Publications

2014

• The curvature of optimal control problems with applications to sub-Riemannian geometry
  
  • Rizzi Luca
  , 2014. Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
• A Robust and Entropy-Satisfying Numerical Scheme for Fluid Flows in Discontinuous Nozzles
  
  • Coquel Frédéric
  • Saleh Khaled
  • Seguin Nicolas
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2014, 24 (10), pp.2043-2083. We propose in this work an original finite volume scheme for the system of gas dynamics in a nozzle. Our numerical method is ased on a piecewise constant discretization of the cross-section and on a approximate Riemann solver in the sense of Harten, Lax and van Leer. The solver is obtained by the use of a relaxation approximation that leads to a positive and entropy satisfying numerical scheme for all variation of section, even discontinuous with arbitrary large jumps. To do so, we introduce in the first step of the relaxation solver a singular dissipation measure superposed on the standing wave which enables us to control the approximate speeds of sound and thus, the time step, even for extreme initial data. (10.1142/S0218202514500158)
  DOI : 10.1142/S0218202514500158
• Log-majorization of eigenvalues of matrix polynomials and tropical scaling
  
  • Akian Marianne
  , 2014.
• Getting the most out of it: optimal experiments for parameter estimation of microalgae growth models
  
  • Munoz Tamayo Rafael
  • Martinon Pierre
  • Bougaran Gaël
  • Mairet Francis
  • Bernard Olivier
  Journal of Process Control, Elsevier, 2014, 24 (6), pp.991-1001. Mathematical models are expected to play a pivotal role for driving microalgal production towards a profitable process of renewable energy generation. To render models of microalgae growth useful tools for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. In this paper, we provide guidelines for the design of experiments with high informative content based on optimal experiment techniques to attain an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was formulated and solved with both static (constant light and temperature) and dynamic (time varying light and temperature) approaches. Simulation results indicated that the optimal experiment design allows for a more accurate parameter estimation than that provided by the existing experimental protocol. For its efficacy in terms of the maximum likelihood properties and its practical aspects of implementation, the dynamic approach is recommended over the static approach. (10.1016/j.jprocont.2014.04.021)
  DOI : 10.1016/j.jprocont.2014.04.021
• Shortfall risk minimization in discrete time financial market models
  
  • Frikha Noufel
  SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics, 2014, 5 (1), pp.384–414. In this paper, we study theoretical and computational aspects of risk minimization in financial market models operating in discrete time. To define the risk, we consider a class of convex risk measures defined on $L^{p}$ in terms of shortfall risk. Under simple assumptions, namely the absence of arbitrage opportunity and the non-degeneracy of the price process, we prove the existence of an optimal strategy by performing a dynamic programming argument in a non-Markovian framework. Optimal strategies are shown to satisfy a first order condition involving the constructed Bellman functions. In a Markovian framework, we propose and analyze several algorithms based on Monte Carlo simulations to estimate the shortfall risk and optimal dynamic strategies. Finally, we illustrate our approach by considering several shortfall risk measures and portfolios inspired by energy and financial markets. (10.1137/120903142)
  DOI : 10.1137/120903142
• The horoboundary and isometry group of Thurston's Lipschitz metric
  
  • Walsh Cormac
  , 2014, 19, pp.838. We show that the horofunction boundary of Teichmüller space with Thurston's Lipschitz metric is the same as the Thurston boundary. We use this to determine the isometry group of the Lipschitz metric, apart from in some exceptional cases. We also show that the Teichmüller spaces of different surfaces, when endowed with this metric, are not isometric, again with some possible exceptions of low genus.
• A novel description of FDG excretion in the renal system: application to metformin-treated models
  
  • Garbarino S
  • Caviglia G
  • Sambuceti G
  • Benvenuto F
  • Piana M
  Magnetic Resonance Materials in Physics, Biology and Medicine, Springer Verlag, 2014, 59 (10), pp.25. (10.1088/0031-9155/59/10/2469)
  DOI : 10.1088/0031-9155/59/10/2469
• Long and winding central paths
  
  • Allamigeon Xavier
  • Benchimol Pascal
  • Gaubert Stéphane
  • Joswig Michael
  , 2014. We disprove a continuous analog of the Hirsch conjecture proposed by Deza, Terlaky and Zinchenko, by constructing a family of linear programs with 3r+4 inequalities in dimension 2r+2 where the central path has a total curvature in Ω(2r/r). Our method is to tropicalize the central path in linear programming. The tropical central path is the piecewise-linear limit of the central paths of parameterized families of classical linear programs viewed through logarithmic glasses. We show in particular that the tropical analogue of the analytic center is nothing but the tropical barycenter, i.e., the maximum of a tropical polyhedron. It follows that unlike in the classical case, the tropical central path may lie on the boundary of the tropicalization of the feasible set, and may even coincide with a path of the tropical simplex method. Finally, our counter-example is obtained as a deformation of a family of tropical linear programs introduced by Bezem, Nieuwenhuis and Rodríguez-Carbonell.
• Le système de Ptolémée avec un petit cercle gris clair -l'épicycle- dont le centre décrit un cercle plus grand gris sombre -le déférend- dont le centre est la Terre -sphère bleue
  
  • Colonna Jean-Francois
  , 2014. The Ptolemaic system with a small light grey circle -the epicycle- whose center describes a larger dark grey circle -the deferend- centered on the Earth -blue sphere- (Le système de Ptolémée avec un petit cercle gris clair -l'épicycle- dont le centre décrit un cercle plus grand gris sombre -le déférend- dont le centre est la Terre -sphère bleue-)
• A new sequential algorithm for L2-approximation and application to Monte-Carlo integration
  
  • Gobet Emmanuel
  • Surana Khushboo
  , 2014. We design a new stochastic algorithm (called SALT) that sequentially approximates a given function in L2 w.r.t. a probability measure, using a finite sample of the distribution. By increasing the sets of approximating functions and the simulation effort, we compute a L2-approximation with higher and higher accuracy. The simulation effort is tuned in a robust way that ensures the convergence under rather general conditions. Then, we apply SALT to build efficient control variates for accurate numerical integration. Examples and numerical experiments support the mathematical analysis.
• Optimal Strokes for Driftless Swimmers: A General Geometric Approach
  
  • Chambrion Thomas
  • Giraldi Laetitia
  • Munnier Alexandre
  , 2014. This paper presents a unified geometric approach for the optimiza- tion of the shape deformations of the so-called driftless swimmers. The class of driftless swimmers includes, among other, isolated swimmers in an infinite 3D Stokes flow (case of micro-swimmers in viscous fluids) or isolated swimmers in an infinite 2D or 3D potential flow. A general framework is introduced, in which five usual nonlinear optimization problems related with the maximization of the traveled distance with a constrained energy consumption) are stated. We prove the existence of regular minimizers under generic con- trollability assumptions. The results are illustrated with an in-depth study of the isolated swimmer with two degrees of freedom in a 2D potential flow.
• Probabilistic Atlas and Geometric Variability Estimation to Drive Tissue Segmentation
  
  • Xu Hao
  • Thirion Bertrand
  • Allassonnière Stéphanie
  Statistics in Medicine, Wiley-Blackwell, 2014, 33 (20), pp.24. Computerized anatomical atlases play an important role in medical image analysis. While an atlas usually refers to a standard or mean image also called template, that presumably represents well a given population, it is not enough to characterize the observed population in detail. A template image should be learned jointly with the geometric variability of the shapes represented in the observations. These two quantities will in the sequel form the atlas of the corresponding population. The geometric variability is modelled as deformations of the template image so that it fits the observations. In this paper, we provide a detailed analysis of a new generative statistical model based on dense deformable templates that represents several tissue types observed in medical images. Our atlas contains both an estimation of probability maps of each tissue (called class) and the deformation metric. We use a stochastic algorithm for the estimation of the probabilistic atlas given a dataset. This atlas is then used for atlas-based segmentation method to segment the new images. Experiments are shown on brain T1 MRI datasets. (10.1002/sim.6156)
  DOI : 10.1002/sim.6156
• Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations
  
  • Bayen Térence
  • Bonnans J. Frederic
  • Silva Francisco J.
  Transactions of the American Mathematical Society, American Mathematical Society, 2014, 366 (4), pp.2063--2087. In this article we consider an optimal control problem of a semi-linear elliptic equation, with bound constraints on the control. Our aim is to characterize local quadratic growth for the cost function $J$ in the sense of strong solutions. This means that the function $J$ growths quadratically over all feasible controls whose associated state is close enough to the nominal one, in the uniform topology. The study of strong solutions, classical in the Calculus of Variations, seems to be new in the context of PDE optimization. Our analysis, based on a decomposition result for the variation of the cost, combines Pontryagin's principle and second order conditions. While these two ingredients are known, we use them in such a way that we do not need to assume that the Hessian of Lagrangian of the problem is a Legendre form, or that it is uniformly positive on an extended set of critical directions. (10.1090/S0002-9947-2013-05961-2)
  DOI : 10.1090/S0002-9947-2013-05961-2
• The contraction rate in Thompson part metric of order-preserving flows on a cone - application to generalized Riccati equations
  
  • Gaubert Stéphane
  • Qu Zheng
  Journal of Differential Equations, Elsevier, 2014, 256 (8), pp.2902-2948. We give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This provides an explicit form of a characterization of Nussbaum concerning non order-preserving flows. As an application of this formula, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other natural Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation. (10.1016/j.jde.2014.01.024)
  DOI : 10.1016/j.jde.2014.01.024
• Tropical bounds for eigenvalues of matrices
  
  • Akian Marianne
  • Gaubert Stéphane
  • Marchesini Andrea
  Linear Algebra and its Applications, Elsevier, 2014, 446, pp.281–303. We show that for all k = 1,...,n the absolute value of the product of the k largest eigenvalues of an n-by-n matrix A is bounded from above by the product of the k largest tropical eigenvalues of the matrix |A| (entrywise absolute value), up to a combinatorial constant depending only on k and on the pattern of the matrix. This generalizes an inequality by Friedland (1986), corresponding to the special case k = 1. (10.1016/j.laa.2013.12.021)
  DOI : 10.1016/j.laa.2013.12.021
• Effectivized Holder-logarithmic stability estimates for the Gel'fand inverse problem
  
  • Isaev Mikhail
  • Novikov Roman
  Inverse Problems, IOP Publishing, 2014, 30 (9), pp.19. We give effectivized Holder-logarithmic energy and regularity dependent stability estimates for the Gel'fand inverse boundary value problem in dimension $d=3$. This effectivization includes explicit dependance of the estimates on coefficient norms and related parameters. Our new estimates are given in $L^2$ and $L^\infty$ norms for the coefficient difference and related stability efficiently increases with increasing energy and/or coefficient difference regularity. Comparisons with preceeding results are given.
• Probabilistic atlas statistical estimation with multimodal datasets and its application to atlas based segmentation
  
  • Xu Hao
  , 2014. Computerized anatomical atlases play an important role in medical image analysis. While an atlas usually refers to a standard or mean image also called template, that presumably represents well a given population, it is not enough to characterize the observed population in detail. A template image should be learned jointly with the geometric variability of the shapes represented in the observations. These two quantities will in the sequel form the atlas of the corresponding population. The geometric variability is modelled as deformations of the template image so that it fits the observations. In the first part of the work, we provide a detailed analysis of a new generative statistical model based on dense deformable templates that represents several tissue types observed in medical images. Our atlas contains both an estimation of probability maps of each tissue (called class) and the deformation metric. We use a stochastic algorithm for the estimation of the probabilistic atlas given a dataset. This atlas is then used for atlas-based segmentation method to segment the new images. Experiments are shown on brain T1 MRI datasets. Traditional analyses of Functional Magnetic Resonance Imaging use little anatomical information. The registration of the images to a template is based on the individual anatomy and ignores functional information; subsequently detected activations are not confined to gray matter. In the second part of the work, we propose a statistical model to estimate a probabilistic atlas from functional and T1 MRIs that summarizes both anatomical and functional information and the geometric variability of the population. Registration and Segmentation are performed jointly along the atlas estimation and the functional activity is constrained to the gray matter, increasing the accuracy of the atlas. Inferring protein abundances from peptide intensities is the key step in quantitative proteomics. The inference is necessarily more accurate when many peptides are taken into account for a given protein. Yet, the information brought by the peptides shared by different proteins is commonly discarded. In the third part of the work, we propose a statistical framework based on a hierarchical modeling to include that information. Our methodology, based on a simultaneous analysis of all the quantified peptides, handles the biological and technical errors as well as the peptide effect. In addition, we propose a practical implementation suitable for analyzing large datasets. Compared to a method based on the analysis of one protein at a time (that does not include shared peptides), our methodology proved to be far more reliable for estimating protein abundances and testing abundance changes.
• Comparison of multiobjective gradient-based methods for structural shape optimization
  
  • Giacomini Matteo
  • Désidéri Jean-Antoine
  • Duvigneau Régis
  , 2014, pp.26. This work aims at formulating a shape optimization problem within a multiobjective optimization framework and approximating it by means of the so-called Multiple-Gradient Descent Algorithm (MGDA), a gradient-based strategy that extends classical Steepest-Descent Method to the case of the simultaneous optimization of several criteria. We describe several variants of MGDA and we apply them to a shape optimization problem in linear elasticity using a numerical solver based on IsoGeometric Analysis (IGA). In particular, we study a multiobjective gradient-based method that approximates the gradients of the functionals by means of the Finite Difference Method; kriging-assisted MGDA that couples a statistical model to predict the values of the objective functionals rather than actually computing them; a variant of MGDA based on the analytical gradients of the functionals extracted from the NURBS -based parametrization of the IGA solver. Some numerical simulations for a test case in computational mechanics are carried on to validate the methods and a comparative analysis of the results is presented.
• An approach to improve ill-conditioned steepest descent methods, application to a parabolic optimal control problem via time domain decomposition
  
  • Riahi Mohamed-Kamel
  Applied Mathematics and Computation, Elsevier, 2014. In this paper we present a new steepest-descent type algorithm for convex optimization problems. The method combines a Newton technique together with time domain decomposition in order to achieve the optimal step-length for the given set of descent directions. This is a parallel algorithm, where the parallel tasks turn on the control during a specific time-window and turn it off elsewhere. This new technique significantly improves computational time compared with recognized methods. Convergence analysis of the algorithm is provided for an arbitrary choice of partition. Numerical experiments are presented to illustrate the efficiency of our algorithm.
• Points fixes d’opérateur de Shapley sans paiement et propriétés structurelles des jeux à paiement moyen
  
  • Akian Marianne
  • Gaubert Stephane
  • Hochart Antoine
  , 2014.
• Optimal control problems on stratifiable state constraints sets.
  
  • Hermosilla Cristopher
  • Zidani Hasnaa
  , 2014. We consider an infinite horizon problem with state constraints K : inf Z 1 0 e t'(yx;u(t); u(t))dt u : [ 0 ;+1) ! A measurable yx;u(t) 2 K 8t 0 (P) : where > 0 is fixed and yx;u( ) is a trajectory of the control system ( y_ = f (y; u) a.e. t 0 y(0) = x 2 K We are mainly concerned with a characterization of the value function of (P) as the bilateral solution to a Hamilton-Jacobi-Bellman equation.
• Martingale Optimal Transport and Utility Maximization
  
  • Guillaume Royer
  , 2014. This PhD dissertation presents two independent research topics dealing with contemporary issues from financial mathematics, the second one being composed of two distinct problems. In the first part we study the question of martingale optimal transport, which comes from the questions of no-arbitrage optimal bounds of liabilities. We first consider the question in discret time of the existence of a martingale law with given marginals. This result was first proved by Strassen (1965) and is the starting point of martingale optimal transport. We provide a new proof of this theorem based on utility maximization technics, adapted from a proof of the fundamental theorem of asset pricing by Rogers. We then consider the question of martingale optimal transport in continuous time, introduced in the framework of lookback options by Galichon, Henry-Labordère et Touzi. We first establish a partial duality result concerning the robust superhedging of any contingent claim. For that purpose, we adapt recent technics developed by Neufeld and Nutz in the context of martingale optimal transport. In a second time we study a robust utility maximization of a contingent claim with exponential utility in the context of martingale optimal transport, and we deduce its robust utility indifference price, given that the underlying's dynamic has a constant and well-known sharpe ratio. We prove that this robust utility indifference price is equal to the robust superhedging price. The second part of this disseration considers first the problem of optimal liquidation of an indivisible asset. We study the advantage that an agent can take from having a dynamic trading strategy in an orthogonal asset. The question of its influence on the optimal liquidation rule is asked. We then provide examples illustrating our results. The last chapter of this thesis concerns the utility indifference price of a European option in the context of small transaction costs. We use technics developed by Soner and Touzi to obtain an asymptotic expansion of the Merton value functions with and without the option. These expansions are obtained by using homogenization technics. We formally obtain a system of equations verified by the values involved in the expansion and show rigorously that they are solutions. We then deduce an asymptotic expansion of the utility indifference price.
• Second-order sufficient conditions for strong solutions to optimal control problems
  
  • Bonnans Joseph Frederic
  • Dupuis Xavier
  • Pfeiffer Laurent
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2014, 20 (03), pp.704-724. In this report, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem. (10.1051/cocv/2013080)
  DOI : 10.1051/cocv/2013080
• Où est l'Univers ?
  
  • Colonna Jean-Francois
  , 2014. Where is the Universe ? (Où est l'Univers ?)
• A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging of biological tissues
  
  • Nguyen Dang Van
  , 2014. Diffusion magnetic resonance imaging (dMRI) is a non-invasive imaging technique that gives a measure of the diffusion characteristics of water in biological tissues, notably, in the brain. The hindrances that the microscopic cellular structure poses to water diffusion are statistically aggregated into the measurable macroscopic dMRI signal. Inferring the microscopic structure of the tissue from the dMRI signal allows one to detect pathological regions and to monitor functional properties of the brain. For this purpose, one needs a clearer understanding of the relation between the tissue microstructure and the dMRI signal. This requires novel numerical tools capable of simulating the dMRI signal arising from complex microscopic geometrical models of tissues. We propose such a numerical method based on linear finite elements that allows for a more accurate description of complex geometries. The finite elements discretization is coupled to the adaptive Runge-Kutta Chebyshev time stepping method. This method, which leads to the second order convergence in both time and space, is implemented on FeniCS C++ platform. We also use the mesh generator Salome to work efficiently with multiple-compartment and periodic geometries. Four applications of the method for studying the dMRI signal inside multi-compartment models are considered. In the first application, we investigate the long-time asymptotic behavior of the dMRI signal and show the convergence of the apparent diffusion coefficient to the effective diffusion tensor computed by homogenization. The second application aims to numerically verify that a two-compartment model of cells accurately approximates the three-compartment model, in which the interior cellular compartment and the extracellular space are separated by a finite thickness membrane compartment. The third application consists in validating the macroscopic Karger model of dMRI signals that takes into account compartmental exchange. The last application focuses on the dMRI signal arising from isolated neurons. We propose an efficient one-dimensional model for accurately computing the dMRI signal inside neurite networks in which the neurites may have different radii. We also test the validity of a semi-analytical expression for the dMRI signal arising from neurite networks.