Centre de Mathématiques Appliquées de l'École polytechnique

WINNER of a Riskbook.com Best of 2004 Book Award!During the last decade, financial models based on jump processes have acquired increasing popularity in risk management and option pricing. Much has been published on the subject, but the technical nature of most papers makes them difficult for nonspecialists to understand, and the mathematic

Evolutionary computation techniques have received a great deal of attention regarding their potential as optimization techniques for complex numerical functions. However, they have not produced a significant breakthrough in the area of nonlinear programming due to the fact that they have not addressed the issue of constraints in a systematic way. Only recently have several methods been proposed for handling nonlinear constraints by evolutionary algorithms for numerical optimization problems; however, these methods have several drawbacks, and the experimental results on many test cases have been disappointing. In this paper we (1) discuss difficulties connected with solving the general nonlinear programming problem; (2) survey several approaches that have emerged in the evolutionary computation community; and (3) provide a set of 11 interesting test cases that may serve as a handy reference for future methods.

In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.

This paper introduces a new class of bases, called bandelet bases, which decompose the image along multiscale vectors that are elongated in the direction of a geometric flow. This geometric flow indicates directions in which the image gray levels have regular variations. The image decomposition in a bandelet basis is implemented with a fast subband-filtering algorithm. Bandelet bases lead to optimal approximation rates for geometrically regular images. For image compression and noise removal applications, the geometric flow is optimized with fast algorithms so that the resulting bandelet basis produces minimum distortion. Comparisons are made with wavelet image compression and noise-removal algorithms.

It is control that turns scientific knowledge into useful technology: in physics and engineering it provides a systematic way for driving a dynamical system from a given initial state into a desired target state with minimized expenditure of energy and resources. As one of the cornerstones for enabling quantum technologies, optimal quantum control keeps evolving and expanding into areas as diverse as quantum-enhanced sensing, manipulation of single spins, photons, or atoms, optical spectroscopy, photochemistry, magnetic resonance (spectroscopy as well as medical imaging), quantum information processing and quantum simulation. In this communication, state-of-the-art quantum control techniques are reviewed and put into perspective by a consortium of experts in optimal control theory and applications to spectroscopy, imaging, as well as quantum dynamics of closed and open systems. We address key challenges and sketch a roadmap for future developments.

The state of the art in real-time programming is briefly reviewed. The synchronous approach is then introduced informally and its possible impact on the design of real-time and reactive systems is discussed. The authors present and discuss the application fields and the principles of synchronous programming. The major concern of the synchronous approach is to base synchronous programming languages on mathematical models. This makes it possible to handle compilation, logical correctness proofs, and verification of real-time programs in a formal way, leading to a clean and precise methodology for design and programming.>

As the number of single-cell transcriptomics datasets grows, the natural next step is to integrate the accumulating data to achieve a common ontology of cell types and states. However, it is not straightforward to compare gene expression levels across datasets and to automatically assign cell type labels in a new dataset based on existing annotations. In this manuscript, we demonstrate that our previously developed method, scVI, provides an effective and fully probabilistic approach for joint representation and analysis of scRNA-seq data, while accounting for uncertainty caused by biological and measurement noise. We also introduce single-cell ANnotation using Variational Inference (scANVI), a semi-supervised variant of scVI designed to leverage existing cell state annotations. We demonstrate that scVI and scANVI compare favorably to state-of-the-art methods for data integration and cell state annotation in terms of accuracy, scalability, and adaptability to challenging settings. In contrast to existing methods, scVI and scANVI integrate multiple datasets with a single generative model that can be directly used for downstream tasks, such as differential expression. Both methods are easily accessible through scvi-tools.

We solve the initial and boundary condition problem for a general first order quasilinear equation in several space variables by using a vanishing viscosity method and give a definition which characterizes the obtained solution.

A scattering transform defines a locally translation invariant representation which is stable to time-warping deformation. It extends MFCC representations by computing modulation spectrum coefficients of multiple orders, through cascades of wavelet convolutions and modulus operators. Second-order scattering coefficients characterize transient phenomena such as attacks and amplitude modulation. A frequency transposition invariant representation is obtained by applying a scattering transform along log-frequency. State-the-of-art classification results are obtained for musical genre and phone classification on GTZAN and TIMIT databases, respectively.

We present a simple model of a stock market where a random communication structure between agents generically gives rise to heavy tails in the distribution of stock price variations in the form of an exponentially truncated power law, similar to distributions observed in recent empirical studies of high-frequency market data. Our model provides a link between two well-known market phenomena: the heavy tails observed in the distribution of stock market returns on one hand and herding behavior in financial markets on the other hand. In particular, our study suggests a relation between the excess kurtosis observed in asset returns, the market order flow, and the tendency of market participants to imitate each other.

We study a level-set method for numerical shape optimization of elastic structures. Our approach combines the level-set algorithm of Osher and Sethian with the classical shape gradient. Although this method is not specifically designed for topology optimization, it can easily handle topology changes for a very large class of objective functions. Its cost is moderate since the shape is captured on a fixed Eulerian mesh.

The study of islands as model systems has played an important role in the development of evolutionary and ecological theory. The 50th anniversary of MacArthur and Wilson's (December 1963) article, 'An equilibrium theory of insular zoogeography', was a recent milestone for this theme. Since 1963, island systems have provided new insights into the formation of ecological communities. Here, building on such developments, we highlight prospects for research on islands to improve our understanding of the ecology and evolution of communities in general. Throughout, we emphasise how attributes of islands combine to provide unusual research opportunities, the implications of which stretch far beyond islands. Molecular tools and increasing data acquisition now permit re-assessment of some fundamental issues that interested MacArthur and Wilson. These include the formation of ecological networks, species abundance distributions, and the contribution of evolution to community assembly. We also extend our prospects to other fields of ecology and evolution - understanding ecosystem functioning, speciation and diversification - frequently employing assets of oceanic islands in inferring the geographic area within which evolution has occurred, and potential barriers to gene flow. Although island-based theory is continually being enriched, incorporating non-equilibrium dynamics is identified as a major challenge for the future.

A large number of imaging problems reduce to the optimization of a cost function, with typical structural properties. The aim of this paper is to describe the state of the art in continuous optimization methods for such problems, and present the most successful approaches and their interconnections. We place particular emphasis on optimal first-order schemes that can deal with typical non-smooth and large-scale objective functions used in imaging problems. We illustrate and compare the different algorithms using classical non-smooth problems in imaging, such as denoising and deblurring. Moreover, we present applications of the algorithms to more advanced problems, such as magnetic resonance imaging, multilabel image segmentation, optical flow estimation, stereo matching, and classification.

In a financial market model with constraints on the portfolios, define the price for a claim C as the smallest real number p such that sup π E[ U ( X T x + p , π − C )]≥ sup π E[ U ( X T x , π )], where U is the negative exponential utility function and X x , π is the wealth associated with portfolio π and initial value x . We give the relations of this price with minimal entropy or fair price in the flavor of Karatzas and Kou (1996) and superreplication. Using dynamical methods, we characterize the price equation, which is a quadratic Backward SDE, and describe the optimal wealth and portfolio. Further use of Backward SDE techniques allows for easy determination of the pricing function properties.

We introduce a class of multifractal processes, referred to as multifractal random walks (MRWs). To our knowledge, it is the first multifractal process with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascadelike multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by four parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.

We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e. we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.

We first present multicomponent flow models derived from the kinetic theory of gases. We then investigate the symmetric hyperbolic-parabolic structure of the resulting system of partial differential equations and discuss the Cauchy problem for smooth solutions. We also address the existence of deflagration waves also termed anchored waves. We further indicate related models which have a similar hyperbolic-parabolic structure, notably the Saint-Venant system with a temperature equation as well as the equations governing chemical equilibrium flows. We next investigate multicomponent ionized and magnetized flow models with anisotropic transport fluxes which have a different mathematical structure. We finally discuss numerical algorithms specifically devoted to complex chemistry flows, in particular the evaluation of multicomponent transport properties, as well as the impact of multicomponent transport. 1

We present a finite difference method for solving parabolic partial integro-differential equations with possibly singular kernels which arise in option pricing theory when the random evolution of the underlying asset is driven by a Lévy process or, more generally, a time-inhomogeneous jump-diffusion process. We discuss localization to a finite domain and provide an estimate for the localization error under an integrability condition on the Lévy measure. We propose an explicit-implicit finite difference scheme which can be used to price European and barrier options in such models. We study stability and convergence of the scheme proposed and, under additional conditions, provide estimates on the rate of convergence. Numerical tests are performed with smooth and nonsmooth initial conditions.

We are concerned with the numerical resolution of backward stochastic differential equations. We propose a new numerical scheme based on iterative regressions on function bases, which coefficients are evaluated using Monte Carlo simulations. A full convergence analysis is derived. Numerical experiments about finance are included, in particular, concerning option pricing with differential interest rates.

In this paper we improve an earlier result by Bukhgeim and Uhlmann We follow the general strategy of [1] but use a richer set of solutions to the Dirichlet problem. This implies a similar result for the problem of Electrical Impedance Tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary.