look at the wavelet basis for nonparametric estimation of density and regression functions for continuous functional stationary processes in Hilbert space. The mean integrated squared error for a small subset is established. We employ a martingale approach to obtain the asymptotic properties of these wavelet estimators. These findings are established under rather broad assumptions. All we assume about the data is that it is ergodic, but beyond that, we make no assumptions. In this paper, the mean integrated squared error findings in the independence or mixing setting were generalized to the ergodic setting. The theoretical results presented in this study are (or will be) valuable resources for various cutting-edge functional data analysis applications. Applications include conditional distribution, conditional quantile, entropy, and curve discrimination. (Sultana Didi), Sultana Didi

[hal-04389539] Nonparametric Recursive Kernel Type Eestimators for the Moment Generating Function Under Censored Data

We are mainly concerned with kernel-type estimators for the moment-generating function in the present paper. More precisely, we establish the central limit theorem with the characterization of the bias and the variance for the nonparametric recursive kernel-type estimators for the moment-generating function under some mild conditions in the censored data setting. Finally, we investigate the methodology's performance for small samples through a short simulation study. (Salim Bouzebda), Salim Bouzebda

[hal-02459255] Metabolic Flux Analysis in Isotope Labeling Experiments Using the Adjoint Approach

[...] (Stéphane Mottelet), Stéphane Mottelet

[hal-04337080] Singularity Removal for 3D Elliptic Problems with Variable Coefficients and Line Sources

Three dimensional elliptic problems with variable coefficients and line Dirac sources arise in a number of fields. The lack of regularity on the solution prompts users to turn towards alternative variational formulations. Rather than using weighted Sobolev spaces, we prefer the dual variational formulation written in the Hilbertian Lebesgue space, the one used by G. Stampacchia [Séminaire Jean Leray, 1964]. The key work is to show a singular/regular expansion where the singularity of the potential is fully expressed by a convolution formula, based on the Green kernel of the Laplacian. The correction term restores the boundary condition and fits with the standard variational formulation of Poisson equation (in the Sobolev space H^1). We intend to develop a thorough analysis of the proposed expansion while avoiding stringent assumptions on the conductivities. Sharp technical tools, as those developed in [E. Di-Giorgi, Mem. Accad. Sci. Torino. 1957] and [N. G. Meyers Ann. Scuo. Norm. Sup. Pisa, 1963], are necessary in the proofs. (Eya Bejaoui), Eya Bejaoui

[hal-03294727] The jamming constant of uniform random graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a “parking process” on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given–possibly unbounded–degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald. (Paola Bermolen), Paola Bermolen

[hal-04033438] Robust augmented energy a posteriori estimates for Lipschitz and strongly monotone elliptic problems

In this paper, we design a posteriori estimates for finite element approximations of nonlinear elliptic problems satisfying strong-monotonicity and Lipschitz-continuity properties. These estimates include, and build on, any iterative linearization method that satisfies a few clearly identified assumptions; this encompasses the Picard, Newton, and Zarantonello linearizations. The estimates give a guaranteed upper bound on an augmented energy difference (reliability with constant one), as well as a lower bound (efficiency up to a generic constant). We prove that for the Zarantonello linearization, this generic constant only depends on the space dimension, the mesh shape regularity, and possibly the approximation polynomial degree in four or more space dimensions, making the estimates robust with respect to the strength of the nonlinearity. For the other linearizations, there is only a computable dependence on the local variation of the linearization operators. We also derive similar estimates for the usual energy difference that depend locally on the nonlinearity and improve the established bound. Numerical experiments illustrate and validate the theoretical results, for both smooth and singular solutions. (André Harnist), André Harnist

[hal-04180133] Non parametric observation driven HMM

[...] (Hanna Bacave), Hanna Bacave

[hal-04171324] Extensions of the empirical interpolation method to vector-valued functions

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. (Florian de Vuyst), Florian de Vuyst

[hal-04129681] Accounting for inspection errors and change in maintenance behaviour

We propose a way to account for inspection errors in a particular framework. We consider a situation where the lifetime of a system depends essentially of a particular part. A deterioration of this part is regarded as an unacceptable state for the safety of the system and a major renewal is deemed necessary. Thus the statistical analysis of the deterioration time distribution of this part is of primary interest for the preventive maintenance of the system. In this context, we faced the following problem. In the early life of the system, unwarranted renewals of the part are decided upon, caused by overly cautious behaviour. Such unnecessary renewals make the statistical analysis of deterioration time data difficult and can induce an underestimation of the mean life of the part. To overcome this difficulty, we propose to regard the problem as an incomplete data model. We present its estimation under the maximum likelihood methodology. Numerical experiments show that this approach eliminates the pessimistic bias in the estimation of the mean life of the part. We also present a Bayesian analysis of the problem which can be useful in a small sample setting. (Gilles Celeux), Gilles Celeux

[tel-04122066] Local matching algorithms on the configuration model

The present thesis constructs an alternative framework to online matching algorithms on large graphs. Using the configuration model to mimic the degree distributions of large networks, we are able to build algorithms based on local matching policies for nodes. Thus, we are allowed to predict and approximate the performances of a class of matching policies given the degree distributions of the initial network. Towards this goal, we use a generalization of the differential equation method to measure valued processes. Throughout the text, we provide simulations and a comparison to the seminal work of Karp, Vazirani and Vazirani based on the prevailing viewpoint in online bipartite matching. (Mohamed Habib Aliou Diallo Aoudi), Mohamed Habib Aliou Diallo Aoudi

[hal-04120816] Non-intrusive reduced order models for partitioned fluid-structure interactions

The main goal of this research is to develop a data-driven reduced order model (ROM) strategy from high-fidelity simulation result data of a full order model (FOM). The goal is to predict at lower computational cost the time evolution of solutions of Fluid-Structure Interaction (FSI) problems. For some FSI applications like tire/water interaction, the FOM solid model (often chosen as quasistatic) can take far more computational time than the HF fluid one. In this context, for the sake of performance one could only derive a reduced-order model for the structure and try to achieve a partitioned HF fluid solver coupled with a ROM solid one. In this paper, we present a datadriven partitioned ROM on a study case involving a simplified 1D-1D FSI problem representing an axisymmetric elastic model of an arterial vessel, coupled with an incompressible fluid flow. We derive a purely data-driven solid ROM for FOM fluid-ROM structure partitioned coupling and present early results. (Azzeddine Tiba), Azzeddine Tiba

[hal-00260732] A method to compute the transition function of a piecewise deterministic Markov process with application to reliability

We study the time evolution of an increasing stochastic process governed by a first-order stochastic differential system. This defines a particular piecewise deterministic Markov process (PDMP). We consider a Markov renewal process (MRP) associated to the PDMP and its Markov renewal equation (MRE) which is solved in order to obtain a closed-form solution of the transition function of the PDMP. It is then applied in the framework of survival analysis to evaluate the reliability function of a given system. We give a numerical illustration and we compare this analytical solution with the Monte Carlo estimator. (Julien Chiquet), Julien Chiquet

[hal-03564379] Lebesgue Induction and Tonelli’s Theorem in Coq

Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions. (Sylvie Boldo), Sylvie Boldo

[hal-03105815] Lebesgue integration. Detailed proofs to be formalized in Coq

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. (François Clément), François Clément

[hal-03889276] A Coq Formalization of Lebesgue Induction Principle and Tonelli’s Theorem

Lebesgue integration is a well-known mathematical tool, used for instance in probability theory, real analysis, and numerical mathematics. Thus, its formalization in a proof assistant is to be designed to fit different goals and projects. Once the Lebesgue integral is formally defined and the first lemmas are proved, the question of the convenience of the formalization naturally arises. To check it, a useful extension is Tonelli's theorem, stating that the (double) integral of a nonnegative measurable function of two variables can be computed by iterated integrals, and allowing to switch the order of integration. This article describes the formal definition and proof in Coq of product sigma-algebras, product measures and their uniqueness, the construction of iterated integrals, up to Tonelli's theorem. We also advertise the Lebesgue induction principle provided by an inductive type for nonnegative measurable functions. (Sylvie Boldo), Sylvie Boldo

[hal-03888607] Analysis of a one dimensional energy dissipating free boundary model with nonlinear boundary conditions. Existence of global weak solutions

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of global weak solutions to the problem. For this, a semi-discrete in time minimizing movements scheme à la De Giorgi is introduced. First, the existence of solutions to the scheme is established and then, using a priori estimates, it is proved that as the time step goes to zero these solutions converge up to extraction towards a weak solution to the free boundary model. (Benoît Merlet), Benoît Merlet

[hal-03882839] Analysis of Lavrentiev-Finite Element Methods for Data Completion Problems

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. (Faker Ben Belgacem), Faker Ben Belgacem

[hal-03879762] AptaMat : a matrix-based algorithm to compare single-stranded oligonucleotides secondary structures

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. (Thomas Binet), Thomas Binet

[hal-03858196] Uniqueness’ Failure for the Finite Element Cauchy-Poisson’s Problem

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. (F Ben Belgacem), F Ben Belgacem

[tel-03774522] Semiparametric M-estimators and their applications to multiple change-point problems

In this dissertation we are concerned with semiparametric models. These models have success and impact in mathematical statistics due to their excellent scientific utility and intriguing theoretical complexity. In the first part of the thesis, we consider the problem of the estimation of a parameter θ, in Banach spaces, maximizing some criterion function which depends on an unknown nuisance parameter h, possibly infinite-dimensional. We show that the m out of n bootstrap, in a general setting, is weakly consistent under conditions similar to those required for weak convergence of the non smooth M-estimators. In this framework, delicate mathematical derivations will be required to cope with estimators of the nuisance parameters inside non-smooth criterion functions. We then investigate an exchangeable weighted bootstrap for function-valued estimators defined as a zero point of a function-valued random criterion function. The main ingredient is the use of a differential identity that applies when the random criterion function is linear in terms of the empirical measure. A large number of bootstrap resampling schemes emerge as special cases of our settings. Examples of applications from the literature are given to illustrate the generality and the usefulness of our results. The second part of the thesis is devoted to the statistical models with multiple change-points. The main purpose of this part is to investigate the asymptotic properties of semiparametric M-estimators with non-smooth criterion functions of the parameters of multiple change-points model for a general class of models in which the form of the distribution can change from segment to segment and in which, possibly, there are parameters that are common to all segments. Consistency of the semiparametric M-estimators of the change-points is established and the rate of convergence is determined. The asymptotic normality of the semiparametric M-estimators of the parameters of the within-segment distributions is established under quite general conditions. We finally extend our study to the censored data framework. We investigate the performance of our methodologies for small samples through simulation studies. (Anouar Abdeldjaoued Ferfache), Anouar Abdeldjaoued Ferfache

[hal-03714164] Study of an entropy dissipating finite volume scheme for a nonlocal cross-diffusion system

In this paper we analyse a finite volume scheme for a nonlocal version of the Shigesada-Kawazaki-Teramoto (SKT) cross-diffusion system. We prove the existence of solutions to the scheme, derive qualitative properties of the solutions and prove its convergence. The proofs rely on a discrete entropy-dissipation inequality, discrete compactness arguments, and on the novel adaptation of the so-called duality method at the discrete level. Finally, thanks to numerical experiments, we investigate the influence of the nonlocality in the system: on convergence properties of the scheme, as an approximation of the local system and on the development of diffusive instabilities. (Maxime Herda), Maxime Herda

[hal-03700112] Fast calibration of weak FARIMA models

In this paper, we investigate the asymptotic properties of Le Cam's one-step estimator for weak Fractionally AutoRegressive Integrated Moving-Average (FARIMA) models. For these models, noises are uncorrelated but neither necessarily independent nor martingale differences errors. We show under some regularity assumptions that the onestep estimator is strongly consistent and asymptotically normal with the same asymptotic variance as the least squares estimator. We show through simulations that the proposed estimator reduces computational time compared with the least squares estimator. An application for providing remotely computed indicators for time series is proposed. (Samir Ben Hariz), Samir Ben Hariz

[tel-03530823] Properties of words and competing risk processes under semi-Markov hypothesis

Our thesis is dedicated in big part, to solving certain problems in biology (biologic sequences and lifespan using the competing risk framework) under semi-Markovian hypothesis. In recent years, computing the properties of words through random sequences has become a topic of interest in the intersection between mathematics and biology. In the literature, a vast number of methods have tackled this problem under the assumption that sequences of symbols are modeled by Markov processes. Nevertheless, the markovian hypothesis has some disadvantages. In Markov processes, the sojourn time is modeled by the exponential (geometric) distribution in continuous (discrete) time. By contrast, in semi-Markov processes the sojourn time in a state can be modeledby any probability law. Therefore, in order to propose a more general approach to compute the properties of words through a random sequence, in this PhD work weconsider that biological sequences are modeled by semi-Markovian discrete processes. We also compute the average number of times that the elements from a specific set of words appear through a sequence of letters. To achieve our goal, we use the strong law of large numbers and we provide the central limit theorem. To prove the applicationof our proposed model, we find a particular enzyme in a bacteriophage DNA sequence. Competing risk problems conform another interesting topic in the lifespan domain. In general, competing risk problems have been dealt with a statistic approach. In this thesis, we present competing risk models within a semi-Markov framework. We consider continuous and discrete time semi-Markov processes with a finite number of transient and absorbing states. Each absorbing state represents a failure mode (in reliability of a system) or a cause of death of an individual (in survival analysis). We express the probability that a failure occurs at a given time due to a unique cause. We give the joint distribution of the life time and the failure cause via the transition function of the semi-Markov process in continuous and discrete-time respectively. Some examples are given for illustration. We also present a method for solving continuous time Markovian renewal equations based on well-established algorithms in their discrete time corresponding counter parts. The great advantage drawn by this approach is that the in finite series of the renewal function, in continuous time, is replaced, in discrete time, by a finite series. Results for error estimation are also established. To illustrate this approach we propose a digital application concerning cyber-attacks where the functions of conditional transitions are of the Weibull type. (Brenda Ivette Garcia Maya), Brenda Ivette Garcia Maya

[hal-03273118] A Hybrid High-Order method for incompressible flows of non-Newtonian fluids with power-like convective behaviour

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. (Daniel Castanon Quiroz), Daniel Castanon Quiroz

[hal-03471095] A Coq Formalization of Lebesgue Integration of Nonnegative Functions

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. (Sylvie Boldo), Sylvie Boldo

[hal-03194113] A Coq Formalization of Lebesgue Integration of Nonnegative Functions

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. (Sylvie Boldo), Sylvie Boldo

[hal-03234445] Thin layer approximations in mechanical structures : The Dirichlet boundary condition case

[...] (Frédérique Le Louër), Frédérique Le Louër

[hal-02512652] Analytical preconditioners for Neumann elastodynamic Boundary Element Methods

Recent works in the Boundary Element Method (BEM) community have been devoted to the derivation of fast techniques to perform the matrix vector product needed in the iterative solver. Fast BEMs are now very mature. However, it has been shown that the number of iterations can significantly hinder the overall efficiency of fast BEMs. The derivation of robust preconditioners is now inevitable to increase the size of the problems that can be considered. Analytical precon-ditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to propose new analytical preconditioners to treat Neumann exterior scattering problems in 2D and 3D elasticity. These preconditioners are local approximations of the adjoint Neumann-to-Dirichlet map. We propose three approximations with different orders. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). An analytical spectral study confirms the expected behavior of the preconditioners, i.e., a better eigenvalue clustering especially in the elliptic part contrary to the standard CFIE of the first-kind. We provide various 2D numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be independent of the density of discretization points per wavelength which is not the case of the standard CFIE. In addition, it is less sensitive to the frequency. (Stéphanie Chaillat), Stéphanie Chaillat

[hal-03003563] A priori identifiability : An overview on definitions and approaches

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. (Floriane Anstett-Collin), Floriane Anstett-Collin

[hal-03168254] A general stochastic matching model on multigraphs

We extend the general stochastic matching model on graphs introduced in [13], to matching models on multigraphs, that is, graphs with self-loops. The evolution of the model can be described by a discrete time Markov chain whose positive recurrence is investigated. Necessary and sufficient stability conditions are provided, together with the explicit form of the stationary probability in the case where the matching policy is 'First Come, First Matched'. (Jocelyn Begeot), Jocelyn Begeot

[hal-02921498] A Game Theoretic Approach for Privacy Preserving Model in IoT-Based Transportation

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. (Arbia Riahi Sfar), Arbia Riahi Sfar

[hal-00305487] On the strong approximation and functional limit laws for the increments of the non-overlapping k-spacings processes

The first aim of the present paper, is to establish strong approximations of the uniform non-overlapping k-spacings process extending the results of Aly et al. (1984). Our methods rely on the invariance principle in Mason and van Zwet (1987). The second goal, is to generalize the Dindar (1997) results for the increments of the spacings quantile process to the uniforme non-overlapping k-spacings quantile process. We apply the last result to characterize the limit laws of functionals of the increments k-spacings quantile process. (Salim Bouzebda), Salim Bouzebda

[hal-02274493] A posteriori estimates distinguishing the error components and adaptive stopping criteria for numerical approximations of parabolic variational inequalities

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. (Jad Dabaghi), Jad Dabaghi

[hal-01666845] Adaptive inexact semismooth Newton methods for the contact problem between two membranes

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. (Jad Dabaghi), Jad Dabaghi

[hal-01349456] Approche d’un territoire de montagne : occupations humaines et contexte pédo-sédimentaire des versants du col du Petit-Saint-Bernard, de la Préhistoire à l’Antiquité

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 (Pierre-Jérôme Rey), Pierre-Jérôme Rey

[hal-01823261] Predictive spatio-temporal model for spatially sparse global solar radiation data

This paper introduces a new approach for the forecasting of solar radiation series at a located station for very short time scale. We built a multivariate model in using few stations (3 stations) separated with irregular distances from 26 km to 56 km. The proposed model is a spatio temporal vector autoregressive VAR model specifically designed for the analysis of spatially sparse spatio-temporal data. This model differs from classic linear models in using spatial and temporal parameters where the available pre-dictors are the lagged values at each station. A spatial structure of stations is defined by the sequential introduction of predictors in the model. Moreover, an iterative strategy in the process of our model will select the necessary stations removing the uninteresting predictors and also selecting the optimal p-order. We studied the performance of this model. The metric error, the relative root mean squared error (rRMSE), is presented at different short time scales. Moreover, we compared the results of our model to simple and well known persistence model and those found in literature. (Maïna André), Maïna André

[hal-02445223] Inverse problem for a coupling model of reaction-diffusion and ordinary differential equations systems. Application to an epidemiological model

This paper investigates an identifiability method for a class of systems of reaction diffusion equations in the L^2 framework. This class is composed of a master system of ordinary differential equations coupled with a slave system of diffusion equations. It can model two populations, the second one being diffusive contrary to the first one. The identifiability method is based on an elimination procedure providing relations called input-output polynomials and linking the unknown parameters , the inputs and the outputs of the model. These polynomials can also be used to estimate the parameters as shown in this article. To our best knowledge, such an identifiability method and a parameter estimation procedure have not yet been explored for such a system in the L^2 framework. This work is applied on an epidemiological model describing the propagation of the chikungunya in a local population. (Nathalie Verdière), Nathalie Verdière

[hal-01919067] A posteriori error estimates for a compositional two-phase flow with nonlinear complementarity constraints

In this work, we develop an a-posteriori-steered algorithm for a compositional two-phase flow with exchange of components between the phases in porous media. As a model problem, we choose the two-phase liquid-gas flow with appearance and disappearance of the gas phase formulated as a system of nonlinear evolutive partial differential equations with nonlinear complementarity constraints. The discretization of our model is based on the backward Euler scheme in time and the finite volume scheme in space. The resulting nonlinear system is solved via an inexact semismooth Newton method. The key ingredient for the a posteriori analysis are the discretization, linearization, and algebraic flux reconstructions allowing to devise estimators for each error component. These enable to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver whenever the corresponding error components do not affect significantly the overall error. Numerical experiments are performed using the Newton-min algorithm as well as the Newton-Fischer-Burmeister algorithm in combination with the GMRES iterative linear solver to show the efficiency of the proposed adaptive method. (Ibtihel Ben Gharbia), Ibtihel Ben Gharbia

[hal-01817823] Binacox : automatic cut-point detection in high-dimensional Cox model with applications in genetics

We introduce the binacox, a prognostic method to deal with the problem of detect- ing multiple cut-points per features in a multivariate setting where a large number of continuous features are available. The method is based on the Cox model and com- bines one-hot encoding with the binarsity penalty, which uses total-variation regular- ization together with an extra linear constraint, and enables feature selection. Original nonasymptotic oracle inequalities for prediction (in terms of Kullback-Leibler diver- gence) and estimation with a fast rate of convergence are established. The statistical performance of the method is examined in an extensive Monte Carlo simulation study, and then illustrated on three publicly available genetic cancer datasets. On these high- dimensional datasets, our proposed method signi cantly outperforms state-of-the-art survival models regarding risk prediction in terms of the C-index, with a computing time orders of magnitude faster. In addition, it provides powerful interpretability from a clinical perspective by automatically pinpointing signi cant cut-points in relevant variables. (Simon Bussy), Simon Bussy

[hal-01993267] Identifiability and identification of a pollution source in a river by using a semi-discretized model

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. (Nathalie Verdière), Nathalie Verdière

[hal-02176154] Global representation and multi-scale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

For each pair ε = (ε 1 , ε 2) of positive parameters, we define a perforated domain Ω ε by making a small hole of size ε 1 ε 2 in an open regular subset Ω of R n (n ≥ 3). The hole is situated at distance ε 1 from the outer boundary ∂Ω of the domain. Then, when ε → (0, 0) both the size of the hole and its distance from ∂Ω tend to zero, but the size shrinks faster than the distance. In such perforated domain Ω ε we consider a Dirichlet problem for the Laplace equation and we denote by u ε its solution. Our aim is to represent the map that takes ε to u ε in term of real analytic functions of ε defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of u ε to suitable subsets of Ω ε , we prove a global representation formula that holds on the whole of Ω ε. Such a formula allows to rigorously justify multi-scale expansions, which we subsequently construct. (Virginie Bonnaillie-Noël), Virginie Bonnaillie-Noël

[hal-01958872] Solving coupled problems of lumped parameter models in a platform for severe accidents in nuclear reactors

This paper focuses on solving coupled problems of lumped parameter models. Such problems are of interest for the simulation of severe accidents in nuclear reactors: these coarse-grained models allow for fast calculations for statistical analysis used for risk assessment and solutions of large problems when considering the whole severe accident scenario. However, this modeling approach has several numerical flaws. Besides, in this industrial context, computational efficiency is of great importance leading to various numerical constraints. The objective of this research is to analyze the applicability of explicit coupling strategies to solve such coupled problems and to design implicit coupling schemes allowing stable and accurate computations. The proposed schemes are theoretically analyzed and tested within CEA's procor platform on a problem of heat conduction solved with coupled lumped parameter models and coupled 1D models. Numerical results are discussed and allow us to emphasize the benefits of using the designed coupling schemes instead of the usual explicit coupling schemes. (Louis Viot), Louis Viot

[hal-02025747] Optimal input design for parameter estimation in a bounded-error context for nonlinear dynamical systems

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. (Carine Jauberthie), Carine Jauberthie

[cea-02023046] Aerosols released during the laser cutting of a Fukushima Daiichi debris simulant

One of the important challenges for the decommissioning of the damaged reactors of the Fukushima Daiichi Nuclear Power Plant is the safe retrieval of the fuel debris or corium. It is especially primordial to investigate the cutting conditions for air configuration and for underwater configuration at different water levels. Concerning the cutting techniques, the laser technique is well adapted to the cutting of expected material such as corium that has an irregular shape and heterogeneous composition. A French consortium (ONET Technologies, CEA and IRSN) is being subsidized by the Japanese government to implement R&D related to the laser cutting of Fukushima Daiichi fuel debris and related to dust collection technology. Debris simulant have been manufactured in the PLINIUS platform to represent Molten Core Concrete Interaction as estimated from Fukushima Daiichi calculations. In this simulant, uranium is replaced by hafnium and the major fission products have been replaced by their natural isotopes. During laser cutting experiments in the DELIA facility, aerosols have been collected thanks to filters and impactors. The collected aerosols have been analyzed. Both chemical analysis (dissolution + ICP MS and ICP AES) and microscopic analyses (SEM EDS) will be presented and discussed. These data provide insights on the expected dust releases during cutting and can be converted to provide radioactivity estimates. They have also been successfully compared to thermodynamic calculations with the NUCLEA database. (Christophe Journeau), Christophe Journeau

[hal-01700663] A Lagrange multiplier method for a discrete fracture model for flow in porous media

In this work we present a novel discrete fracture model for single-phase Darcy flow in porous media with fractures of co-dimension one, which introduces an additional unknown at the fracture interface. Inspired by the fictitious domain method this Lagrange multiplier couples fracture and matrix domain and represents a local exchange of the fluid. The multipliers naturally impose the equality of the pressures at the fracture interface. The model is thus appropriate for domains with fractures of permeability higher than that in the surrounding bulk domain. In particular the novel approach allows for independent, regular meshing of fracture and matrix domain and therefore avoids the generation of small elements. We show existence and uniqueness of the weak solution of the continuous primal formulation. Moreover we discuss the discrete inf-sup condition of two different finite element formulations. Several numerical examples verify the accuracy and convergence of proposed method. (Markus Köppel), Markus Köppel

[hal-01761591] A stabilized Lagrange multiplier finite-element method for flow in porous media with fractures

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. (Markus Köppel), Markus Köppel

[hal-01800481] Diffusion Problems in Multi-layer Media with Nonlinear Interface Contact Resistance

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. (F Ben Belgacem), F Ben Belgacem

[hal-01914536] Optimal initial state for fast parameter estimation in nonlinear dynamical systems

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. (Qiaochu Li), Qiaochu Li

[hal-01906872] Convergence of a positive nonlinear control volume finite element scheme for an anisotropic seawater intrusion model with sharp interfaces

We consider a degenerate parabolic system modelling the flow of fresh and saltwater in an anisotropic porous medium in the context of seawater intrusion. We propose and analyze a nonlinear Control Volume Finite Element scheme. This scheme ensures the nonnegativity of the discrete solution without any restriction on the mesh and on the anisotropy tensor. Moreover It also provides a control on the entropy. Based on these nonlinear stability results, we show that the scheme converges towards a weak solution to the problem. Numerical results are provided to illustrate the behavior of the model and of the scheme. (Ahmed Ait Hammou Oulhaj), Ahmed Ait Hammou Oulhaj

[hal-01635178] Estimation of the Expected Number of Earthquake Occurrences Based on Semi-Markov Models

The present paper aims at the introduction of the semi-Markov model in continuous time as a candidate model for the description of seismicity patterns in time domain in the Northern Aegean Sea (Greece). Estimators of the semi-Markov kernels, Markov renewal functions and transition functions are calculated through a nonparametric method. Moreover , the hitting times for spatial occurrence of the strongest earthquakes as well as the confidence intervals of certain important indicators are estimated. Firstly, the classification of model states is based on earthquakes magnitude. The instantaneous earthquake occurrence rate between the states of the model as well as the total earthquake occurrence rate are calculated. In order to increase the consistency between the model and the process of earthquake generation, seismotectonic features have been incorporated as an important component in the model. Therefore, a new classification of states is proposed which combines both magnitude and fault orientation states. This model which takes into account seismotectonic features contributes significantly to the seismic hazard assessment in the region under study. The model is applied to earthquake catalogues for the Northern Aegean Sea, an area that accommodates high seismicity, being a key structure from the seismotec-tonic point of view. (Irene Votsi), Irene Votsi

[hal-01557190] An intrinsic Proper Generalized Decomposition for parametric symmetric elliptic problems

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. (Mejdi Azaiez), Mejdi Azaiez

[hal-01523020] Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics

The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles. (Stéphanie Chaillat), Stéphanie Chaillat

[hal-01412768] Modeling spatial spread of an epidemiologic model in a spatially continuous domain

In this paper, we study a Chikungunya epidemic transmission model which describes an epidemic disease transmitted by Aedes mosquitoes. This model includes the spatial mobility of humans which is probably a factor that has influenced the re-emergence of several diseases. Assuming that the spatial mobility of humans is random described as Brownian random motion, an original model including a reaction-diffusion system is proposed. Since the displacement of mosquitoes is limited to a few meters, compared with humans, one can ignore mosquitoes mobility. Therefore, the complete model is composed of a reaction-diffusion system coupled with ordinary differential equations (ODEs). In this paper, we prove the existence and uniqueness, the positivity and boundedness of the global solution for the model and give some numerical simulations. (Shousheng Zhu), Shousheng Zhu

[hal-01344090] The Lax–Milgram theorem. A detailed proof to be formalized in Coq

To obtain the highest confidence on the correction of numerical simulation programs implementing the finite element method, one has to formalize the mathematical notions and results that allow to establish the soundness of the method. The Lax-Milgram theorem may be seen as one of those theoretical cornerstones: under some completeness and coercivity assumptions, it states existence and uniqueness of the solution to the weak formulation of some boundary value problems. The purpose of this document is to provide the formal proof community with a very detailed pen-and-paper proof of the Lax-Milgram theorem. (François Clément), François Clément

[hal-01267024] Generalized Impedance Boundary Conditions and Shape Derivatives for 3D Helmholtz Problems

This paper is concerned with the shape sensitivity analysis of the solution to the Helmholtz transmission problem for three dimensional sound-soft or sound-hard obstacles coated by a thin layer. This problem can be asymptotically approached by exterior problems with an improved condition on the exterior boundary of the coated obstacle, called Generalised Impedance Boundary Condition (GIBC). Using a series expansion of the Laplacian operator in the neighborhood of the exterior boundary, we retrieve the first order GIBCs characterizing the presence of an interior thin layer with either a constant or a variable thickness. The first shape derivative of the solution to the original Helmholtz transmission problem solves a new thin layer transmission problem with non vanishing jumps across the exterior and the interior boundary of the thin layer. In the special case of thin layers with a constant thickness, we show that we can interchange the first order differentiation with respect to the shape of the exterior boundary and the asymptotic approximation of the solution. Numerical experiments are presented to highlights the various theoretical results. (Djalil Kateb), Djalil Kateb

[hal-01263494] Résolution d’un Problème de Cauchy en EEG

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. (Abdellatif El-Badia), Abdellatif El-Badia

[hal-01203280] Bayesian optimal adaptive estimation using a sieve prior

We derive rates of contraction of posterior distributions on non-parametric models resulting from sieve priors. The aim of the study was to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter is, for example, a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate. (Julyan Arbel), Julyan Arbel

[hal-01187242] Approximate local Dirichlet-to-Neumann map for three-dimensional time-harmonic elastic waves

It has been proven that the knowledge of an accurate approximation of the Dirichlet-to-Neumann (DtN) map is useful for a large range of applications in wave scattering problems. We are concerned in this paper with the construction of an approximate local DtN operator for time-harmonic elastic waves. The main contributions are the following. First, we derive exact operators using Fourier analysis in the case of an elastic half-space. These results are then extended to a general three-dimensional smooth closed surface by using a local tangent plane approximation. Next, a regularization step improves the accuracy of the approximate DtN operators and a localization process is proposed. Finally, a first application is presented in the context of the On-Surface Radiation Conditions method. The efficiency of the approach is investigated for various obstacle geometries at high frequencies. (Stéphanie Chaillat), Stéphanie Chaillat

[hal-01152319] A domain derivative-based method for solving elastodynamic inverse obstacle scattering problems

The present work is concerned with the shape reconstruction problem of isotropic elastic inclusions from far-field data obtained by the scattering of a finite number of time-harmonic incident plane waves. This paper aims at completing the theoretical framework which is necessary for the application of geometric optimization tools to the inverse transmission problem in elastodynamics. The forward problem is reduced to systems of boundary integral equations following the direct and indirect methods initially developed for solving acoustic transmission problems. We establish the Fréchet differentiability of the boundary to far-field operator and give a characterization of the first Fréchet derivative and its adjoint operator. Using these results we propose an inverse scattering algorithm based on the iteratively regularized Gauß Newton method and show numerical experiments in the special case of star-shaped obstacles. (Frédérique Le Louër), Frédérique Le Louër

[hal-01063083] Karhunen-Loève’s Series Truncation for Bivariate Functions

Karhunen-Loève's decompositions (KLD) or the proper orthogonal decompositions (POD) of bivariate functions are revisited in this work. We investigate the truncation error first for regular functions and try to improve and sharpen bounds found in the literature. However it happens that (KL)-series expansions are in fact more sensitive to the liability of fields to approximate to be well represented by a small sum of products of separated variables functions. We consider this very issue for some interesting fields solutions of partial differential equations such as the transient heat problem and Poisson's equation. The main tool to state approximation bounds is linear algebra. We show how the singular value decomposition underlying the (KL)-expansion is connected to the spectrum of some Gram matrices. Deriving estimates on the truncation error is thus strongly tied to the spectral properties of these Gram matrices which are structured matrices with low displacement ranks. (Mejdi Azaïez), Mejdi Azaïez

[hal-01026447] Ill-Conditioning versus Ill-Posedness for the Boundary Controllability of the Heat Equation

Ill-posedness and/or Ill-conditioning are features users have to deal with appropriately in the controllability of diffusion problems for secure and reliable outputs. We investigate those issues in the case of a boundary Dirichlet control, in an attempt to underline the origin of the troubles arising in the numerical computations and to shed some light on the difficulties to obtain good quality simulations. The exact controllability is severely ill-posed while, in spite of its well-posedness, the null-controllability turns out to be very badly ill-conditioned. Theoretical and numerical results are stated on the heat equation in one dimension to illustrate the specific instabilities of each problem. The main tools used here are first a characterization of the subspace where the HUM control lies and the study of the spectrum of some structured matrices, of Pick and Löwner type, obtained from the Fourier calculations on the state and adjoint equations. (Faker Ben Belgacem), Faker Ben Belgacem

[hal-01026457] Cauchy Matrices in the Observation of Diffusion Equations

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. (Faker Ben Belgacem), Faker Ben Belgacem

[hal-01005515] Finite element methods for the temperature in composite media with contact resistance

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. (Faker Ben Belgacem), Faker Ben Belgacem

[hal-00780735] Shape optimization methods for the Inverse Obstacle Problem with generalized impedance boundary conditions

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. (Fabien Caubet), Fabien Caubet

[hal-00780730] Stability of critical shapes for the drag minimization problem in Stokes flow

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. (Fabien Caubet), Fabien Caubet

[hal-00780379] Fast spectral methods for the shape identification problem of a perfectly conducting obstacle

We are concerned with fast methods for the numerical implementation of the direct and inverse scattering problems for a perfectly conducting obstacle. The scattering problem is usually reduced to a single uniquely solvable modified combined-field integral equation (M-CFIE). For the numerical solution of the M-CFIE we propose a new high-order spectral algorithm by transporting this equation on the unit sphere via the Piola transform. The inverse problem is formulated as a nonlinear least squares problem for which the iteratively regularized Gauss-Newton method is applied to recover an approximate solution. Numerical experiments are presented in the special case of star-shaped obstacles. (Frédérique Le Louër), Frédérique Le Louër

[inria-00625293] Exact MLE and asymptotic properties for nonparametric semi-Markov models

This article concerns maximum-likelihood estimation for discrete time homogeneous nonparametric semi-Markov models with finite state space. In particular, we present the exact maximum-likelihood estimator of the semi-Markov kernel which governs the evolution of the semi-Markov chain (SMC). We study its asymptotic properties in the following cases: (i) for one observed trajectory, when the length of the observation tends to infinity, and (ii) for parallel observations of independent copies of an SMC censored at a fixed time, when the number of copies tends to infinity. In both cases, we obtain strong consistency, asymptotic normality, and asymptotic efficiency for every finite dimensional vector of this estimator. Finally, we obtain explicit forms for the covariance matrices of the asymptotic distributions. (Samis Trevezas), Samis Trevezas

[hal-00731856] On the necessity of Nitsche term

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. (Gaël Dupire), Gaël Dupire

[hal-00731528] On the necessity of Nitsche term. Part II : An alternative approach

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. (Jean-Paul Boufflet), Jean-Paul Boufflet

[inria-00628032] MagnetoHemoDynamics in Aorta and Electrocardiograms

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). (Vincent Martin), Vincent Martin

[hal-00699171] Modeling Pollutant Emissions of Diesel Engine based on Kriging Models : a Comparison between Geostatistic and Gaussian Process Approach

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. (Sébastien Castric), Sébastien Castric

[hal-00696683] Identifiability and identification of a pollution source in a river by using a semi-discretized model

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. (Nathalie Verdiere), Nathalie Verdiere

[hal-00696660] An efficient algorithm for testing nonlinear dynamical model identifiability

This paper is concerned by the analysis of nonlinear controlled or uncontrolled dynamical model identifiability. The proposed approach is based on the construction of an input-output ideal. The aim is to develop an algorithm which gives identifiability results from this approach. Differential algebra theory allows realization of such a project. In order to state the algorithm, new results of differential algebra must be proved. Then the implementation is done in a symbolic computing language (Lilianne Denis-Vidal), Lilianne Denis-Vidal

[hal-00696659] Two approaches for testing identifiability and corresponding algorithms

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. (Lilianne Denis-Vidal), Lilianne Denis-Vidal

[hal-00696672] A new method for estimating derivatives based on a distribution approach

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. (Nathalie Verdière), Nathalie Verdière

[hal-00684625] Persistency of wellposedness of Ventcel’s boundary value problem under shape deformations

Ventcel boundary conditions are second order di erential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to well-posed variational problems under a sign condition of the coe cient. This is achieved when physical situations are considered. Nevertheless, situations where this condition is violated appeared in several recent works where absorbing boundary conditions or equivalent boundary conditions on rough surface are sought for numerical purposes. The well-posedness of such problems was recently investigated : up to a countable set of parameters, existence and uniqueness of the solution for the Ventcel boundary value problem holds without the sign condition. However, the values to be avoided depend on the domain where the boundary value problem is set. In this work, we address the question of the persistency of the solvability of the boundary value problem under domain deformation. (Marc Dambrine), Marc Dambrine

[hal-00678036] A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid

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. (Fabien Caubet), Fabien Caubet

[hal-00592282] Shape derivatives of boundary integral operators in electromagnetic scattering. Part II : Application to scattering by a homogeneous dielectric obstacle

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. The latter are typically bounded on the space of tangential vector fields of mixed regularity $TH\sp{-\frac{1}{2}}(\Div_{\Gamma},\Gamma)$. Using Helmholtz decomposition, we can base their analysis on the study of pseudo-differential integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity. We also give a characterization of the first shape derivative of the solution of the dielectric scattering problem as a solution of a new electromagnetic scattering problem. (Martin Costabel), Martin Costabel

[hal-00592280] Shape derivatives of boundary integral operators in electromagnetic scattering. Part I : Shape differentiability of pseudo-homogeneous boundary integral operators.

In this paper we study the shape differentiability properties of a class of boundary integral operators and of potentials with weakly singular pseudo-homogeneous kernels acting between classical Sobolev spaces, with respect to smooth deformations of the boundary. We prove that the boundary integral operators are infinitely differentiable without loss of regularity. The potential operators are infinitely shape differentiable away from the boundary, whereas their derivatives lose regularity near the boundary. We study the shape differentiability of surface differential operators. The shape differentiability properties of the usual strongly singular or hypersingular boundary integral operators of interest in acoustic, elastodynamic or electromagnetic potential theory can then be established by expressing them in terms of integral operators with weakly singular kernels and of surface differential operators. (Martin Costabel), Martin Costabel

[hal-00664429] Uniqueness for an ill-posed reaction-dispersion model. Application to organic pollution in stream-waters

We are concerned with the inverse problem of detecting sources in a coupled diffusion-reaction system. This problem arises from the Biochemical Oxygen Demand-Dissolved Oxygen model(\footnote{The acronym BOD-DO model is currently used.}) governing the interaction between organic pollutants and the oxygen available in stream waters. The sources we consider are point-wise and simulate stationary or moving pollution sources. The ultimate objective is to obtain their discharge location and recover their output rate from accessible measurements of DO when BOD measurements are difficult and time consuming to obtain. It is, as a matter of fact, the most realistic configuration. The subject to address here is the identifiability of these sources, in other words to determine if the observations uniquely determine the sources. The key tool is the study of coupled parabolic systems derived after restricting the global model to regions at the exterior of the observations. The absence of any prescribed condition on the BOD density is compensated by data recorded on the DO which provide over-determined Cauchy boundary conditions. Now, the first step toward the identifiability of the sources is precisely to recover the BOD at the observation points (of DO). This may be achieved by handling and solving the coupled systems. Unsurprisingly, they turn out to be ill-posed. That issue is investigated first. Then, we state a uniqueness result owing to a suitable saddle-point variational framework and to Pazy's uniqueness Theorem. This uniqueness complemented by former identifiability results proved in [2011, Inverse problems] for scalar reaction-diffusion equations yields the desired identifiability for the global model. (Faker Ben Belgacem), Faker Ben Belgacem

[inria-00576524] Maximum likelihood estimation for general hidden semi-Markov processes with backward recurrence time dependence

This article concerns the study of the asymptotic properties of the maximum likelihood estimator (MLE) for the general hidden semi-Markov model (HSMM) with backward recurrence time dependence. By transforming the general HSMM into a general hidden Markov model, we prove that under some regularity conditions, the MLE is strongly consistent and asymptotically normal. We also provide useful expressions for the asymptotic covariance matrices, involving the MLE of the conditional sojourn times and the embedded Markov chain of the hidden semi-Markov chain. (Samis Trevezas), Samis Trevezas

[hal-00439221] On the Kleinman-Martin integral equation method for electromagnetic scattering by a dielectric body

The interface problem describing the scattering of time-harmonic electromagnetic waves by a dielectric body is often formulated as a pair of coupled boundary integral equations for the electric and magnetic current densities on the interface Γ. In this paper, following an idea developed by Kleinman and Martin for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over Γ. for a single unknown density. One knows that such boundary integral formulations of the Maxwell equations are not uniquely solvable when the exterior wave number is an eigenvalue of an associated interior Maxwell boundary value problem. We obtain four different families of integral equations for which we can show that by choosing some parameters in an appropriate way, they become uniquely solvable for all real frequencies. We analyze the well-posedness of the integral equations in the space of finite energy on smooth and non-smooth boundaries. (Martin Costabel), Martin Costabel

[hal-00565512] Two stage approaches for modeling pollutant emission of diesel engine based on Kriging model

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 (El Hassane Brahmi), El Hassane Brahmi

[inria-00561601] Modeling fractures as interfaces with nonmatching grids

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. (Najla Frih), Najla Frih

[hal-00534134] Uniqueness results for diagonal hyperbolic systems with large and monotone data

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. (Ahmad El Hajj), Ahmad El Hajj

[inria-00543014] Analysis of a stabilized finite element method for fluid flows through a porous interface

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. (Alfonso Caiazzo), Alfonso Caiazzo

[inria-00468804] Variance Estimation in the Central Limit Theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the MLE estimator for the stationary distribution. (Samis Trevezas), Samis Trevezas

[hal-00461144] On the unilateral contact between membranes Part 2 : A posteriori analysis and numerical experiments

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. (Faker Ben Belgacem), Faker Ben Belgacem

[hal-00453948] Shape derivatives of boundary integral operators in electromagnetic scattering

We develop the shape derivative analysis of solutions to the problem of scattering of time-harmonic electromagnetic waves by a bounded penetrable obstacle. Since boundary integral equations are a classical tool to solve electromagnetic scattering problems, we study the shape differentiability properties of the standard electromagnetic boundary integral operators. Using Helmholtz decomposition, we can base their analysis on the study of scalar integral operators in standard Sobolev spaces, but we then have to study the Gâteaux differentiability of surface differential operators. We prove that the electromagnetic boundary integral operators are infinitely differentiable without loss of regularity and that the solutions of the scattering problem are infinitely shape differentiable away from the boundary of the obstacle, whereas their derivatives lose regularity on the boundary. We also give a characterization of the first shape derivative as a solution of a new electromagnetic scattering problem. (Martin Costabel), Martin Costabel

[tel-00421863] Optimisation de forme d’antennes lentilles intégrées aux ondes millimétriques

Les antennes lentilles sont des dispositifs ayant pour support les ondes électromagnétiques et sont constituées d'une source primaire et d'un système focalisant diélectrique. La montée en importance récente d'applications en ondes millimétriques (exemple : radars d'assistance et d'aide à la conduite), nécessite la construction d'antennes lentilles de quelques centimètres qui répondent à des cahiers des charges spécifiques à chaque cas. L'une des problématiques à résoudre consiste à déterminer la forme optimale de la lentille étant données : (i) les caractéristiques de la source primaire, (ii) les caractéristiques en rayonnement fixées. Ce projet de thèse vise à développer de nouveaux outils pour l'optimisation de forme en utilisant une formulation intégrale du problème. Cette thèse s'articule en deux parties. Dans la première nous avons construit plusieurs formulations intégrales pour le problème de diffraction diélectrique en utilisant une approche par équation intégrale surfacique. Dans la seconde nous avons étudié les dérivées de forme des opérateurs intégraux standard en électromagnétisme dans le but de les incorporer dans un algorithme d'optimisation de forme. (Frédérique Le Louër), Frédérique Le Louër

[hal-00387808] Spectral discretization of Darcy’s equations with pressure dependent porosity

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. (Mejdi Azaïez), Mejdi Azaïez

[tel-00372423] Problèmes inverses de sources et lien avec l’Electro-Encéphalo-Graphie

Ce travail porte sur un problème inverse de sources dipolaires et son application à l'identification des sources de l'activité cérébrale telle qu'elle peut être mesurée par l'Electro-Encéphalo-Graphie (EEG). Des résultats d'identifiabilité et de stabilité ont été établis. Par ailleurs, une étude du problème de Cauchy en 3D, motivée par l'application de la méthode d'identification dite "algébrique", a été faite à l'aide de la méthode itérative introduite par Kozlov, Maz'ya et Fomin et au moyen des équations intégrales de frontières. En outre, une autre méthode basée sur une fonctionnelle coût de type Kohn et Vogelius a été considérée pour l'identification des sources et dont les résultats numériques sont avérés plus performants que ceux donnés par la méthode des moindres carrés. (Maha Farah), Maha Farah

[hal-00222765] Inégalités de Calderon-Zygmund, Potentiels et Transformées de Riesz dans des Espaces avec Poids

[...] (Chérif Amrouche), Chérif Amrouche

[hal-00140368] LEPISME

We present a first version of a software dedicated to an application of a classical nonlinear control theory problem to the study of compartmental models in biology. The software is being developed over a new free computer algebra library dedicated to differential and algebraic elimination. (François Boulier), François Boulier

[hal-00140211] On second order shape optimization methods for electrical impedance tomography

This paper is devoted to the analysis of a second order method for recovering the \emph{a priori} unknown shape of an inclusion $\omega$ inside a body $\Omega$ from boundary measurement. This inverse problem - known as electrical impedance tomography - has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state $u$ with respect to perturbations of the shape of the interface $\partial\omega$, then we choose a cost function in order to recover the geometry of $\partial \omega$ and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer. (Lekbir Afraites), Lekbir Afraites

[inria-00136971] Numerical simulation of blood flows through a porous interface

We propose a model for a medical device, called a stent, designed for the treatment of cerebral aneurysms. The stent consists of a grid, immersed in the blood flow and located at the inlet of the aneurysm. It aims at promoting a clot within the aneurysm. The blood flow is modelled by the incompressible Navier-Stokes equations and the stent by a dissipative surface term. We propose a stabilized finite element method for this model and we analyse its convergence in the case of the Stokes equations. We present numerical results for academical test cases, and on a realistic aneurysm obtained from medical imaging. (Miguel Angel Fernández), Miguel Angel Fernández

[hal-00112170] The mortar spectral element method in domains of operators Part II : The curl operator and the vector potential problem

The mortar spectral element method is a domain decomposition technique that allows for discretizing second- or fourth-order elliptic equations when set in standard Sobolev spaces.he aim of this paper is to extend this method to problems formulated in the space of square-integrable vector fields with square-integrable curl.We consider the problem of computing the vector potential associated with a divergence- free function in dimension 3 and propose a discretization of it. The numerical analysis of the discrete problem is performed and numerical experiments are presented, they turn out to be in good coherency with the theoretical results. (Mjedi Azaïez), Mjedi Azaïez


In this work, we consider singular perturbations of the boundary of a smooth domain. We describe the asymptotic behavior of the solution uε of a second order elliptic equation posed in the perturbed domain with respect to the size parameter ε of the deformation. We are also interested in the variations of the energy functional. We propose a numerical method for the approximation of uε based on a multiscale superposition of the unperturbed solution u0 and a profile defined in a model domain. We conclude with numerical results. (Marc Dambrine), Marc Dambrine

[hal-00084598] A remark on precomposition on $\sH^{1/2}(S^1)$ and $\eps$-identifiability of disks in tomography.

We consider the inverse conductivity problem with one measurement for the equation $div((\sigma_1+(\sigma_2-\sigma_1)\chi_D)\nabla{u})=0$ determining the unknown inclusion $D$ included in $\Omega$. We suppose that $\Omega$ is the unit disk of $\mathbb{R}^2$. With the tools of the conformal mappings, of elementary Fourier analysis and also the action of some quasi-conformal mapping on the Sobolev space $\sH^{1/2}(S^1)$, we show how to approximate the Dirichlet-to-Neumann map when the original inclusion $D$ is a $\varepsilon-$ approximation of a disk. This enables us to give some uniqueness and stability results. (Marc Dambrine), Marc Dambrine

[tel-00011838] Identifiabilité de systèmes d’équations aux dérivées partielles semi-discrétisées et applications à l’identifiabilité paramétrique de modèles en pharmacocinétique et en pollution.

Avant d'estimer les paramètres intervenant dans des systèmes dynamiques, linéaires ou non-linéaires, contrôlés ou non contrôlés, il est important d'effectuer une étude d'identifiabilité, c'est à dire si, à partir des données expérimentales, les paramètres étudiés sont uniques ou non. Plusieurs méthodes ont été développées ces dernières années, en particulier une qui est basée sur l'algèbre différentielle. Celle-ci a conduit à un algorithme utilisant le package Diffalg implémenté sous Maple et permettant de tester l'identifiabilité de systèmes d'équations différentielles. Les résultats obtenus à partir de cette étude permettent de mettre en place des méthodes numériques pour obtenir une première estimation des paramètres, ceci sans aucune connaissance à priori de leur valeur. Cette première estimation peut alors être utilisée comme point de départ d'algorithmes itératifs spécialisés dans l'étude des problèmes mal posés : la régularisation de Tikhonov. Dans cette thèse, deux modèles non linéaires en pharmacocinétique de type Michaelis-Menten ont tout d'abord été étudiés. Ensuite, nous nous sommes intéressés à un modèle de pollution décrit par une équation aux dérivées partielles parabolique. Le terme source à identifier était modélisé par le produit de la fonction débit avec la masse de Dirac, de support la position de la source polluante. Le but du travail était de fournir une première estimation de la source polluante. Après avoir obtenu l'identifiabilité du problème continu, nous avons étudié l'identifiabilité d'un problème approché en nous appuyant sur les méthodes d'algèbre différentielle. Celui-ci a été obtenu en approchant la masse de Dirac par une fonction gaussienne et en discrétisant ensuite le système en espace. Les résultats d'identifiabilité ont été obtenus quel que soit le nombre de points de discrétisation en espace. De cette étude théorique, nous en avons déduit des algorithmes numériques donnant une première estimation des paramètres à identifier. (Nathalie Verdière), Nathalie Verdière

[hal-00020177] Conformal mappings and shape derivatives for the transmission problem with a single measurement.

In the present work, we consider the inverse conductivity problem of recovering inclusion with one measurement. First, we use conformal mapping techniques for determining the location of the anomaly and estimating its size. We then get a good initial guess for quasi-Newton type method. The inverse problem is treated from the shape optimization point of view. We give a rigorous proof for the existence of the shape derivative of the state function and of shape functionals. We consider both Least Squares fitting and Kohn and Vogelius functionals. For the numerical implementation, we use a parametrization of shapes coupled with a boundary element method. Several numerical exemples indicate the superiority of the Kohn and Vogelius functional over Least Squares fitting. (Lekbir Afraites), Lekbir Afraites