# Publications sur H.A.L.

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### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Jean-Paul Boufflet), Jean-Paul Boufflet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Gaël Dupire), Gaël Dupire

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Virginie Bonnaillie-Noël), Virginie Bonnaillie-Noël

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (F Ben Belgacem), F Ben Belgacem

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaïez), Mejdi Azaïez

### [hal-00820289] An Ill-posed Parabolic Evolution System for Dispersive Deoxygenation-Reaeration in Waters

We consider an inverse problem that arises in the management of water resources and pertains to the analysis of the surface waters pollution by organic matter. Most of physical models used by engineers derive from various additions and corrections to enhance the earlier deoxygenation-reaeration model proposed by Streeter and Phelps in 1925, the unknowns being the biochemical oxygen demand (BOD) and the dissolved oxygen (DO) concentrations. The one we deal with includes Taylor's dispersion to account for the heterogeneity of the contamination in all space directions. The system we obtain is then composed of two reaction-dispersion equations. The particularity is that both Neumann and Dirichlet boundary conditions are available on the DO tracer while the BOD density is free of any condition. In fact, for real-life concerns, measurements on the dissolved oxygen are easy to obtain and to save. In the contrary, collecting data on the biochemical oxygen demand is a sensitive task and turns out to be a long-time process. The global model pursues the reconstruction of the BOD density, and especially of its flux along the boundary. Not only this problem is plainly worth studying for its own interest but it can be also a mandatory step in other applications such as the identification of the pollution sources location. The non-standard boundary conditions generate two difficulties in mathematical and computational grounds. They set up a severe coupling between both equations and they are cause of ill-posedness for the data reconstruction problem. Existence and stability fail. Identifiability is therefore the only positive result one can seek after ; it is the central purpose of the paper. We end by some computational experiences to assess the capability of the mixed finite element capability in the missing data recovery (on the biochemical oxygen demand).

ano.nymous@ccsd.cnrs.fr.invalid (Mejdi Azaïez), Mejdi Azaïez

### [tel-00165782] Modélisation et estimation des processus de dégradation avec application en fiabilité des structures

Nous décrivons le niveau de dégradation caractéristique d'une structure à l'aide d'un processus stochastique appelé processus de dégradation. La dynamique de ce processus est modélisée par un système différentiel à environnement markovien. Nous étudions la fiabilité du système en considérant la défaillance de la structure lorsque le processus de dégradation dépasse un seuil fixe. Nous obtenons la fiabilité théorique à l'aide de la théorie du renouvellement markovien. Puis, nous proposons une procédure d'estimation des paramètres des processus aléatoires du système différentiel. Les méthodes d'estimation et les résultats théoriques de la fiabilité, ainsi que les algorithmes de calcul associés, sont validés sur des données simulés. Notre méthode est appliquée à la modélisation d'un mécanisme réel de dégradation, la propagation des fissures, pour lequel nous disposons d'un jeu de données expérimental.

ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Julyan Arbel), Julyan Arbel

### [hal-00937113] An extremal eigenvalue problem for the Wentzell-Laplace operator

We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain $\Omega$, involving only geometrical informations. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

ano.nymous@ccsd.cnrs.fr.invalid (Marc Dambrine), Marc Dambrine

### [tel-02470901] Analysis of an elasto-visco-plastic model describing dislocation dynamics

In this thesis, we are interested in the theoretical and numerical analysis o the dynamics of dislocation densities, where dislocations are crystalline defects appearing at the microscopic scale in metallic alloys. Particularly, the study of the Groma-Czikor-Zaiser model (GCZ) and the study of the Groma-Balog model (GB) are considered. The first is actually a system of parabolic type equations, where as, the second is a system of non-linear Hamilton-Jacobi equations. Initially, we demonstrate an existence and uniqueness result of a regular solution using a comparison principle and a fixed point argument for the GCZ model. Next, we establish a time-based global existence result for the GB model, based on notions of discontinuous viscosity solutions and a new estimate of total solution variation, as well as finite velocity propagation of the governed equations. This result is extended also to the cas of general Hamilton-Jacobi equation systems. Finally, we propose a semi-explicit numerical scheme allowing the discretization of the GB model. Based on the theoretical study, we prove that the discrete solution converges toward the continuous solution, as well as an estimate of error between the continuous solution and the numerical solution has been established. Simulations showing the robustness of the numerical scheme are also presented.

ano.nymous@ccsd.cnrs.fr.invalid (Vivian Rizik), Vivian Rizik

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Maïna André), Maïna André

### [hal-01294471] SENSITIVITY ANALYSIS FOR THE EEG MODEL IN NEONATES WITH RESPECT TO VARIATIONS OF THE CONDUCTIVITY

A mathematical model for the forward problem in electroencephalographic (EEG) source localization in neonates is proposed. The model is able to take into account the presence and ossification process of fontanels which are characterized by a variable conductivity. A subtraction approach is used to deal with the singularity in the source term, and existence and uniqueness results are proved for the continuous problem. Discretization is performed with 3D Finite Elements of type P1 and error estimates are proved in the energy (H 1-)norm. Numerical simulations for a three-layer spherical model as well as for a realistic neonatal head model have been obtained and corroborate the theoretical results. A mathematical tool related to the concept of Gâteau derivatives is introduced which is able to measure the sensitivity of the electric potential with respect to small variations in the fontanel conductivity. Numerical simulations attest that the presence of fontanels in neonates does have an impact on EEG measurements. The present work is an essential preamble to the numerical analysis of the corresponding EEG source reconstruction.

ano.nymous@ccsd.cnrs.fr.invalid (M Darbas), M Darbas

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière

### [hal-01919067] A posteriori error estimates and adaptive stopping criteria 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.

ano.nymous@ccsd.cnrs.fr.invalid (Ibtihel Ben Gharbia), Ibtihel Ben Gharbia

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Qiaochu Li), Qiaochu Li

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Samis Trevezas), Samis Trevezas

### [cea-02360117] Estimating Stochastic Dynamical Systems Driven by a Continuous-Time Jump Markov Process

We discuss the use of a continuous-time jump Markov process as the driving process in stochastic differential systems. Results are given on the estimation of the infinitesimal generator of the jump Markov process, when considering sample paths on random time intervals. These results are then applied within the framework of stochastic dynamical systems modeling and estimation. Numerical examples are given to illustrate both consistency and asymptotic normality of the estimator of the infinitesimal generator of the driving process. We apply these results to fatigue crack growth modeling as an example of a complex dynamical system, with applications to reliability analysis.

ano.nymous@ccsd.cnrs.fr.invalid (Julien Chiquet), Julien Chiquet

### [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. 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 complement by any iterative linear algebraic solver. We then derive an a posteriori estimate on the error between the exact solution 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 our estimates is also established, meaning that we prove them equivalent with the error up to a generic constant, except for a typically small contact term. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.

### [hal-01939854] A New Algorithm of Proper Generalized Decomposition for Parametric Symmetric Elliptic Problems

We introduce a new algorithm of proper generalized decomposition (PGD) for parametric symmetric elliptic partial differential equations. For any given dimension, we prove the existence of an optimal subspace of at most that dimension which realizes the best approximation---in the mean parametric norm associated to the elliptic operator---of the error between the exact solution and the Galerkin solution calculated on the subspace. This is analogous to the best approximation property of the proper orthogonal decomposition (POD) subspaces, except that in our case the norm is parameter-dependent. We apply a deflation technique to build a series of approximating solutions on finite-dimensional optimal subspaces, directly in the online step, and we prove that the partial sums converge to the continuous solution in the mean parametric elliptic norm. We show that the standard PGD for the considered parametric problem is strongly related to the deflation algorithm introduced in this paper. This opens the possibility of computing the PGD expansion by directly solving the optimization problems that yield the optimal subspaces.

ano.nymous@ccsd.cnrs.fr.invalid (M. Azaïez), M. Azaïez

### [insu-00159752] EBSD investigation of SiC for HTR fuel particles

Electron back-scattering diffraction (EBSD) can be successfully performed on SiC coatings for HTR fuel particles. EBSD grain maps obtained from thick and thin unirradiated samples are presented, along with pole figures showing textures and a chart showing the distribution of grain aspect ratios. This information is of great interest, and contributes to improving the process parameters and ensuring the reproducibility of coatings

ano.nymous@ccsd.cnrs.fr.invalid (D. Helary), D. Helary

### [hal-01006087] Multidimensional Independent Component Analysis with Higher-order cumulant matrices for vector sources with possibly differing dimensions

The paper addresses the separation of multidimensional sources, with possibly different dimensions, by means of higher-order cumulant matrices. First, it is rigorously proved, in a general setting, that contracted cumulant matrices of any order are all block-diagonalizable in the same basis. Second, a family of joint block-diagonalization algorithms is proposed that separate multidimensional sources by combining contracted cumulant matrices of arbitrary orders. Third, a specific solution is given to determine the source dimensions when they are unknown but all different. The performances of the proposed algorithms are compared between them and with algorithms of the literature based on orders 3 and 6.

ano.nymous@ccsd.cnrs.fr.invalid (Hanany Ould-Baba), Hanany Ould-Baba

### [hal-01006074] Concise formulae for the cumulant matrices of a random vector

Concise formulae are given for the cumulant matrices of a random vector up to order 6. In addition to usual matrix operations, they involve only the Kronecker product, the vec operator, and the commutation matrix. Orders 5 and 6 are provided here for the first time; the same method as provided in the paper can be applied to compute higher orders. An immediate consequence of these formulae is to return 1) upper bounds on the rank of the cumulant matrices and 2) the expression of the sixth-order moment matrix of a Gaussian vector. Due to their conciseness, the proposed formulae also have a computational advantage as compared to the repeated use of Leonov and Shiryaev formula.

ano.nymous@ccsd.cnrs.fr.invalid (Hanany Ould-Baba), Hanany Ould-Baba

### [hal-01635222] Reliability and probability of first occurred failure for discrete-time semi-Markov systems

In this chapter, we present the empirical estimation of some reliability measures, such as the rate of occurrence of failures and the steady-state availability, for a discrete-time semi-Markov system. The probability of first occurred failure is introduced and estimated. A numerical application is given to illustrate the strong consistency of these estimators.

### [hal-02153384] Hypotheses testing and posterior concentration rates for semi-Markov processes

In this paper, we adopt a nonparametric Bayesian approach and investigate the asymptotic behavior of the posterior distribution in continuous time and general state space semi-Markov processes. In particular, we obtain posterior concentration rates for semi-Markov kernels. For the purposes of this study, we construct robust statistical tests between Hellinger balls around semi-Markov kernels and present some specifications to particular cases, including discrete-time semi-Markov processes and finite state space Markov processes. The objective of this paper is to provide sufficient conditions on priors and semi-Markov kernels that enable us to establish posterior concentration rates.

ano.nymous@ccsd.cnrs.fr.invalid (V Barbu), V Barbu

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Irene Votsi), Irene Votsi

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Carine Jauberthie), Carine Jauberthie

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Nathalie Verdière), Nathalie Verdière

### [hal-01391578] A Coq formal proof of the Lax–Milgram theorem

The Finite Element Method is a widely-used method to solve numerical problems coming for instance from physics or biology. 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 sound-ness 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. This article presents the full formal proof of the Lax–Milgram theorem in Coq. It requires many results from linear algebra, geometry, functional analysis , and Hilbert spaces.

ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo

### [hal-01581807] Preuve formelle du théorème de Lax–Milgram

Résumé du papier "A Coq formal proof of the Lax-Milgram Theorem", CPP 2017.

ano.nymous@ccsd.cnrs.fr.invalid (Sylvie Boldo), Sylvie Boldo

### [hal-01070701] Implementation of an adaptive energy-efficient MAC protocol in OMNeT++/MiXiM

In recent years, many MAC protocols for wireless sensor networks have been proposed and most of them are evaluated using Matlab simulator and/or network simulators (OMNeT++, NS2, etc). However, most of them have a static behavior and few network simulations are available for adaptive protocols. Specially, in OMNeT++/MiXiM, there are few energy efficient MAC protocols for WSNs (B-MAC & L-MAC) and no adaptive ones. To this end, the TAD-MAC (Traffic Aware Dynamic MAC) protocol has been simulated in OMNeT++ with the MiXiM framework and implementation details are given in this paper. The simulation results have been used to evaluate the performance of TAD-MAC through comparisons with B-MAC and L-MAC protocols.

ano.nymous@ccsd.cnrs.fr.invalid (Van-Thiep Nguyen), Van-Thiep Nguyen

### [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.

### [hal-02182974] Characterization of palladium species after γ-irradiation of a TBP–alkane–Pd(NO 3 ) 2 system

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ano.nymous@ccsd.cnrs.fr.invalid (Bénédicte Simon), Bénédicte Simon

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Christophe Journeau), Christophe Journeau

### [hal-00018493] A Stochastic EM algorithm for a semiparametric mixture model

Recently several authors considered finite mixture models with semi-/non-parametric component distributions. Identifiability of such model parameters is generally not obvious, and when it occurs, inference methods are rather specific to the mixture model under consideration. In this paper we propose a generalization of the EM algorithm to semiparametric mixture models. Our approach is methodological and can be applied to a wide class of semiparametric mixture models. The behavior of the EM type estimators we propose is studied numerically through several Monte Carlo experiments but also by comparison with alternative methods existing in the literature. In addition to these numerical experiments we provide applications to real data showing that our estimation methods behaves well, that it is fast and easy to be implemented.

ano.nymous@ccsd.cnrs.fr.invalid (Laurent Bordes), Laurent Bordes

### [hal-01788510] A comparative review of soil charcoal data : Spatiotemporal patterns of origin and long-term dynamics of Western European nutrient-poor grasslands

The nutrient-poor grasslands of Western Europe are of major conservation concern because land use changes threaten their high biodiversity. Studies assessing their characteristics show that their past and ongoing dynamics are strongly related to human activities. Yet, the initial development patterns of this specific ecosystem remain unclear. Here, we examine findings from previous paleoecological investigations performed at local level on European grassland areas ranging from several hundred square meters to several square kilometers. Comparing data from these locally relevant studies at a regional scale, we investigate these grasslands' spatiotemporal patterns of origin and long-term dynamics. The study is based on taxonomic identification and radiocarbon AMS dating of charcoal pieces from soil/soil sediment archives of nutrient-poor grasslands in Mediterranean and temperate Western Europe (La Crau plain, Mont Lozère, Grands Causses, Vosges Mountains, Franconian Alb, and Upper-Normandy region). We address the following questions: (1) What are the key determinants of the establishment of these nutrient-poor grasslands? (2) What temporal synchronicities might there be? and (3) What is the spatial scale of these grasslands' past dynamics? The nutrient-poor grasslands in temperate Western Europe are found to result from the first anthropogenic woodland clearings during the late Neolithic, revealed by fire events in mesophilious mature forests. In contrast, the sites with Mediterranean affinities appear to have developed at earlier plant successional stages (pine forest, matorral), established before the first human impacts in the same period. However, no general pattern of establishment and dynamics of the nutrient-poor grasslands could be identified. Local mechanisms appear to be the key determinants of the dynamics of these ecosystems. Nevertheless, this paleoecological synthesis provides insights into past climate or human impacts on present-day vegetation.

ano.nymous@ccsd.cnrs.fr.invalid (Vincent Robin), Vincent Robin

### [hal-01525249] Shape sensitivity analysis for elastic structures with generalized impedance boundary conditions of the Wentzell type -Application to compliance minimization

This paper focuses on Generalized Impedance Boundary Conditions (GIBC) with second order derivatives in the context of linear elasticity and general curved interfaces. A condition of the Wentzell type modeling thin layer coatings on some elastic structure is obtained through an asymptotic analysis of order one of the transmission problem at the thin layer interfaces with respect to the thickness parameter. We prove the well-posedness of the approximate problem and the theoretical quadratic accuracy of the boundary conditions. Then we perform a shape sensitivity analysis of the GIBC model in order to study a shape optimization/optimal design problem. We prove the existence and characterize the first shape derivative of this model. A comparison with the asymptotic expansion of the first shape derivative associated to the original thin layer transmission problem shows that we can interchange the asymptotic and shape derivative analysis. Finally we apply these results to the compliance minimization problem. We compute the shape derivative of the compliance in this context and present some numerical simulations.

ano.nymous@ccsd.cnrs.fr.invalid (Fabien Caubet), Fabien Caubet

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Mjedi Azaïez), Mjedi Azaïez

### [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.

ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem

### [hal-00781238] Mixed finite element discretization of a model for organic pollution in waters Part I. The problem and its discretization

We consider a mixed reaction diffusion system describing the organic pollution in stream-waters. It may be viewed as the static version of Streeter-Phelps equations relating the Biochemical Oxygen Demand and Dissolved Oxygen to which dispersion terms are added. In this work, we propose a mixed variational formulation and prove its well-posedness. Next, we develop two finite element discretizations of this problem and establish optimal a priori error estimates for the second discrete problem.

ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem

### [hal-00815297] Direct electrochemical reduction of solid uranium oxide in molten fluoride salts

The direct electrochemical reduction of UO2 solid pellets was carried out in LiF-CaF2 (+ 2 mass. % Li2O) at 850°C. An inert gold anode was used instead of the usual reactive sacrificial carbon anode. In this case, oxidation of oxide ions present in the melt yields O2 gas evolution on the anode. Electrochemical characterisations of UO2 pellets were performed by linear sweep voltammetry at 10mV/s and reduction waves associated to oxide direct reduction were observed at a potential 150mV more positive in comparison to the solvent reduction. Subsequent, galvanostatic electrolyses runs were carried out and products were characterised by SEM-EDX, EPMA/WDS and XRD. In one of the runs, uranium oxide was partially reduced and three phases were observed: non reduced UO2 in the centre, pure metallic uranium on the external layer and an intermediate phase representing the initial stage of reduction taking place at the grain boundaries. In another run, the UO2 sample was fully reduced. Due to oxygen removal, the U matrix had a typical coral-like structure which is characteristic of the pattern observed after the electroreduction of solid oxides.

ano.nymous@ccsd.cnrs.fr.invalid (Mathieu Gibilaro), Mathieu Gibilaro

### [hal-01394849] Strong approximations for the $p$-fold integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the $p$-fold integrated empirical process, $p$ being a fixed positive integer. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on results of Koml\'os, Major and Tusn\'ady (1975). Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of integrated empirical processes when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of integrated empirical processes.

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

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ano.nymous@ccsd.cnrs.fr.invalid (Chérif Amrouche), Chérif Amrouche

### [hal-01157178] Some asymptotic results for the integrated empirical process with applications to statistical tests

The main purpose of this paper is to investigate the strong approximation of the integrated empirical process. More precisely, we obtain the exact rate of the approximations by a sequence of weighted Brownian bridges and a weighted Kiefer process. Our arguments are based in part on the Komlós et al. (1975)'s results. Applications include the two-sample testing procedures together with the change-point problems. We also consider the strong approximation of the integrated empirical process when the parameters are estimated. Finally, we study the behavior of the self-intersection local time of the partial sum process representation of the integrated empirical process.Reference: Koml\'os, J., Major, P. and Tusn\'ady, G. (1975). An approximation of partial sums of independent RV's and the sample DF. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 32, 111-131.

### [hal-01023384] A Finite Element Method for the Boundary Data Recovery in an Oxygen-Balance Dispersion Model

The inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional transport model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the outfall point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequences of these non-standard boundary conditions is that dispersive transport equations on both oxygen tracers are strongly coupled and the resulting system becomes ill-posed. The main purpose is a finite element space-discretization of the variational problem put under a non-symmetric mixed form. Combining analytical calculations, numerical computations and theoretical justifications, we try to elucidate the characteristics related to the ill-posedness of this data completion dynamical problem and understand its mathematical structure.

ano.nymous@ccsd.cnrs.fr.invalid (Faker Ben Belgacem), Faker Ben Belgacem

### [hal-01784139] Stein’s method for diffusive limit of Markov processes

The invariance principle for M/M/1 and M/M/∞ queues states that when properly renormalized (i.e. rescaled and centered), the Markov processes which describe these systems both converge to a diffusive limit when the driving parameters go to infinity: a killed Brownian motion in the former case and an Ornstein-Uhlenbeck process for the latter. The purpose of this paper is to assess the rate of convergence in these diffusion approximations. To this end, we extend to these contexts, the functional Stein's method introduced for the Brownian approximation of Poisson processes.

ano.nymous@ccsd.cnrs.fr.invalid (Eustache Besançon), Eustache Besançon

### [hal-01136619] Guaranteed State and Parameter Estimation for Nonlinear Dynamical Aerospace Models

This paper deals with parameter and state estimation in a bounded-error context for uncertain dynamical aerospace models when the input is considered optimized or not. In a bounded-error context, perturbations are assumed bounded but otherwise unknown. The parameters to be estimated are also considered bounded. The tools of the presented work are based on a guaranteed numerical set integration solver of ordinary differential equations combined with adapted set inversion computation. The main contribution of this work consists in developing procedures for parameter estimation whose performance is highly related with the input of system. In this paper, a comparison with a classical non-optimized input is proposed.

ano.nymous@ccsd.cnrs.fr.invalid (Qiaochu Li), Qiaochu Li