Laboratoire d’Analyse et de Mathématiques Appliquées

Using the ‘monotonicity trick’ introduced by Struwe, we derive a generic theorem. It says that for a wide class of functionals, having a mountain-pass (MP) geometry, almost every functional in this class has a bounded Palais-Smale sequence at the MP level. Then we show how the generic theorem can be used to obtain, for a given functional, a special Palais–Smale sequence possessing extra properties that help to ensure its convergence. Subsequently, these abstract results are applied to prove the existence of a positive solution for a problem of the form We assume that the functional associated to (P) has an MP geometry. Our results cover the case where the nonlinearity f satisfies (i) f ( x, s ) s −1 → a ∈)0, ∞) as s →+∞; and (ii) f ( x, s ) s –1 is non decreasing as a function of s ≥ 0, a.e. x → ℝ N .

Introduction The Paradigm of Structural Analysis Social Relationships and Networks Personal Networks and Local Circles Graph Theory Equivalence and Cohesion Social Capital Power and Centrality Dynamics Multiple Affiliations

Abstract We consider the local and global Cauchy problem for the generalized Korteweg-de Vries equation, with initial data in homogeneous and nonhomogeneous Besov spaces. This allows us to slightly extend known results on this problem. Furthermore we prove existence and uniqueness of self-similar solutions.

A predictive-maintenance structure for a gradually deteriorating single-unit system (continuous time/continuous state) is presented in this paper. The proposed decision model enables optimal inspection and replacement decision in order to balance the cost engaged by failure and unavailability on an infinite horizon. Two maintenance decision variables are considered: the preventive replacement threshold and the inspection schedule based on the system state. In order to assess the performance of the proposed maintenance structure, a mathematical model for the maintained system cost is developed using regenerative and semi-regenerative processes theory. Numerical experiments show that the s-expected maintenance cost rate on an infinite horizon can be minimized by a joint optimization of the replacement threshold and the a periodic inspection times. The proposed maintenance structure performs better than classical preventive maintenance policies which can be treated as particular cases. Using the proposed maintenance structure, a well-adapted strategy can automatically be selected for the maintenance decision-maker depending on the characteristics of the wear process and on the different unit costs. Even limit cases can be reached: for example, in the case of expensive inspection and costly preventive replacement, the optimal policy becomes close to a systematic periodic replacement policy. Most of the classical maintenance strategies (periodic inspection/replacement policy, systematic periodic replacement, corrective policy) can be emulated by adopting some specific inspection scheduling rules and replacement thresholds. In a more general way, the proposed maintenance structure shows its adaptability to different possible characteristics of the maintained single-unit system.

The periodic unfolding method was introduced in 2002 in [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104] (with the basic proofs in [Proceedings of the Narvik Conference 2004, GAKUTO Internat. Ser. Math. Sci. Appl. 24, Gakkōtosho, Tokyo, 2006, pp. 119–136]). In the present paper we go into all the details of the method and include complete proofs, as well as several new extensions and developments. This approach is based on two distinct ideas, each leading to a new ingredient. The first idea is the change of scale, which is embodied in the unfolding operator. At the expense of doubling the dimension, this allows one to use standard weak or strong convergence theorems in $L^p$ spaces instead of more complicated tools (such as two-scale convergence, which is shown to be merely the weak convergence of the unfolding; cf. Remark 2.15). The second idea is the separation of scales, which is implemented as a macro-micro decomposition of functions and is especially suited for the weakly convergent sequences of Sobolev spaces. In the framework of this method, the proofs of most periodic homogenization results are elementary. The unfolding is particularly well-suited for multiscale problems (a simple backward iteration argument suffices) and for precise corrector results without extra regularity on the data. A list of the papers where these ideas appeared, at least in some preliminary form, is given with a discussion of their content. We also give a list of papers published since the publication [Cioranescu, Damlamian, and Griso, C.R. Acad. Sci. Paris, Ser. 1, 335 (2002), pp. 99–104], and where the unfolding method has been successfully applied.

A novel approach to periodic homogenization is proposed, based on an unfolding method, which leads to a fixed domain problem (without singularly oscillating coefficients). This method is elementary in nature and applies to cases of periodic multi-scale problems in domains with or without holes (including truss-like structures).

The classical Łojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, and tame geometry. This paper provides alternative characterizations of this type of inequality for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Łojasiewicz inequality (hereby called the Kurdyka-Łojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus partial-differential f"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mi mathvariant="normal"> ∂ </mml:mi> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">-\partial f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka-Łojasiewicz inequality is satisfied. Further characterizations in terms of <italic>talweg</italic> lines —a concept linked to the location of the less steepest points at the level sets of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> — and integrability conditions are given. In the convex case these results are significantly reinforced, allowing us in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C squared"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">C^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> function in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka-Łojasiewicz inequality.

The objective of this paper is twofold: first, to assess the potential of radar data for tropical vegetation cartography and, second, to evaluate the contribution of different polarimetric indicators that can be derived from a fully polarimetric data set. Because of its ability to take numerous and heterogeneous parameters into account, such as the various polarimetric indicators under consideration, a support vector machine (SVM) algorithm is used in the classification step. The contribution of the different polarimetric indicators is estimated through a greedy forward and backward method. Results have been assessed with AIRSAR polarimetric data polarimetric data acquired over a dense tropical environment. The results are compared to those obtained with the standard Wishart approach, for single frequency and multifrequency bands. It is shown that, when radar data do not satisfy the Wishart distribution, the SVM algorithm performs much better than the Wishart approach, when applied to an optimized set of polarimetric indicators.

We prove concentration inequalities for some classes of Markov chains and $\Phi$-mixing processes, with constants independent of the size of the sample, that extend the inequalities for product measures of Talagrand. The method is based on information inequalities put forwardby Marton in case of contracting Markov chains. Using a simple duality argument on entropy, our results also include the family of logarithmic Sobolev inequalities for convex functions. Applications to bounds on supremum of dependent empirical processes complete this work.

This paper deals with the fixed sampling interval case for stochastic volatility models. We consider a two-dimensional diffusion process (Y&lt;sub&gt;t&lt;/sub&gt;, V&lt;sub&gt;t&lt;/sub&gt;), where only (Y&lt;sub&gt;t&lt;/sub&gt;) is observed at n discrete times with regular sampling interval ͉. The unobserved coordinate (V&lt;sub&gt;t&lt;/sub&gt;) is ergodic and rules the diffusion coefficient (volatility) of (Y&lt;sub&gt;t&lt;/sub&gt;). We study the ergodicity and mixing properties of the observations (Y&lt;sub&gt;i͉&lt;/sub&gt;). For this purpose, we first present a thorough review of these properties for stationary diffusions. We then prove that our observations can be viewed as a hidden Markov model and inherit the mixing properties of (V&lt;sub&gt;t&lt;/sub&gt;). When the stochastic differential equation of (V&lt;sub&gt;t&lt;/sub&gt;) depends on unknown parameters, we derive moment-type estimators of all the parameters, and show almost sure convergence and a central limit theorem at rate n&lt;sup&gt;1/2&lt;/sup&gt;&lt;i/&gt;. Examples of models coming from finance are fully treated. We focus on the asymptotic variances of the estimators and establish some links with the small sampling interval case studied in previous papers.

Luminescent lanthanide complexes display unrivalled spectroscopic properties, which place them in a special category in the luminescent toolbox. Their long-lived line-like emission spectra are the cornerstones of numerous analytical applications ranging from ultrasensitive homogeneous fluoroimmunoassays to the study of molecular interactions in living cells with multiplexed microscopy. However, achieving such minor miracles is a result of years of synthetic efforts and spectroscopic studies to understand and gather all the necessary requirements for the labels to be efficient. This feature article intends to survey these criteria and to discuss some of the most important examples reported in the literature, before explaining in detail some of the applications of luminescent lanthanide labels to bioanalysis and luminescence microscopy. Finally, the emphasis will be put on some recent applications that hold great potential for future biosensing.

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the methods (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

A new grid method for computing the Snell envelope of a function of an $\mathbb{R}^d$-valued simulatable Markov chain $(X_k)_{0\lambda \leq k\lambda \leq n}$ is proposed. (This is a typical nonlinear problem that cannot be solved by the standard Monte Carlo method.) Every $X_k$ is replaced by a `quantized approximation' $\widehat{X}_k$ taking its values in a grid $\Gamma_k$ of size $N_k$. The $n$ grids and their trans\-ition probability matrices form a discrete tree on which a pseudo-Snell envelope is devised by mimicking the regular dynamic programming formula. Using the quantization theory of random vectors, we show the existence of a set of optimal grids, given the total number $N$ of elementary $\mathbb{R}^d$-valued quantizers. A recursive stochastic gradient algorithm, based on simulations of $(X_k)_{0\lambda \leq k \lambda \leq n}$, yields these optimal grids and their transition probability matrices. Some a priori error estimates based on the $L^p$-quantization errors $\|X_k-\widehat X_k\|_{_p}$ are established. These results are applied to the computation of the Snell envelope of a diffusion approximated by its (Gaussian) Euler scheme. We apply these result to provide a discretization scheme for reflected backward stochastic differential equations. Finally, a numerical experiment is carried out on a two-dimensional American option pricing problem.

Models with a vanishing anisotropic viscosity in the vertical direction are of relevance for the study of turbulent flows in geophysics. This motivates us to study the two-dimensional Boussinesq system with horizontal viscosity in only one equation. In this paper, we focus on the global existence issue for possibly large initial data. We first examine the case where the Navier–Stokes equation with no vertical viscosity is coupled with a transport equation. Second, we consider a coupling between the classical two-dimensional incompressible Euler equation and a transport–diffusion equation with diffusion in the horizontal direction only. For both systems, we construct global weak solutions à la Leray and strong unique solutions for more regular data. Our results rest on the fact that the diffusion acts perpendicularly to the buoyancy force.

Abstract In this paper, we study the quasineutral limit of an Euler-Poisson system arising from plasma physics i.e. the limit when the Debye length tends to of a non linear hyperbolic system coupled with a non linear elliptic equation.The proof uses pseudodiiferential energy estimates techniques, in order to justify classical limits in small time, for strong solutions.

We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. Theproof is achieved applying the abstract Cauchy--Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433--461], as we do not require analyticity of the data with respect to the normal variable.

We compare three different stochastic versions of the EM algorithm: The Stochastic EM algorithm (SEM), the “Simulated Annealing” EM algorithm (SAEM) and the Monte Carlo EM algorithm (MCEM). We focus particularly on the mixture of distributions problem. In this context, we investigate the practical behaviour of these algorithms through intensive Monte Carlo numerical simulations and a real data study. We show that, for some particular mixture situations, the SEM algorithm is almost always preferable to the EM and “simulated annealing” versions SAEM and MCEM. For some severely overlapping mixtures, however, none of these algorithms can be confidently used. Then, SEM can be used as an efficient data exploratory tool for locating significant maxima of the likelihood function. In the real data case, we show that the SEM stationary distribution provides a contrasted view of the loglikelihood by emphasizing sensible maxima.

We present here the quantization method which is well‐adapted for the pricing and hedging of American options on a basket of assets. Its purpose is to compute a large number of conditional expectations by projection of the diffusion on optimal grids designed to minimize the (square mean) projection error ( Graf and Luschgy 2000 ). An algorithm to compute such grids is described. We provide results concerning the orders of the approximation with respect to the regularity of the payoff function and the global size of the grids. Numerical tests are performed in dimensions 2, 4, 5, 6, 10 with American style exchange options. They show that theoretical orders are probably pessimistic.

This paper develops a local regularization operator on triangular or quadrilateral finite elements built on structured or unstructured meshes. This operator is a variant of the regularization operator of Clément; however, ours is constructed via a local projection in a reference domain. We prove in this paper that it has the same optimal approximation properties as the standard interpolation operator, and we present some applications.

We consider the mean‐variance hedging problem when the risky assets price process is a continuous semimartingale. The usual approach deals with self‐financed portfolios with respect to the primitive assets family. By adding a numéraire as an asset to trade in, we show how self‐financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims. We introduce the hedging numéraire and relate it to the variance‐optimal martingale measure. Using this numéraire both as a deflator and to extend the primitive assets family, we are able to transform the original mean‐variance hedging problem into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the Galtchouk–Kunita–Watanabe projection theorem under a martingale measure for the hedging numéraire extended assets family. This gives in turn an explicit description of the optimal hedging strategy for the original mean‐variance hedging problem.