Publications

2016

• 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.
• 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
• 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
• 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.
• Discrete Hammersley's Lines with sources and sinks
  
  • Basdevant A-L
  • Enriquez Nathanaël
  • 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.
• 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.
• 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.
• Robust domain decomposition methods for non-symmetric problems
  
  • Bovet Christophe
  • Spillane Nicole
  • Parret-Fréaud Augustin
  • Gosselet Pierre
  , 2016. no abstract
• 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.
• 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.
• 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
• 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
• New transmission condition accounting for diffusion anisotropy in thin layer applied to diffusion MRI
  
  • Caubet Fabien
  • Haddar Houssen
  • Li Jing Rebecca
  • Nguyen Dang Van
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2016 (51), pp.1279--1301. 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 also prove the theoretical quadratic accuracy of the ADTC. Finally, numerical tests validate these results and show that our model 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
• Jan de Leeuw and the French School of Data Analysis
  
  • Husson François
  • Josse Julie
  • Saporta Gilbert
  Journal of Statistical Software, University of California, Los Angeles, 2016, 73 (6), pp.16 p.. The Dutch and the French schools of data analysis differ in their approaches to the question: How does one understand and summarize the information contained in a data set? The commonalities and discrepancies between the schools are explored here with a focus on methods dedicated to the analysis of categorical data, which are known either as homogeneity analysis (HOMALS) or multiple correspondence analysis (MCA). (10.18637/jss.v073.i06)
  DOI : 10.18637/jss.v073.i06
• Spectral analysis and the Aharonov-Bohm effect on certain almost-Riemannian manifolds
  
  • Boscain Ugo
  • Prandi Dario
  • Seri Marcello
  Communications in Partial Differential Equations, Taylor & Francis, 2016, 41 (1), pp.32–50. We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. As for general almost-Riemannian structures (under certain technical hypothesis), the singular set acts as a barrier for the evolution of the heat and of a quantum particle, although geodesics can cross it. This is a consequence of the self-adjointness of the Laplace-Beltrami operator on each connected component of the manifolds without the singular set. We get explicit descriptions of the spectrum, of the eigenfunctions and their properties. In particular in both cases we get a Weyl law with dominant term $E\log E$. We then study the effect of an Aharonov-Bohm non-apophantic magnetic potential that has a drastic effect on the spectral properties. Other generalized Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator. (10.1080/03605302.2015.1095766)
  DOI : 10.1080/03605302.2015.1095766
• Universal polarization domain walls in optical fibers as topological bit-entities for data transmission
  
  • Garnier Josselin
  • Gilles M.
  • Rahmani M.
  • Picozzi A.
  • Guasoni M.
  • Fatome J.
  , 2016. (10.1364/ACOFT.2016.JW6A.6)
  DOI : 10.1364/ACOFT.2016.JW6A.6
• Inverse Scattering Theory and Transmission Eigenvalues
  
  • Cakoni Fioralba
  • Colton David
  • Haddar Houssem
  , 2016, 88.
• 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
• 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.
• Quadratic BSDEs with jumps: Related nonlinear expectations
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  Stochastics and Dynamics, World Scientific Publishing, 2016, 16 (4), pp.1650012. In this article, we follow the study of quadratic backward SDEs with jumps,that is to say for which the generator has quadratic growth in the variables (z, u), started in our accompanying paper [15]. Relying on the existence and uniqueness result of [15], we define the corresponding g-expectations and study some of their properties. We obtain in particular a non-linear Doob-Meyer decomposition for g-submartingales and a downcrossing inequality which implies their regularity in time. As a consequence of these results, we also obtain a converse comparison theorem for our class of BSDEs. Finally, we provide a dual representation for the corresponding dynamic risk measures, and study the properties of their inf-convolution, giving several explicit examples. (10.1142/S021949371650012X)
  DOI : 10.1142/S021949371650012X
• Korn-Poincare inequalities for functions with a small jump set
  
  • Chambolle Antonin
  • Conti Sergio
  • Francfort Gilles A
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2016, 65 (4), pp.1373 - 1399. (10.1512/iumj.2016.65.5852)
  DOI : 10.1512/iumj.2016.65.5852
• A two-pool model to describe the IVIM cerebral perfusion
  
  • Fournet Gabrielle
  • Li Jing-Rebecca
  • Cerjanic Alex M
  • Sutton Bradley P
  • Ciobanu Luisa
  • Le Bihan Denis
  Journal of Cerebral Blood Flow and Metabolism, Nature Publishing Group, 2016. IntraVoxel Incoherent Motion (IVIM) is a magnetic resonance imaging (MRI) technique capable of measuring perfusion-related parameters. In this manuscript, we show that the mono-exponential model commonly used to process IVIM data might be challenged, especially at short diffusion times. Eleven rat datasets were acquired at 7T using a diffusion-weighted pulsed gradient spin echo sequence with b-values ranging from 7 to 2500 s/mm2 at three diffusion times. The IVIM signals, obtained by removing the diffusion component from the raw MR signal, were fitted to the standard mono-exponential model, a bi-exponential model and the Kennan model. The Akaike information criterion used to find the best model to fit the data demonstrates that, at short diffusion times, the bi-exponential IVIM model is most appropriate. The results obtained by comparing the experimental data to a dictionary of numerical simulations of the IVIM signal in microvascular networks support the hypothesis that such a bi-exponential behavior can be explained by considering the contribution of two vascular pools: capillaries and somewhat larger vessels. (10.1177/0271678X16681310)
  DOI : 10.1177/0271678X16681310
• Stacking sequence and shape optimization of laminated composite plates via a level-set method
  
  • Allaire Grégoire
  • Delgado Gabriel
  Journal of the Mechanics and Physics of Solids, Elsevier, 2016, 97, pp.168-196. We consider the optimal design of composite laminates by allowing a variable stacking sequence and in-plane shape of each ply. In order to optimize both variables we rely on a decomposition technique which aggregates the constraints into one unique constraint margin function. Thanks to this approach, a rigorous equivalent bi-level optimization problem is established. This problem is made up of an inner level represented by the combinatorial optimization of the stacking sequence and an outer level represented by the topology and geometry optimization of each ply. We propose for the stacking sequence optimization an outer approximation method which iteratively solves a set of mixed integer linear problems associated to the evaluation of the constraint margin function. For the topology optimization of each ply, we lean on the level set method for the description of the interfaces and the Hadamard method for boundary variations by means of the computation of the shape gradient. Numerical experiments are performed on an aeronautic test case where the weight is minimized subject to different mechanical constraints, namely compliance, reserve factor and buckling load.
• Second order analysis of state-constrained control-affine problems
  
  • Aronna Maria Soledad
  • Bonnans J. Frederic
  • Goh Bean San
  Mathematical Programming, Series A, Springer, 2016, 160 (1), pp.115-147. In this article we establish new second order necessary and suffi-cient optimality conditions for a class of control-affine problems with a scalar control and a scalar state constraint. These optimality conditions extend to the constrained state framework the Goh transform, which is the classical tool for obtaining an extension of the Legendre condition.