Publications

2015

• Market activity and price impact throughout time scales
  
  • Jaisson Thibault
  , 2015. This thesis tackles several issues raised by the multi-scale properties of financial data. It consists of four connected parts which can however be read independently. In the first part, we introduce the Rough Fractional Stochastic Volatility (RFSV) model. In this framework, the logarithm of the volatility is a fractional Ornstein-Uhlenbeck process with long mean reversion time scale and small Hurst index. We show that this model reproduces numer- ous stylized facts of financial data such as implied volatility properties, the autocorrelation structure and the regularity of the volatility. In particular, the RFSV sheds some new light on the supposed long memory property of volatility. The second part deals with the long-time behavior of Hawkes and autoregressive processes which are nearly unstable, that is whose kernels have a norm close to one. We show that in the case where the kernel has a light tail, nearly unstable Hawkes processes asymptotically behaves as integrated Cox-Ingersoll-Ross (CIR) processes. In the case where it has a power law shape, depending on the power law exponent, the limiting process asymptotically exhibits either long memory or a rough behavior. This enables us to use the modeling of the order flow as a Hawkes process to obtain a microstructural foundation of the rough nature of volatility. Similar results are obtained for autoregressive processes. In the third part of this thesis, we propose two estimation procedures for the kernels of Hawkes processes. These algorithms are in particular relevant to the case of slowly decreasing Hawkes kernels which corresponds to what is observed on financial data. The first method relies on the inversion of the Wiener-Hopf equation while the second is based on a likelihood maximization via a stochastic gradient ascent. Finally, we are interested in deriving, under very general assumptions, consequences of market efficiency on the structure of price impact. We prove the existence of a model independent fair price with respect to which the average ex post gain of limit orders must be equal to zero. We finally show that, under a few additional assumptions, price impact must be proportional to the market anticipation of the order flow imbalance.
• Estimating the Template in the Total Space with the Fréchet Mean on Quotient Spaces may have a Bias: a Case Study on Vector Spaces Quotiented by the Group of Translations
  
  • Allassonnière Stéphanie
  • Devilliers Loïc
  • Pennec Xavier
  , 2015, pp.131-142. When we have a deformation group acting on a vector space of observations, these data are not anymore elements of our space but rather orbits for the group action we consider. If the data are generated from an unknown template with noise, to estimate this template, one may want to minimize the variance in the quotient set. In this article we study statistics on a particular quotient space. We prove that the expected value of a random variable in our vector space mapped in the quotient space is different from the Fréchet mean in the quotient space when the observations are noisy.
• Mixed-effects model for the spatiotemporal analysis of longitudinal manifold-valued data
  
  • Schiratti Jean-Baptiste
  • Allassonnière Stéphanie
  • Colliot Olivier
  • Durrleman Stanley
  , 2015. In this work , we propose a generic hierarchical spatiotem-poral model for longitudinal manifold-valued data , which consist in repeated measurements over time for a group of individuals. This model allows us to estimate a group-average trajectory of progression , considered as a geodesic of a given Riemannian manifold. Individual trajectories of progression are obtained as random variations , which consist in parallel shifting and time reparametrization , of the average trajectory. These spatiotemporal tranformations allow us to characterize changes in the direction and in the pace at which trajectories are followed. We propose to estimate the parameters of the model using a stochastic approximation of the expectation-maximization (EM) algorithm , the Monte Carlo Markov Chain Stochastic Approximation EM (MCMC SAEM) algorithm. This generic spatiotemporal model is used to analyze the temporal progression of a family of biomarkers. This progression model estimates a normative scenario of the progressive impairments of several cognitive functions , considered here as biomarkers , during the course of Alzheimer ' s disease. The estimated average trajectory provides a normative scenario of disease progression. Random effects provide unique insights into the variations in the ordering and timing of the succession of cognitive impairments across different individuals .
• Homogenization of composite ferromagnetic materials
  
  • Alouges François
  • Di Fratta Giovanni
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2015, 471 (2182), pp.20150365. (10.1098/rspa.2015.0365)
  DOI : 10.1098/rspa.2015.0365
• A scalable algebraic method to infer quadratic invariants of switched systems
  
  • Allamigeon Xavier
  • Gaubert Stéphane
  • Goubault Eric
  • Putot Sylvie
  • Stott Nikolas
  , 2015. We present a new numerical abstract domain based on ellip- soids designed for the formal verification of switched linear systems. Unlike the existing approaches, this domain does not rely on a user-given template. We overcome the diffi- culty that ellipsoids do not have a lattice structure by ex- hibiting a canonical operator over-approximating the union. This operator is the only one which permits to perform anal- yses that are invariant with respect to a linear transforma- tion of state variables. Moreover, we show that this operator can be computed efficiently using basic algebraic operations on positive semidefinite matrices. We finally develop a fast non-linear power-type algorithm, which allows one to de- termine sound quadratic invariants on switched systems in a tractable way, by solving fixed point problems over the space of ellipsoids. We test our approach on several bench- marks, and compare it with the standard techniques based on linear matrix inequalities, showing an important speedup on typical instances. (10.1109/EMSOFT.2015.7318262)
  DOI : 10.1109/EMSOFT.2015.7318262
• DLA based compressed sensing for high resolution MR microscopy of neuronal tissue
  
  • Nguyen Khieu-Van
  • Li Jing-Rebecca
  • Radecki Guillaume
  • Ciobanu Luisa
  Journal of Magnetic Resonance, Elsevier, 2015, 259, pp.186-191. (10.1016/j.jmr.2015.08.012)
  DOI : 10.1016/j.jmr.2015.08.012
• Path dependent partial differential equation: theory and applications
  
  • Ren Zhenjie
  , 2015. In the previous works, Ekren, Keller, Touzi & Zhang [35] and Ekren, Touzi & Zhang [37, 38], the new notion of viscosity solutions to path dependent PDEs is introduced, and a wellposedness theory is proved by a ‘path-frozen’ argument. This new notion generalizes that of viscosity solutions to PDEs developed intensively in the years of 80’s and 90’s, and can be used to characterize the value function of non-Markovian stochastic control problem. In this thesis, we report the recent development of the new theory. We improve the argument for the comparison result, and provide a PDE-style Perron’s method for proving the existence of viscosity solutions to semi- linear path dependent PDEs. As in the seminar work of Barles and Souganidis [4] in the context of PDEs, we show that a family of numerical schemes satisfying the so- called monotonicity condition provides numerical solutions converging to viscosity solutions of fully nonlinear path dependent PDEs. Further, we develop a notion of elliptic path dependent PDEs, and provide a wellposedness theory by following the lines of arguments in [38]. This thesis also includes some interesting applications of path dependent PDEs. One of them is on the large deviations of non-Markovian dif- fusion. As Fleming used the stability of viscosity solutions of PDEs to establish the large deviation principle in Markovian case (see [51]), we use the theory of backward stochastic differential equations and that of path dependent PDEs to generalize his result for non-Markovian diffusions. Moreover, the large deviation result is applied to investigate the short maturity asymptotics of the implied volatility surface in financial mathematics. Finally, a study of dual algorithm for stochastic control pro- blems is presented. As Monte-Carlo simulations for the stochastic control problems provide low-biased estimate, a dual algorithm offer upper bounds of the true values. The idea of ‘path-frozen’ is exploited to give a dual representation of non-Markovian stochastic control problems.
• Experimental optimization of upwind sail trim compared to computational FSI results
  
  • Sacher Matthieu
  • Hauville Frédéric
  • Duvigneau Régis
  • Le Maitre Olivier
  • Aubin Nicolas
  • Durand Mathieu
  , 2015.
• Explicit reconstruction of Riemann surface with given boundary in complex projective space
  
  • Agaltsov Alexey
  • Khenkine Guennadi
  The Journal of Geometric Analysis, Springer, 2015, 25 (4), pp.2450–2473. In this paper we propose a numerically realizable method for reconstruction of a complex curve with known boundary and without compact components in complex projective space. (10.1007/s12220-014-9522-1)
  DOI : 10.1007/s12220-014-9522-1
• Second order Pontryagin's principle for stochastic control problems
  
  • Bonnans Joseph Frederic
  , 2015, pp.19. We discuss stochastic optimal control problems whose volatility does not depend on the control, and which have finitely many equality and inequality constraints on the expected value of functions of the final state, as well as control constraints. The main result is a proof of necessity of some second order optimality conditions involving Pontryagin multipliers.
• Modélisation probabiliste et éco-évolutionnaire des communautés proies-prédateurs
  
  • Costa Manon
  , 2015. Cette thèse porte sur la modélisation mathématique et l'étude rigoureuse de l'impact de la sélection naturelle sur les communautés proies-prédateurs. Dans une première partie, nous étudions la coévolution de phénotypes des proies et des prédateurs sous les hypothèses des dynamiques adaptatives (grande population, mutations rares et de petite amplitude). A l'aide de différentes limites d'échelle d'un processus microscopique, nous introduisons successivement un processus de saut pur décrivant les états d'équilibres successifs de la dynamique coévolutive en fonction de l'arrivée des mutations des proies ou des prédateurs, puis un système de deux équations différentielles couplées représentant l'évolution des phénotypes lorsque les mutations sont de faible amplitude. Nous illustrons ces résultats sur un modèle écologique prenant en compte plusieurs types de défenses des proies. Dans une seconde partie, nous nous intéressons à des communautés dans lesquelles les dynamiques démographiques et évolutionnaires des prédateurs sont plus rapides que celles de leurs proies (e.g. arbres-insectes). Nous modélisons la communauté par un processus déterministe par morceaux (PDMP) dans lequel les proies évoluent selon un processus de naissance et mort et les prédateurs selon une équation différentielle logistique. Ce processus décrit les dynamiques démographiques de la communauté lorsque la population de prédateurs est infinie et nous étudions son comportement stationnaire. Dans une asymptotique de petite masse des prédateurs, le processus lent-rapide converge vers une processus moyenné, dans lequel la population de prédateurs est toujours à son équilibre démographique. Afin de prendre en compte l'évolution phénotypique de la population de prédateurs, nous considérons un processus lent-rapide en dimension infinie constitué d'un processus de naissance et mort couplé avec la solution d'une équation de réaction diffusion. Nous étudions la convergence, dans une limite de petite masse des prédateurs, du processus des proies vers un processus de naissance et mort dépendant uniquement de l'équilibre stationnaire de la population de prédateurs.
• Construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography
  
  • Chesnel Lucas
  • Hyvönen Nuutti
  • Staboulis Stratos
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2015. We explain how to build invisible isotropic conductivity perturbations of the unit conductivity in the framework of the point electrode model for two-dimensional electrical impedance tomography. The theoretical approach, based on solving a fixed point problem, is constructive and allows the implementation of an algorithm for approximating the invisible perturbations. The functionality of the method is demonstrated via numerical examples.
• Qualitative methods for heterogeneous media
  
  • Audibert Lorenzo
  , 2015. This thesis focuses on non destructive testing of concrete using ultrasonic waves, and thus examines imaging in complex heterogeneous media. We assume that measurements are multistatic, which means that we record the total field on different points by using several sources. For this type of data we wish to build methods that are able to image inclusions or defects that contributed to the measured field. We focus in this work on the extension of so called sampling methods to deal with the over-mentioned application where the main additional difficulty is the lack of knowledge of the reference media (media without defects, also referred to as background media). The first part of this thesis consists of a new theoretical analysis of the Linear Sampling Method leading to new mathematically sound formulation of this method. Such analysis is done in the framework of regularization theory, and our main contribution is to provide and analyze a regularization term that ensures exact characterization of the shape in terms of measured data. We also prove that one is able to reconstruct from regularized solutions a sequence of functions that strongly converges to the solution of the so-called interior transmission problem. This result gives a central place to the interior transmission problem as it allows describing the asymptotic behaviour of our regularized problem. More importantly it also allows us to compare solutions coming from two different datasets. Based on the result of this comparison, we manage to produce an image of the connected components of the background that contain the defects appearing between two measurement campaigns and this is regardless of background “microstructure”. This strategy is well suited for applications to concrete- like backgrounds as shown on several numerical examples with realistic concrete-like microstructures. Finally, we extend our theoretical results to the case of limited aperture, anisotropic medium and elastic waves, which correspond to the real physics of the ultrasounds.
• Curve-and-Surface Evolutions for Image Processing
  
  • Mercier Gwenael
  , 2015. The goal of this manuscript was to study several problems which appear in image processing and which involves hypersurfaces of the Euclidian space R^n. Denoising a image basically consists in smoothing its lines. This smoothing can appear either as a minimizer of a suitable functional or results from a regularizing flow on the level sets of the image. In this thesis, we study two examples of these approaches. In the first chapter, we smooth by minimization. More precisely, we work on generalizations of the procedure suggested by Rudin Osher and Fatemi, which penalizes the total variation. We prove that under different assumptions on the domain, on the way to link the image to the data and on the choice of the total variation (isotropic, anisotropic,...), the continuity of the source image is preserved by the minimizing procedure. In Chapter 2, we study Mean curvature flow and add some obstacles which constraint the evolution. We choose the level-set approach: the surface is the preimage of 0 by a function which therefore satisfies a PDE. We prove existence and uniqueness of a (viscosity) solution for this equation. and study its asymptotic in time using comparison with a discrete minimizing scheme. In Chapter 3 (with M. Novaga), we add some information to the result of Chapter 2 by focusing on the geometric formulation of the mean curvature flow with obstacles. We follow the approach by Ecker and Huisken to show that there exists a unique solution of the motion in short times. Finally, in the last chapter (with M. Novaga and P. Pozzi), we make a first step towards the understanding of crystalline motion. Restricted to the planar framework, we show (using an approximation by a smooth motion) that there exists a short time of existence for an anisotropic curvature motion of an immersed curve.
• Assimilation de Données et Mesures Primaires REP
  
  • Mercier Thibaud
  , 2015. Le circuit primaire d'un réacteur à eau pressurisée (REP) est un système thermohydraulique complexe, présentant des champs hétérogènes de puissances, températures et débits dans des conditions extrêmes : l'instrumentation in situ est limitée en nombre, localisation et précision. De ce fait, la connaissance de ces paramètres de fonctionnement est impactée par des incertitudes de représentativité notamment, prises en compte dans la conception et intégrées dans les protections d'exploitation. Dans ce contexte, EDF R&D cherche à évaluer les apports potentiels de l'Assimilation de Données, un ensemble d'outils mathématiques très utilisé en géosciences qui permettent de corriger des estimations issus de modèles à l'aide de mesures provenant du système réel modélisé, pour une meilleure caractérisation (justesse, incertitudes) des paramètres du point de fonctionnement primaire nominal. Dans cette thèse, nous définissons un modèle semi-empirique 0D adapté au niveau de description usuellement choisi par les exploitants, et développons une méthodologie (de type Monte-Carlo et inspirée des Méthodes d'Ensemble) pour utiliser ce modèle dans un cadre d'Assimilation de Données. L'application de cette méthodologie à des données simulées permet d'évaluer la réduction des incertitudes pesant sur les paramètres-clés : les résultats sont au-delà des espérances initiales mais nécessitent des hypothèses fortes impliquant un prétraitement soigneux des données d'entrée.
• Performance evaluation of an emergency call center: tropical polynomial systems applied to timed Petri nets
  
  • Allamigeon Xavier
  • Boeuf Vianney
  • Gaubert Stéphane
  , 2015, 9268. (10.1007/978-3-319-22975-1_2)
  DOI : 10.1007/978-3-319-22975-1_2
• BocopHJB 1.0.1 – User Guide
  
  • Bonnans Frédéric
  • Giorgi Daphné
  • Heymann Benjamin
  • Martinon Pierre
  • Tissot Olivier
  , 2015, pp.24. The original Bocop package implements a local optimization method. The optimal control problem is approximated by a finite dimensional optimization problem (NLP) using a time discretization (the direct transcription approach). The NLP problem is solved by the well known software Ipopt, using sparse exact derivatives computed by Adol-C. The second package BocopHJB implements a global optimization method. Similarly to the Dynamic Programming approach, the optimal control problem is solved in two steps. First we solve the Hamilton-Jacobi-Bellman equation satisfied by the value fonction of the problem. Then we simulate the optimal trajectory from any chosen initial condition. The computational effort is essentially taken by the first step, whose result, the value fonction, can be stored for subsequent trajectory simulations.
• Non line of sight signal analysis: Investigation of interferometry modes over urban area
  
  • Nouvel Jean-François
  • Dupuis Xavier
  • Lesturgie Marc
  , 2015, pp.pp 582-586. Due to complex geometry (urban canyons generate multiple scattering) SAR images are difficult to interpretate on urban areas. In order to characterize Non Line Of Sight (NLOS) scattering phenomenon at Ka band, measurement campaigns have been performed using the ONERA BUSARD motor glider and a dedicated Ka band SAR sensor. From 2011 to 2014 a large number of measurements flights have been done, leading to the acquisition of a radar dataset that covers different geometries and different acquisition modes: Single channel, cross-track or along-track interferometry modes. NLOS radar echoes have been investigated on two types of materials: concrete and metallic walls and different geometries of acquisition. NLOS echoes observation has been validated in various cases and interferometry modes have been proved to be efficient in NLOS echoes detection and validation. (10.1109/APSAR.2015.7306276)
  DOI : 10.1109/APSAR.2015.7306276
• Ergodicity conditions for zero-sum games
  
  • Akian Marianne
  • Gaubert Stephane
  • Hochart Antoine
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.31. A basic question for zero-sum repeated games consists in determining whether the mean payoff per time unit is independent of the initial state. In the special case of "zero-player" games, i.e., of Markov chains equipped with additive functionals, the answer is provided by the mean ergodic theorem. We generalize this result to repeated games. We show that the mean payoff is independent of the initial state for all state-dependent perturbations of the rewards if and only if an ergodicity condition is verified. The latter is characterized by the uniqueness modulo constants of non-linear harmonic functions (fixed point of the recession operator of the Shapley operator), or, in the special case of stochastic games with finite action spaces and perfect information, by a reachability condition involving conjugated subsets of states in directed hypergraphs. We show that the ergodicity condition for games only depend on the support of the transition probability, and that it can be checked in polynomial time when the number of states is fixed. (10.3934/dcds.2015.35.3901)
  DOI : 10.3934/dcds.2015.35.3901
• Stratified discontinuous differential equations and sufficient conditions for robustness
  
  • Hermosilla Cristopher
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, 35 (9), pp.23. This paper is concerned with state-constrained discontinuous ordinary differential equations for which the corresponding vector field has a set of singularities that forms a stratification of the state domain. Existence of solutions and robustness with respect to external perturbations of the righthand term are investigated. Moreover, notions of regularity for stratifications are discussed. (10.3934/dcds.2015.35.4415)
  DOI : 10.3934/dcds.2015.35.4415
• Learning the intensity of time events with change-points
  
  • Alaya Mokhtar Z.
  • Gaïffas Stéphane
  • Guilloux Agathe
  IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2015, 61 (9), pp.5148-5171. We consider the problem of learning the inhomogeneous intensity of a counting process, under a sparse segmentation assumption. We introduce a weighted total-variation penalization, using data-driven weights that correctly scale the penalization along the observation interval. We prove that this leads to a sharp tuning of the convex relaxation of the segmentation prior, by stating oracle inequalities with fast rates of convergence, and consistency for change-points detection. This provides first theoretical guarantees for segmentation with a convex proxy beyond the standard i.i.d signal + white noise setting. We introduce a fast algorithm to solve this convex problem. Numerical experiments illustrate our approach on simulated and on a high-frequency genomics dataset.
• Can Magnetic Multilayers Propel Artificial Microswimmers Mimicking Sperm Cells?
  
  • Alouges François
  • Desimone Antonio
  • Giraldi Laetitia
  • Zoppello Marta
  Soft Robotics, Mary Ann Liebert, Inc., 2015, 2 (3), pp.117-128. (10.1089/soro.2015.0007)
  DOI : 10.1089/soro.2015.0007
• Majorization inequalities for valuations of eigenvalues using tropical algebra
  
  • Akian Marianne
  , 2015. We consider a matrix with entries over the field of Puiseux series, equipped with its non-archimedean valuation (the leading exponent). We establish majorization inequalities relating the sequence of the valuations of the eigenvalues of a matrix with the tropical eigenvalues of its valuation matrix (the latter is obtained by taking the valuation entrywise). We also show that, generically in the leading coefficients of the Puiseux series, the precise asymptotics of eigenvalues, eigenvectors and condition numbers can be determined. For this, we apply diagonal scalings constructed from the dual variables of a parametric optimal assignment constructed from the valuation matrix. Next, we establish an archimedean analogue of the above inequalities, which applies to matrix polynomials with coefficients in the field of complex numbers, equipped with the modulus as its valuation. This talk covers joint works with Ravindra Bapat, Stéphane Gaubert, Andrea Marchesini, and Francoise Tisseur.
• Supertropical SLn
  
  • Izhakian Zur
  • Niv Adi
  • Rowen Louis
  , 2015. Extending earlier work on supertropical adjoints and applying symmetrization, we provide a symmetrized supertropical version SLn of the special linear group, which we partition into submonoids, based on " quasi-identity " matrices, and we display maximal sub-semigroups of SLn. We also study the monoid generated by SLn. Several illustrative examples are given of unexpected behavior. We describe the action of elementary matrices on SLn, which enables one to connect different matrices in SLn, but in a weaker sense than the classical situation.
• Tropical diagonal scaling for asymptotic eigenvalue problems
  
  • Marchesini Andrea
  , 2015. We study the behaviour of the eigenvalues of a parametric matrix polynomial P in a neighbourhood of zero. If we suppose that the entries of P have Puiseux series expansion, we can build an auxiliary matrix polynomial Q whose entries are the leading exponents of those of P. We show that preconditioning P via a diagonal scaling based on the tropical eigenvalues of Q can improve conditioning and backward error of the eigenvalues.