Université Paris Dauphine-PSL

The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

Abstract Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual Riesz bases of compactly supported wavelets. We give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions, and we present a sufficient conditions for the decay of their Fourier transforms. We study the regularity of these biorthogonal bases. We provide several families of examples, all symmetric (corresponding to “linear phase” filters). In particular we can construct symmetric biorthogonal wavelet bases with arbitraily high preassigned regularity; we also show how to construct symmetric biorthogonal wavelet bases “close” to a (nonsymmetric) orthonormal basis.

We present here a new method for solving minimization problems in unbounded domains. We first derive a general principle showing the equivalence between the compactness of all minimizing sequences and some strict sub-additivity conditions. The proof is based upon a compactness lemma obtained with the help of the notion of concentration function of a measure. We give various applications to problems arising in Mathematical Physics. Résumé Nous présentons ici une méthode nouvelle de résolution des problèmes de minimisation dans des domaines non bornés. Nous commençons par établir un principe général montrant l’équivalence entre la compacité de toutes les suites minimisantes et certaines conditions de sous-additivité stricte. La démonstration s’appuie sur un lemme de compacité obtenu à l’aide de la notion de fonction de concentration d’une mesure. Nous donnons diverses applications à des problèmes de physique mathématique.

A stable algorithm is proposed for image restoration based on the 'mean curvature motion' equation. Existence and uniqueness of the 'viscosity' solution of the equation are proved, a L∞ stable algorithm is given, experimental results are shown, and the subjacent vision model is compared with those introduced recently by several vision researchers. The algorithm presented appears to be the sharpest possible among the multiscale image smoothing methods preserving uniqueness and stability.

OBJECTIVE: Most current electroencephalography (EEG)-based brain-computer interfaces (BCIs) are based on machine learning algorithms. There is a large diversity of classifier types that are used in this field, as described in our 2007 review paper. Now, approximately ten years after this review publication, many new algorithms have been developed and tested to classify EEG signals in BCIs. The time is therefore ripe for an updated review of EEG classification algorithms for BCIs. APPROACH: We surveyed the BCI and machine learning literature from 2007 to 2017 to identify the new classification approaches that have been investigated to design BCIs. We synthesize these studies in order to present such algorithms, to report how they were used for BCIs, what were the outcomes, and to identify their pros and cons. MAIN RESULTS: We found that the recently designed classification algorithms for EEG-based BCIs can be divided into four main categories: adaptive classifiers, matrix and tensor classifiers, transfer learning and deep learning, plus a few other miscellaneous classifiers. Among these, adaptive classifiers were demonstrated to be generally superior to static ones, even with unsupervised adaptation. Transfer learning can also prove useful although the benefits of transfer learning remain unpredictable. Riemannian geometry-based methods have reached state-of-the-art performances on multiple BCI problems and deserve to be explored more thoroughly, along with tensor-based methods. Shrinkage linear discriminant analysis and random forests also appear particularly useful for small training samples settings. On the other hand, deep learning methods have not yet shown convincing improvement over state-of-the-art BCI methods. SIGNIFICANCE: This paper provides a comprehensive overview of the modern classification algorithms used in EEG-based BCIs, presents the principles of these methods and guidelines on when and how to use them. It also identifies a number of challenges to further advance EEG classification in BCI.

We investigate the relative importance of diffusion and jumps in a new jump diffusion model for asset returns. In contrast to the standard modelling of jumps for asset returns, the jump component of our process can display finite or infinite activity, and finite or infinite variation. Empirical investigations of time series indicate that index dynamics are essentially devoid of a diffusion component, while this component may be present in the dynamics of individual stocks. This result leads to the conjecture that the risk-neutral process should be free of a diffusion component for both indices and individual stocks. Empirical investigation of options data tends to confirm this conjecture. We conclude that the statistical and risk-neutral processes for indices and stocks tend to be pure jump processes of innite activity and finite variation.

This paper is the second part of a work devoted to the study of variational problems (with constraints) in functional spaces defined on domains presenting some (local) form of invariance by a non-compact group of transformations like the dilations in \mathbb R^N . This contains for example the class of problems associated with the determination of extremal functions in inequalities like Sovolev inequalities, convolution or trace inequalities... We show how the concentration-compactness principle and method introduced in the so-called locally compact case are to be modified in order to solve these problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.

In this paper (sequel of Part 1) we investigate further applications of the concentration-compactness principle to the solution of various minimization problems in unbounded domains. In particular we present here the solution of minimization problems associated with nonlinear field equations. Résumé Dans cette deuxième partie, nous examinons de nouvelles applications du principe de concentration-compacité à la résolution dedivers problèmes de minimization dans des ouverts non bornés. En particulier nous résolvons des problèmes de minimisation associés aux équations de champ non linéaires.

After the study made in the locally compact case for variational problems with some translation invariance, we investigate here the variational problems (with constraints) for example in \mathbb R^N where the invariance of \mathbb R^N by the group of dilatations creates some possible loss of compactness. This is for example the case for all the problems associated with the determination of extremal functions in functional inequalities (like for example the Sobolev inequalities). We show here how the concentration-compactness principle has to be modified in order to be able to treat this class of problems and we present applications to Functional Analysis, Mathematical Physics, Differential Geometry and Harmonic Analysis.

The use of energy-minimizing curves, known as "snakes" to extract features of interest in images has been introduced by Kass, Witkin and Terzopoulos (1987). A balloon model was introduced by Cohen (1991) as a way to generalize and solve some of the problems encountered with the original method. A 3-D generalization of the balloon model as a 3-D deformable surface, which evolves in 3-D images, is presented. It is deformed under the action of internal and external forces attracting the surface toward detected edgels by means of an attraction potential. We also show properties of energy-minimizing surfaces concerning their relationship with 3-D edge points. To solve the minimization problem for a surface, two simplified approaches are shown first, defining a 3-D surface as a series of 2-D planar curves. Then, after comparing finite-element method and finite-difference method in the 2-D problem, we solve the 3-D model using the finite-element method yielding greater stability and faster convergence. This model is applied for segmenting magnetic resonance images.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

This volume discusses an in-depth theory of function spaces in an Euclidean setting, including several new features, not previously covered in the literature. In particular, it develops a unified theory of anisotropic Besov and Bessel potential spaces on Euclidean corners, with infinite-dimensional Banach spaces as targets. It especially highlights the most important subclasses of Besov spaces, namely Slobodeckii and Hölder spaces. In this case, no restrictions are imposed on the target spaces, except for reflexivity assumptions in duality results. In this general setting, the author proves sharp embedding, interpolation, and trace theorems, point-wise multiplier results, as well as Gagliardo-Nirenberg estimates and generalizations of Aubin-Lions compactness theorems. The results presented pave the way for new applications in situations where infinite-dimensional target spaces are relevant – in the realm of stochastic differential equations, for example.

Abstract This book deals with a new generation of econometric methods leading to criterion functions without simple analytical expression. The difficulty often comes from the presence of integrals of large dimension in the probability density function or in the moments, and the idea is to circumvent this numerical difficulty by an approach based on simulation. The main methods considered are the methods of Simulated Moments, Simulated Maximum Likelihood, Simulated Pseudo‐Maximum Likelihood, Simulated Non‐Linear Least Squares, and Indirect Inference. These methods are applied to Limited Dependent Variables Models, to Financial Series, and to Switching Regime Models.

We present a simple, purely analytic method for proving the convergence of a wide class of approximation schemes to the solution of fully non linear second-order elliptic or parabolic PDE. Roughly speaking, we prove that any monotone, stable and consistent scheme converges to the correct solution provided that there exists a comparison principle for the limiting equation. This method is based on the notion of viscosity solution of Crandall and Lions and it gives completely new results concerning the convergence of numerical schemes for stochastic differential games.

This paper proposes new definitions of lower and upper approximations, which are basic concepts of the rough set theory. These definitions follow naturally from the concept of ambiguity introduced in this paper. The new definitions are compared to the classical definitions and are shown to be more general, in the sense that they are the only ones which can be used for any type of indiscernibility or similarity relation.

Nous introduisons ici une approche générale pour modéliser des jeux avec un très grand nombre de joueurs. Plus précisément, nous considérons des équilibres de Nash à N joueurs pour des problèmes stochastiques en temps long et déduisons rigoureusement les équations de type « champ moyen » quand N tend vers l'infini. Nous prouvons également des résultats généraux d'unicité et établissons la limite déterministe.

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean‐reverting square root process. The model for the mean‐reverting time change is then generalized to include non‐Gaussian models that are solutions to Ornstein‐Uhlenbeck equations driven by one‐sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean‐corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean‐corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one‐dimensional marginal distributions of the level of the process at each future date.

The deviance information criterion (DIC) introduced by Spiegelhalter et al.(2002) for model assessment and model comparison is directly inspired by linear and generalised linear models, but it is open to different possible variations in the setting of missing data models, depending in particular on whether or not the missing variables are treated as parameters. In this paper, we reassess the criterion for such models and compare different DIC constructions, testing the behaviour of these various extensions in the cases of mixtures of distributions and random effect models.

"The quadratic cost optimal control problem for systems described by linear ordinary differential equations occupies a central role in the study of control systems both from a theoretical and design point of view. The study of this problem over an infinite time horizon shows the beautiful interplay between optimality and the qualitative properties of systems such as controllability, observability, stabilizability, and detectability. This theory is far more difficult for infinite dimensional systems such as those with time delays and distributed parameter systems. ... Part I reviews basic optimal control and game theory of finite dimensional systems, which serves as an introduction to the book. Part II deals with time evolution of some generic controlled infinite dimensional systems and contains a fairly complete account of semigroup theory. It incorporates interpolation theory and exhibits the role of semigroup theory in delay differential and partial differential equations. Part III studies the generic qualitative properties of controlled systems. Parts IV and V examine the optimal control of systems when performance is measured via a quadratic cost. Boundary control of parabolic and hyperbolic systems and exact controllability are also covered"--P. [4] of cover.

Abstract The COVID-19 pandemic is impacting human activities, and in turn energy use and carbon dioxide (CO 2 ) emissions. Here we present daily estimates of country-level CO 2 emissions for different sectors based on near-real-time activity data. The key result is an abrupt 8.8% decrease in global CO 2 emissions (−1551 Mt CO 2 ) in the first half of 2020 compared to the same period in 2019. The magnitude of this decrease is larger than during previous economic downturns or World War II. The timing of emissions decreases corresponds to lockdown measures in each country. By July 1st, the pandemic’s effects on global emissions diminished as lockdown restrictions relaxed and some economic activities restarted, especially in China and several European countries, but substantial differences persist between countries, with continuing emission declines in the U.S. where coronavirus cases are still increasing substantially.

We discuss problems the null hypothesis significance testing (NHST) paradigm poses for replication and more broadly in the biomedical and social sciences as well as how these problems remain unresolved by proposals involving modified p-value thresholds, confidence intervals, and Bayes factors. We then discuss our own proposal, which is to abandon statistical significance. We recommend dropping the NHST paradigm—and the p-value thresholds intrinsic to it—as the default statistical paradigm for research, publication, and discovery in the biomedical and social sciences. Specifically, we propose that the p-value be demoted from its threshold screening role and instead, treated continuously, be considered along with currently subordinate factors (e.g., related prior evidence, plausibility of mechanism, study design and data quality, real world costs and benefits, novelty of finding, and other factors that vary by research domain) as just one among many pieces of evidence. We have no desire to “ban” p-values or other purely statistical measures. Rather, we believe that such measures should not be thresholded and that, thresholded or not, they should not take priority over the currently subordinate factors. We also argue that it seldom makes sense to calibrate evidence as a function of p-values or other purely statistical measures. We offer recommendations for how our proposal can be implemented in the scientific publication process as well as in statistical decision making more broadly.