Publications

2016

• Self-adjoint extensions and stochastic completeness of the Laplace–Beltrami operator on conic and anticonic surfaces
  
  • Boscain Ugo
  • Prandi Dario
  Journal of Differential Equations, Elsevier, 2016, 260 (4), pp.3234–3269. We study the evolution of the heat and of a free quantum particle (described by the Schrödinger equation) on two-dimensional manifolds endowed with the degenerate Riemannian metric $ds^2=dx^2+|x|^{-2\alpha}d\theta^2$, where $x\in \mathbb{R}$, $\theta\in\mathbb{T}$ and the parameter $\alpha\in\mathbb{R}$. For $\alpha\le-1$ this metric describes cone-like manifolds (for $\alpha=-1$ it is a flat cone). For $\alpha=0$ it is a cylinder. For $\alpha\ge 1$ it is a Grushin-like metric. We show that the Laplace-Beltrami operator $\Delta$ is essentially self-adjoint if and only if $\alpha\notin(-3,1)$. In this case the only self-adjoint extension is the Friedrichs extension $\Delta_F$, that does not allow communication through the singular set $\{x=0\}$ both for the heat and for a quantum particle. For $\alpha\in(-3,-1]$ we show that for the Schrödinger equation only the average on $\theta$ of the wave function can cross the singular set, while the solutions of the only Markovian extension of the heat equation (which indeed is $\Delta_F$) cannot. For $\alpha\in(-1,1)$ we prove that there exists a canonical self-adjoint extension $\Delta_B$, called bridging extension, which is Markovian and allows the complete communication through the singularity (both of the heat and of a quantum particle). Also, we study the stochastic completeness (i.e., conservation of the $L^1$ norm for the heat equation) of the Markovian extensions $\Delta_F$ and $\Delta_B$, proving that $\Delta_F$ is stochastically complete at the singularity if and only if $\alpha\le -1$, while $\Delta_B$ is always stochastically complete at the singularity. (10.1016/j.jde.2015.10.011)
  DOI : 10.1016/j.jde.2015.10.011
• The Newtonian Potential and the Demagnetizing Factors of the General Ellipsoid
  
  • Di Fratta Giovanni
  Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2016, 472 (2190), pp.20160197. The objective of this paper is to present a modern and concise new derivation for the explicit expression of the interior and exterior Newtonian potential generated by homogeneous ellipsoidal domains in $\mathbb{R}^N$ (with $N \geqslant 3$). The very short argument is essentially based on the application of Reynolds transport theorem in connection with Green-Stokes integral representation formula for smooth functions on bounded domains of $\mathbb{R}^N$, which permits to reduce the N-dimensional problem to a 1-dimensional one. Due to its high physical relevance, a separate section is devoted to the derivation of the demagnetizing factors of the general ellipsoid which are one of the most fundamental quantities in ferromagnetism. (10.1098/rspa.2016.0197)
  DOI : 10.1098/rspa.2016.0197
• Approximation of Markov semigroups in total variation distance
  
  • Bally Vlad
  • Rey Clément
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2016, 21 (none). (10.1214/16-EJP4079)
  DOI : 10.1214/16-EJP4079
• Thickness control in structural optimization via a level set method
  
  • Allaire Grégoire
  • Jouve François
  • Michailidis Georgios
  Structural and Multidisciplinary Optimization, Springer Verlag, 2016, 53, pp.1349-1382. In the context of structural optimization via a level-set method we propose a framework to handle geometric constraints related to a notion of local thickness. The local thickness is calculated using the signed distance function to the shape. We formulate global constraints using integral functionals and compute their shape derivatives. We discuss diff erent strategies and possible approximations to handle the geometric constraints. We implement our approach in two and three space dimensions for a model of linearized elasticity. As can be expected, the resulting optimized shapes are strongly dependent on the initial guesses and on the speci fic treatment of the constraints since, in particular, some topological changes may be prevented by those constraints.
• Sub-Riemannian curvature in contact geometry
  
  • Agrachev Andrei
  • Barilari Davide
  • Rizzi Luca
  The Journal of Geometric Analysis, Springer, 2016. We compare different notions of curvature on contact sub-Riemannian manifolds. In particular we introduce canonical curvatures as the coefficients of the sub-Riemannian Jacobi equation. The main result is that all these coefficients are encoded in the asymptotic expansion of the horizontal derivatives of the sub-Riemannian distance. We explicitly compute their expressions in terms of the standard tensors of contact geometry. As an application of these results, we obtain a sub-Riemannian version of the Bonnet-Myers theorem that applies to any contact manifold. (10.1007/s12220-016-9684-0)
  DOI : 10.1007/s12220-016-9684-0
• Empirical Regression Method for Backward Doubly Stochastic Differential Equations
  
  • Bachouch Achref
  • Gobet Emmanuel
  • Matoussi Anis
  SIAM/ASA Journal on Uncertainty Quantification, ASA, American Statistical Association, 2016, 4 (1), pp.358-379. In this paper we design a numerical scheme for approximating Backward Doubly Stochastic Differential Equations (BDSDEs for short) which represent solution to Stochastic Partial Differential Equations (SPDEs). We first use a time-discretization and then, we decompose the value function on a functions basis. The functions are deterministic and depend only on time-space variables, while decomposition coefficients depend on the external Brownian motion B. The coefficients are evaluated through a empirical regression scheme, which is performed conditionally to B. We establish non asymptotic error estimates, conditionally to B, and deduce how to tune parameters to obtain a convergence conditionally and unconditionally to B. We provide numerical experiments as well. (10.1137/15M1022094)
  DOI : 10.1137/15M1022094
• Discrete Hammersley's Lines with sources and sinks
  
  • Basdevant A-L
  • Enriquez N
  • Gerin L
  • Gouéré J-B
  ALEA : Latin American Journal of Probability and Mathematical Statistics, Instituto Nacional de Matemática Pura e Aplicada (Rio de Janeiro, Brasil) [2006-....], 2016, 13 (1), pp.33-52. We introduce two stationary versions of two discrete variants of Hammersley's process in a finite box, this allows us to recover in a unified and simple way the laws of large numbers proved by T. Seppäläinen for two generalized Ulam's problems. As a by-product we obtain an elementary solution for the original Ulam problem. We also prove that for the first process defined on Z, Bernoulli product measures are the only extremal and translation-invariant stationary measures.
• An analog of Chang inversion formula for weighted Radon transforms in multidimensions
  
  • Goncharov Fedor
  • Novikov Roman
  Eurasian Journal of Mathematical and Computer Applications, Eurasian National University, Kazakhstan (Nur-Sultan), 2016, 4 (2), pp.23-32. In this work we study weighted Radon transforms in multidimensions. We introduce an analog of Chang approximate inversion formula for such transforms and describe all weights for which this formula is exact. In addition, we indicate possible tomographical applications of inversion methods for weighted Radon transforms in 3D.
• Wavelet methods for shape perception in electro-sensing
  
  • Waldspurger Irène
  • Ammari Habib
  • Mallat Stéphane
  • Wang Han
  , 2016. This paper aims at presenting a new approach to the electro-sensing problem using wavelets. It provides an efficient algorithm for recognizing the shape of a target from micro-electrical impedance measurements. Stability and resolution capabilities of the proposed algorithm are quantified in numerical simulations. Mathematics Subject Classification (MSC2000): 35R30, 35B30
• Estimation of slowly decreasing Hawkes kernels: application to high-frequency order book dynamics
  
  • Bacry Emmanuel
  • Jaisson Thibault
  • Muzy Jean-François
  Quantitative Finance, Taylor & Francis (Routledge), 2016, pp.1-23. no abstract (10.1080/14697688.2015.1123287)
  DOI : 10.1080/14697688.2015.1123287
• Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs
  
  • Gobet Emmanuel
  • Lopez-Salas Jose
  • Turkedjiev Plamen
  • Vázquez C.
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2016, 38 (6), pp.C652-C677. In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on multicore devices such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization. (10.1137/16M106371X)
  DOI : 10.1137/16M106371X
• Partial Splitting of Longevity and Financial Risks: The Longevity Nominal Choosing Swaptions
  
  • Bensusan Harry
  • El Karoui Nicole
  • Loisel Stéphane
  • Salhi Yahia
  Insurance: Mathematics and Economics, Elsevier, 2016, 68 (May 2016), pp.61-72. In this paper, we introduce a new structured financial product: the so-called Life Nominal Chooser Swaption (LNCS). Thanks to such a contract, insurers could keep pure longevity risk and transfer a great part of interest rate risk underlying annuity portfolios to financial markets. Before the issuance of the contract, the insurer determines a confidence band of survival curves for her portfolio. An interest rate hedge is set up, based on swaption mechanisms. The bank uses this band as well as an interest rate model to price the product. At the end of the first period (e.g. 8 to 10 years), the insurer has the right to enter into an interest rate swap with the bank, where the nominal is adjusted to her (re-forecasted) needs. She chooses (inside the band) the survival curve that better fits her anticipation of future mortality of her portfolio (during 15 to 20 more years, say) given the information available at that time. We use a population dynamics longevity model and a classical two-factor interest rate model %two-factor Heath-Jarrow-Morton (HJM) model for interest rates to price this product. Numerical results show that the option offered to the insurer (in terms of choice of nominal) is not too expensive in many real-world cases. We also discuss the pros and the cons of the product and of our methodology. This structure enables insurers and financial institutions to remain in their initial field of expertise.
• Enhanced Method for Diagnosing Pharmacometric Models: Random Sampling from Conditional Distributions
  
  • Lavielle Marc
  • Ribba Benjamin
  Pharmaceutical Research, American Association of Pharmaceutical Scientists, 2016. Purpose: For nonlinear mixed-effects pharmacometric models, diagnostic approaches often rely on individual parameters, also called empirical Bayes estimates (EBEs), estimated through maximizing conditional distributions. When individual data are sparse, the distribution of EBEs can ``shrink'' towards the same population value, and as a direct consequence, resulting diagnostics can be misleading. Methods: Instead of maximizing each individual conditional distribution of individual parameters, we propose to randomly sample them in order to obtain values better spread out over the marginal distribution of individual parameters. Results: We evaluated, through diagnostic plots and statistical tests, hypothesis related to the distribution of the individual parameters and show that the proposed method leads to more reliable results than using the EBEs. In particular, diagnostic plots are more meaningful, the rate of type I error is correctly controlled and its power increases when the degree of misspecification increases. \textbf{An application to the warfarin pharmacokinetic data confirms the interest of the approach for practical applications}. Conclusions: The proposed method should be implemented to complement EBEs-based approach for increasing the performance of model diagnosis. (10.1007/s11095-016-2020-3)
  DOI : 10.1007/s11095-016-2020-3
• Fixed Rank Kriging for Cellular Coverage Analysis
  
  • Braham Hajer
  • Jemaa Sana Ben
  • Fort Gersende
  • Moulines Éric
  • Sayrac Berna
  IEEE Transactions on Vehicular Technology, Institute of Electrical and Electronics Engineers, 2016, pp.11. Coverage planning and optimization is one of the most crucial tasks for a radio network operator. Efficient coverage optimization requires accurate coverage estimation. This estimation relies on geo-located field measurements which are gathered today during highly expensive drive tests (DT); and will be reported in the near future by users' mobile devices thanks to the 3GPP Minimizing Drive Tests (MDT) feature [1]. This feature consists in an automatic reporting of the radio measurements associated with the geographic location of the user's mobile device. Such a solution is still costly in terms of battery consumption and signaling overhead. Therefore, predicting the coverage on a location where no measurements are available remains a key and challenging task. This paper describes a powerful tool that gives an accurate coverage prediction on the whole area of interest: it builds a coverage map by spatially interpolating geo-located measurements using the Kriging technique. The paper focuses on the reduction of the computational complexity of the Kriging algorithm by applying Fixed Rank Kriging (FRK). The performance evaluation of the FRK algorithm both on simulated measurements and real field measurements shows a good trade-off between prediction efficiency and computational complexity. In order to go a step further towards the operational application of the proposed algorithm, a multicellular use-case is studied. Simulation results show a good performance in terms of coverage prediction and detection of the best serving cell. (10.1109/TVT.2016.2599842)
  DOI : 10.1109/TVT.2016.2599842
• Robust domain decomposition methods for non-symmetric problems
  
  • Bovet Christophe
  • Spillane Nicole
  • Parret-Fréaud Augustin
  • Gosselet Pierre
  , 2016. no abstract
• Local minimization algorithms for dynamic programming equations
  
  • Kalise Dante
  • Kröner Axel
  • Kunisch Karl
  SIAM Journal on Scientific Computing, Society for Industrial and Applied Mathematics, 2016, 38 (3). The numerical realization of the dynamic programming principle for continuous-time optimal control leads to nonlinear Hamilton-Jacobi-Bellman equations which require the minimization of a nonlinear mapping over the set of admissible controls. This minimization is often performed by comparison over a finite number of elements of the control set. In this paper we demonstrate the importance of an accurate realization of these minimization problems and propose algorithms by which this can be achieved effectively. The considered class of equations includes nonsmooth control problems with l1-penalization which lead to sparse controls.
• Generalized analytic functions, Moutard-type transforms and holomorphic maps
  
  • Grinevich Piotr
  • Novikov Roman
  Functional Analysis and Its Applications, Springer Verlag, 2016, 50 (2), pp.150-152. We continue the studies of Moutard-type transforms for generalized analytic functions started in our previous paper hal-01222481v1 . In particular, we suggest an interpretation of generalized analytic functions as spinor fields and show that in the framework of this approach Moutard-type transforms for the aforementioned functions commute with holomorphic changes of variables.
• Moutard transform approach to generalized analytic functions with contour poles
  
  • Grinevich Piotr
  • Novikov Roman
  Bulletin des Sciences Mathématiques, Elsevier, 2016, 140 (6), pp.638–656. We continue studies of Moutard-type transforms for the generalized analytic functions started in hal-01222481v1, hal-01234004v1. In particular, we show that generalized analytic functions with the simplest contour poles can be Moutard transformed to the regular ones, at least, locally. In addition, the later Moutard-type transforms are locally invertible.
• A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures
  
  • Allaire Grégoire
  • Briane Marc
  • Vanninathan Muthusamy
  SeMA Journal: Boletin de la Sociedad Española de Matemática Aplicada, Springer, 2016, 73 (3), pp.237-259. In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation [5] where the homogenized equation can be improved by adding small additional higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects (see e.g. [23], [18], [15]). Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved in [11] that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case. (10.1007/s40324-016-0067-z)
  DOI : 10.1007/s40324-016-0067-z
• Generic controllability of the bilinear Schrödinger equation on 1-D domains: the case of measurable potentials
  
  • Chitour Yacine
  • Sigalotti Mario
  Rendiconti dell'Istituto di Matematica dell'Universita di Trieste: an International Journal of Mathematics, Università di Trieste, 2016, 48, pp.1-17. In recent years, several sufficient conditions for the controllability of the Schrödinger equation have been proposed. In this article, we discuss the genericity of these conditions with respect to the variation of the controlled or the uncontrolled potential. In the case where the Schrödinger equation is set on a domain of dimension one, we improve the results in the literature, removing from the previously known genericity results some unnecessary technical assumptions on the regularity of the potentials.
• New Transmission Condition Accounting For Diffusion Anisotropy In Thin Layers Applied To Diffusion MRI
  
  • Caubet Fabien
  • Haddar Houssem
  • Li Jing-Rebecca
  • Nguyen Dang Van
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016. The Bloch-Torrey Partial Differential Equation (PDE) can be used to model the diffusion Magnetic Resonance Imaging (dMRI) signal in biological tissue. In this paper, we derive an Anisotropic Diffusion Transmission Condition (ADTC) for the Bloch-Torrey PDE that accounts for anisotropic diffusion inside thin layers. Such diffusion occurs, for example, in the myelin sheath surrounding the axons of neurons. This ADTC can be interpreted as an asymptotic model of order two with respect to the layer thickness and accounts for water diffusion in the normal direction that is low compared to the tangential direction. We prove uniform stability of the asymptotic model with respect to the layer thickness and a mass conservation property. We demonstrate the quadratic accuracy of the ADTC by numerical tests and show that it gives a better approximation of the dMRI signal than a simple transmission condition that assumes isotropic diffusion in the layers. (10.1051/m2an/2016060)
  DOI : 10.1051/m2an/2016060
• Stochastic dynamics for adaptation and evolution of microorganisms
  
  • Billiard Sylvain
  • Collet Pierre
  • Ferrière Régis
  • Méléard Sylvie
  • Tran Viet Chi
  , 2018, pp.525-550. We present a model for the dynamics of a population of bacteria with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait with possible mutations. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. We consider a stochastic individual-based pure jump process taking values in the space of point measures, and whose jump events describe the individual reproduction, transfer and death mechanisms. In a large population scale, the stochastic process is proved to converge to the solution of a nonlinear integro-differential equation. When there are only two different traits and no mutation, this equation reduces to a non-standard two-dimensional dynamical system. We show how crucial the forms of the transfer rates are for the long-term behavior of its solutions. We describe the dynamics of invasion and fixation when one of the two traits is initially rare, and compute the invasion probabilities. Then, we study the process under the assumption of rare mutations. We prove that the stochastic process at the mutation time scale converges to a jump process which describes the successive invasions of successful mutants. We show that the horizontal transfer can have a major impact on the distribution of the successive mutational fixations, leading to dramatically different behaviors, from expected evolution scenarios to evolutionary suicide. Simulations are given to illustrate these phenomena.
• An introduction to continuous optimization for imaging
  
  • Chambolle Antonin
  • Pock Thomas
  Acta Numerica, Cambridge University Press (CUP), 2016, 25, pp.161-319. 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. (10.1017/S096249291600009X)
  DOI : 10.1017/S096249291600009X
• Inverse Scattering Theory and Transmission Eigenvalues
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  , 2016, 88.
• The different asymptotic regimes of nearly unstable autoregressive processes
  
  • Rosenbaum Mathieu
  • Jaisson Thibault
  The Fascination of Probability, Statistics and their Applications, 2016, pp.283-301.