Laboratoire de mathématiques appliquées de Compiègne

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We consider a discrete-time semi-Markov process, with a finite state space. Taking a censored history, we obtain empirical estimators for the semi-Markov kernel, semi-Markov transition function, reliability and availability. We study the strong consistency and the asymptotic normality for each estimator.

Abstract In this paper, we define a discrete-time semi-Markov model and propose a computation procedure for solving the corresponding Markov renewal equation, necessary for all our reliability measurements. Then, we compute the reliability and its related measures, and we apply the results to a three-state system. Key Words: Discrete-time semi-Markov processDiscrete-time Markov renewal processMarkov renewal theoryReliabilitySurvival functionMean hitting timesMonte Carlo methodGeneralized ph-distribution Acknowledgments

We consider repairable Multi-state reliability systems with components, the lifetimes and the repair times of which are -independent. The -th component can be either in the complete failure state 0, in the perfect state , or in one of the degradation states . The sojourn time in any of these states is a random variable following a discrete distribution. Thus, the time behavior of each component is described by a discrete-time semi-Markov chain, and the time behavior of the whole system is described by the vector of paired processes of the semi-Markov chain and the corresponding backward recurrence time process. Using recently obtained results concerning the discrete-time semi-Markov chains, we derive basic reliability measures. Finally, we present some numerical results of our proposed approach in specific reliability systems, namely series, parallel, k-out-of-n:F, and consecutive-k-out-of-n:F systems.

In this paper, we consider an inverse source problem for an anisotropic elliptic equation, from boundary measurements. A uniqueness result is established and a local Lipshitz stability, for a linear combination of monopolar and dipolar sources, is discussed. Assuming the number of dipoles bounded by a given integer M, we propose an algebraic algorithm which allows us to estimate the number, the locations and the moments of dipoles. Using special functions, we propose a global Lipschitz stability estimate for dipolar sources.

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.

In the current 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 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 parameterization of shapes coupled with a boundary element method. Several numerical examples indicate the superiority of the Kohn and Vogelius functional over least squares fitting.

While hazard assessment of chemicals can make direct use of descriptive adverse outcome pathways (AOPs), risk assessment requires quantitative relationships from exposure to effect timing and magnitude. To seamlessly integrate the data generated by alternative methods or in vivo testing, quantitative AOPs (qAOPs) providing dose-time-response predictions are more valuable than qualitative AOPs. Here, we compare three approaches to qAOP building: empirical dose-response modeling, Bayesian network (BN) calibration, and systems biology (SB) modeling. These methods were applied to the quantification of a simplified oxidative stress induced chronic kidney disease AOP, on the basis of in vitro data obtained on RPTEC/TERT1 cells exposed to potassium bromate. Effectopedia was used to store the experimental data and the developed models in a unified representation so they can be compared and further analyzed. We argue that despite the fact that dose-response models give adequate fits to the data they should be accompanied by mechanistic SB modeling to gain a proper perspective on the quantification. BNs can be both more precise than dose-response models and simpler than SB models, but more experience with their usage is needed.

This paper is devoted to the analysis of a second order method for recovering the 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 been the focus of much attention during the past few years. However, to the best of our 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.

We consider the inverse problem of identifying multiple moving pollution sources in a linear advection–dispersion–reaction equation. Although we consider the specific application of pollution source identification in surface waters or atmospheric media, this problem has many other important applications in ecological and diffusive systems. We establish an identifiability result using observations on a non-empty subset of the domain boundary and develop an identification method by reformulating the inverse problem into a minimization problem. Finally, we provide numerical results to support the theoretical results.

In this paper, we consider the problem of identifying an unknown source F(x, t) = λ(t)δ(x − S) in the following system: (∂t − D∂xx + V ∂x + R)u(x, t) = F(x, t), 0 < x < , 0 < t < T (∂t − D∂xx + V ∂x + R)v(x, t) = Ru(x, t), 0 < x < , 0 < t < T from measured data [{v(a, t), ∂xv(a, t)}, {v(b, t), ∂xv(b, t)}] for appropriate points a and b. Assuming that the source F became inactive after the time T ∗(i.e. λ(t) = 0 for t T ∗), we prove an identifiability result and propose an identification method. (Some figures in this article are in colour only in the electronic version) 1.

In this paper we compute closed-form expressions for the topological derivative for three-dimensional time-harmonic electromagnetic waves for perfect conductors (Dirichlet condition), electromagnetic cavities (Neumann condition), absorbing obstacles (impedance condition), and dielectric inclusions (transmission conditions). The proofs are based on the computation of shape derivatives followed by asymptotic expansions using Mie series when infinitesimal spheres are considered. An exhaustive gallery of numerical experiments is presented, which demonstrate that the topological derivative is a very powerful tool for the detection of multiple electromagnetic scatterers without a priori information about their number, size, shape, or location. Numerical examples include highly demanding configurations where only a few incident directions and a few observation points (for near-field data) or a few far-field observation directions (for far-field data) are considered.

Two groups of bootstrap methods have been proposed to estimate the statistical properties of positron emission tomography (PET) images by generating multiple statistically equivalent data sets from few data samples. The first group generates resampled data based on a parametric approach assuming that data from which resampling is performed follows a Poisson distribution while the second group consists of nonparametric approaches. These methods either require a unique original sample or a series of statistically equivalent data that can be list-mode files or sinograms. Previous reports regarding these bootstrap approaches suggest different results. This work compares the accuracy of three of these bootstrap methods for 3-D PET imaging based on simulated data. Two methods are based on a unique file, namely a list-mode based nonparametric (LMNP) method and a sinogram based parametric (SP) method. The third method is a sinogram-based nonparametric (SNP) method. Another original method (extended LMNP) was also investigated, which is an extension of the LMNP methods based on deriving a resampled list-mode file by drawings events from multiple original list-mode files. Our comparison is based on the analysis of the statistical moments estimated on the repeated and resampled data. This includes the probability density function and the moments of order 1 and 2. Results show that the two methods based on multiple original data (SNP and extended LMNP) are the only methods that correctly estimate the statistical parameters. Performances of the LMNP and SP methods are variable. Simulated data used in this study were characterized by a high noise level. Differences among the tested strategies might be reduced with clinical data sets with lower noise.

We consider the inverse problem of identifying a moving source in a linear advection–dispersion–reaction equation. The main application, but not the only one, is the identification of an environmental pollution source in a river. An identifiability result is established and an identification method proposed using measurement records at two locations, one upstream and the other downstream from the source. Finally, numerical simulations are performed to assess the identification process.

International audience

The purpose of this work is to use a variational method to identify some of the parameters of one-dimensional models for blood flow in arteries. These parameters can be fit to approach as much as possible some data coming from experimental measurements or from numerical simulations performed using more complex models. A nonlinear least squares approach to parameter estimation was taken, based on the optimization of a cost function. The resolution of such an optimization problem generally requires the efficient and accurate computation of the gradient of the cost function with respect to the parameters. This gradient is computed analytically when the one-dimensional hyperbolic model is discretized with a second order Taylor-Galerkin scheme. An adjoint approach was used. Some preliminary numerical tests are shown. In these simulations, we mainly focused on determining a parameter that is linked to the mechanical properties of the arterial walls, the compliance. The synthetic data we used to estimate the parameter were obtained from a numerical computation performed with a more accurate model: a three-dimensional fluid-structure interaction model. The first results seem to be promising. In particular, it is worth noticing that the estimated compliance which gives the best fit is quite different from the values that are commonly used in practice.

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 $\Gamma$. In this paper, following an idea developed by Kleinman and Martin [SIAM J. Appl. Math., 48 (1988), pp. 307–325] for acoustic scattering problems, we consider methods for solving the dielectric scattering problem using a single integral equation over $\Gamma$ 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 nonsmooth boundaries.

In the present paper, we are mainly concerned with the non parametric estimation of the density as well as the regression function by using orthonormal wavelet bases. We provide the strong uniform consistency properties with rates of these estimators, over compact subsets of , under a general ergodic condition on the underlying processes. We characterize the asymptotic normality of considered wavelet-based estimators, under easily verifiable conditions. The asymptotic properties of these estimators are obtained, by means of the martingale approach.

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

Nonlocal cross-diffusion systems on the torus, arising in population dynamics and neuroscience, are analyzed. The global existence of weak solutions, the weak–strong uniqueness, and the localization limit are proved. The kernels are assumed to be in detailed balance. The proofs are based on entropy estimates coming from Shannon-type and Rao-type entropies, while the weak–strong uniqueness result follows from the relative entropy method. The existence and uniqueness theorems hold for nondifferentiable, only integrable kernels. The associated local cross-diffusion system, derived in the localization limit, is also discussed.