Laboratoire de Mathématiques d'Orsay

This is the first comprehensive reference on trust-region methods, a class of algorithms for the solution of nonlinear nonconvex optimization problems. It is a unified treatment that covers both unconstrained and constrained problems, including many specialised topics and reviews of the literature not easily obtained elsewhere. Written primarily for post-graduates and researchers, the book features an extensive commented bibliography, which contains 972 references by 745 authors. The book also contains several practical comments and an entire chapter devoted to software and implementation issues. Its many illustrations, including nearly 100 figures, helps to make the formal and in-depth treatment of the presented topics more accessible.

We are concerned with different properties of backward stochastic differential equations and their applications to finance. These equations, first introduced by Pardoux and Peng (1990), are useful for the theory of contingent claim valuation, especially cases with constraints and for the theory of recursive utilities, introduced by Duffie and Epstein (1992a, 1992b).

This special issue of Mathematical Structures in Computer Science contains several contributions related to the modern field of Quantum Information and Quantum Computing. The first two papers deal with entanglement. The paper by R. Mosseri and P. Ribeiro presents a detailed description of the two- and three-qubit geometry in Hilbert space, dealing with the geometry of fibrations and discrete geometry. The paper by J.-G.Luque et al . is more algebraic and considers invariants of pure k-qubit states and their application to entanglement measurement.

Abstract This article investigates estimation of finite population totals in the presence of univariate or multivariate auxiliary information. Estimation is equivalent to attaching weights to the survey data. We focus attention on the several weighting systems that can be associated with a given amount of auxiliary information and derive a weighting system with the aid of a distance measure and a set of calibration equations. We briefly mention an application to the case in which the information consists of known marginal counts in a two- or multi-way table, known as generalized raking. The general regression estimator (GREG) was conceived with multivariate auxiliary information in mind. Ordinarily, this estimator is justified by a regression relationship between the study variable y and the auxiliary vector x. But we note that the GREG can be derived by a different route by focusing instead on the weights. The ordinary sampling weights of the kth observation is 1/πk , where πk is the inclusion probability of k. We show that the weights implied by the GREG are as close as possible, according to a given distance measure, to the 1/πk while respecting side conditions called calibration equations. These state that the sample sum of the weighted auxiliary variable values must equal the known population total for that auxiliary variable. That is, the calibrated weights must give perfect estimates when applied to each auxiliary variable. That is a consistency check that appeals to many practitioners, because a strong correlation between the auxiliary variables and the study variable means that the weights that perform well for the auxiliary variable also should perform well for the study variable. The GREG uses the auxiliary information efficiently, so the estimates are precise; however, the individual weights are not always without reproach. For example, negative weights can occur, and in some applications this does not make sense. It is natural to seek the root of the dissatisfaction in the underlying distance measure. Consequently, we allow alternative distance measures that satisfy only a set of minimal requirements. Each distance measure leads, via the calibration equations, to a specific weighting system and thereby to a new estimator. These estimators form a family of calibration estimators. We show that the GREG is a first approximation to all other members of the family; all are asymptotically equivalent to the GREG, and the variance estimator already known for the GREG is recommended for use in any other member of the family. Numerical features of the weights and ease of computation become more than anything else the bases for choosing between the estimators. The reasoning is applied to calibration on known marginals of a two-way frequency table. Our family of distance measures leads in this case to a family of generalized raking procedures, of which classical raking ratio is one.

Rényi divergence is related to Rényi entropy much like Kullback-Leibler divergence is related to Shannon's entropy, and comes up in many settings. It was introduced by Rényi as a measure of information that satisfies almost the same axioms as Kullback-Leibler divergence, and depends on a parameter that is called its order. In particular, the Rényi divergence of order 1 equals the Kullback-Leibler divergence. We review and extend the most important properties of Rényi divergence and Kullback-Leibler divergence, including convexity, continuity, limits of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(\sigma \) </tex-math></inline-formula> -algebras, and the relation of the special order 0 to the Gaussian dichotomy and contiguity. We also show how to generalize the Pythagorean inequality to orders different from 1, and we extend the known equivalence between channel capacity and minimax redundancy to continuous channel inputs (for all orders) and present several other minimax results.

At initialization, artificial neural networks (ANNs) are equivalent to Gaussian processes in the infinite-width limit, thus connecting them to kernel methods. We prove that the evolution of an ANN during training can also be described by a kernel: during gradient descent on the parameters of an ANN, the network function $f_θ$ (which maps input vectors to output vectors) follows the kernel gradient of the functional cost (which is convex, in contrast to the parameter cost) w.r.t. a new kernel: the Neural Tangent Kernel (NTK). This kernel is central to describe the generalization features of ANNs. While the NTK is random at initialization and varies during training, in the infinite-width limit it converges to an explicit limiting kernel and it stays constant during training. This makes it possible to study the training of ANNs in function space instead of parameter space. Convergence of the training can then be related to the positive-definiteness of the limiting NTK. We prove the positive-definiteness of the limiting NTK when the data is supported on the sphere and the non-linearity is non-polynomial. We then focus on the setting of least-squares regression and show that in the infinite-width limit, the network function $f_θ$ follows a linear differential equation during training. The convergence is fastest along the largest kernel principal components of the input data with respect to the NTK, hence suggesting a theoretical motivation for early stopping. Finally we study the NTK numerically, observe its behavior for wide networks, and compare it to the infinite-width limit.

We propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood. For this purpose, the observed data are assigned to unknown clusters using a maximum a posteriori operator. Then, the integrated completed likelihood (ICL) is approximated using the Bayesian information criterion (BIC). Numerical experiments on simulated and real data of the resulting ICL criterion show that it performs well both for choosing a mixture model and a relevant number of clusters. In particular, ICL appears to be more robust than BIC to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data.

International audience

From differential equations to structured population dynamics.- Adaptive dynamics an asymptotic point of view.- Population balance equations: the renewal equation.- Population balance equations: size structure.- Cell motion and chemotaxis.- General mathematical tools.

Abstract. The global and local convergence properties of a class of augmented Lagrangian methods for solving nonlinear programming problems are considered. In such methods, simple bound constraints are treated separately from more general constraints and the stopping rules for the inner minimization algorithm have this in mind. Global convergence is proved, and it is established that a potentially troublesome penalty parameter is bounded away from zero. Key words, constrained optimization, augmented Lagrangian, simple bounds, general constraints AMS(MOS) subject classifications. 65K05, 90C30

The importance of microbial communities (MCs) cannot be overstated. MCs underpin the biogeochemical cycles of the earth's soil, oceans and the atmosphere, and perform ecosystem functions that impact plants, animals and humans. Yet our ability to predict and manage the function of these highly complex, dynamically changing communities is limited. Building predictive models that link MC composition to function is a key emerging challenge in microbial ecology. Here, we argue that addressing this challenge requires close coordination of experimental data collection and method development with mathematical model building. We discuss specific examples where model-experiment integration has already resulted in important insights into MC function and structure. We also highlight key research questions that still demand better integration of experiments and models. We argue that such integration is needed to achieve significant progress in our understanding of MC dynamics and function, and we make specific practical suggestions as to how this could be achieved.

Four human Lactobacillus acidophilus strains were tested for their ability to adhere onto human enterocyte like Caco-2 cells in culture. The LA 1 strain exhibited a high calcium independent adhesive property. This adhesion onto Caco-2 cells required a proteinaceous adhesion promoting factor, which was present in the spent bacterial broth culture supernatant. LA 1 strain also strongly bound to the mucus secreted by the homogeneous cultured human goblet cell line HT29-MTX. The inhibitory effect of LA 1 organisms against Caco-2 cell adhesion and cell invasion by a large variety of diarrhoeagenic bacteria was investigated. As a result, the following dose dependent inhibitions were obtained: (a) against the cell association of enterotoxigenic, diffusely adhering and enteropathogenic Escherichia coli, and Salmonella typhimurium; (b) against the cell invasion by enteropathogenic Escherichia coli, Yersinia pseudotuberculosis, and Salmonella typhimurium. Incubations of L acidophilus LA 1 before and together with enterovirulent E coli were more effective than incubation after infection by E coli.

Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn out to be sub-Riemannian manifolds. Moreover, they come with an algebraic structure: nilpotent Lie groups with dilations. In the classical, Riemannian, case, they are indeed vector spaces, that is, abelian groups with dilations. Actually, the above is true only for regular points. At singular points, instead of nilpotent Lie groups one gets quotient spaces G/H of such groups G.

The subject of geometric numerical integration deals with numerical integrators that preserve geometric properties of the flow of a differential equation, and it explains how structure preservation leads to improved long-time behaviour. This article illustrates concepts and results of geometric numerical integration on the important example of the Störmer–Verlet method. It thus presents a cross-section of the recent monograph by the authors, enriched by some additional material. After an introduction to the Newton–Störmer–Verlet–leapfrog method and its various interpretations, there follows a discussion of geometric properties: reversibility, symplecticity, volume preservation, and conservation of first integrals. The extension to Hamiltonian systems on manifolds is also described. The theoretical foundation relies on a backward error analysis, which translates the geometric properties of the method into the structure of a modified differential equation, whose flow is nearly identical to the numerical method. Combined with results from perturbation theory, this explains the excellent long-time behaviour of the method: long-time energy conservation, linear error growth and preservation of invariant tori in near-integrable systems, a discrete virial theorem, and preservation of adiabatic invariants.

Property (T) is a rigidity property for topological groups, first formulated by D. Kazhdan in the mid 1960's with the aim of demonstrating that a large class of lattices are finitely generated. Later developments have shown that Property (T) plays an important role in an amazingly large variety of subjects, including discrete subgroups of Lie groups, ergodic theory, random walks, operator algebras, combinatorics, and theoretical computer science. This monograph offers a comprehensive introduction to the theory. It describes the two most important points of view on Property (T): the first uses a unitary group representation approach, and the second a fixed point property for affine isometric actions. Via these the authors discuss a range of important examples and applications to several domains of mathematics. A detailed appendix provides a systematic exposition of parts of the theory of group representations that are used to formulate and develop Property (T).

Since the seminal work of P. Anderson in 1958, localization in disordered systems has been the object of intense investigations. Mathematically speaking, the phenomenon can be described as follows: th

We consider a backward stochastic differential equation, whose data (the final condition and the coefficient) are given functions of a jump-diffusion process. We prove that under mild conditions the solution of the BSDE provides a viscosity solution of a system of parabolic integral-partial differential equations. Under an additional assumption, that system of equations is proved to have a unique solution, in a given class of continuous functions

(1995). Controle Exact De Lequation De La Chaleur. Communications in Partial Differential Equations: Vol. 20, No. 1-2, pp. 335-356.

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H1 than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.

OBJECTIVE: The presence of anxiety disorders (AD) in schizophrenia (SZ) is attracting increasing interest. However, published studies have yielded very broad variations in prevalence rates across studies. The current meta-analysis sought to (1) investigate the prevalence of co-occurring AD in SZ by reporting pooled prevalence rates and (2) identify potential sources of variations in reported rates that could guide our efforts to identify and treat these co-occurring disorders in patients with SZ. METHODS: We performed a systematic search of studies reporting prevalence of AD in SZ and related psychotic disorders. Mean prevalence rates and 95% confidence intervals (CIs) were first computed for each disorder. We then examined the impact of potential moderators related to patient sampling or to AD assessment methods on these rates. RESULTS: Fifty-two eligible studies were identified. Pooled prevalence rates and CIs were 12.1% (7.0%-17.1%) for obsessive-compulsive disorders, 14.9% (8.1%-21.8%) for social phobia, 10.9% (2.9%-18.8%) for generalized AD, 9.8% (4.3%-15.4%) for panic disorders, and 12.4% (4.0%-20.8%) for post-traumatic stress disorders. For all disorders, we found significant heterogeneity in rates across studies. This heterogeneity could at least partially be explained by the effect of moderator variables related to patient characteristics or assessment methods. CONCLUSIONS: AD are highly prevalent in SZ, but important variations in rates are observed between studies. This meta-analysis highlights several factors that affect risk for, or detection of AD in SZ, and could, thus, have an important impact on treatment and outcome of SZ patients.