Publications

2016

• 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
• 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.
• Introduction to geodesics in sub-Riemannian geometry
  
  • Agrachev Andrei
  • Barilari Davide
  • Boscain Ugo
  , 2016.
• 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.
• 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
• 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
• 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
• 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.
• Numerical methods for an optimal multiple stopping problem
  
  • Ben Latifa Imène
  • Bonnans Joseph Fréderic
  • Mnif Mohamed
  Stochastics and Dynamics, World Scientific Publishing, 2016, 16 (4), pp.27. This paper deals with numerical solutions to an optimal multiple stopping problem. The corresponding dynamic programing (DP) equation is a variational inequality satisfied by the value function in the viscosity sense. The convergence of the numerical scheme is shown by viscosity arguments. An optimal quantization method is used for computing the conditional expectations arising in the DP equation. Numerical results are presented for the price of swing option and the behavior of the value function. (10.1142/S0219493716500167)
  DOI : 10.1142/S0219493716500167
• Stochastic eco-evolutionary model of a prey-predator community
  
  • Costa Manon
  • Hauzy Céline
  • Loeuille Nicolas
  • Méléard Sylvie
  Journal of Mathematical Biology, Springer, 2016, 72 (3), pp.573-622. We are interested in the impact of natural selection in a prey-predator community. We introduce an individual-based model of the community that takes into account both prey and predator phenotypes. Our aim is to understand the phenotypic coevolution of prey and predators. The community evolves as a multi-type birth and death process with mutations. We first consider the infinite particle approximation of the process without mutation. In this limit, the process can be approximated by a system of differential equations. We prove the existence of a unique globally asymptotically stable equilibrium under specific conditions on the interaction among prey individuals. When mutations are rare, the community evolves on the mutational scale according to a Markovian jump process. This process describes the successive equilibria of the prey-predator community and extends the Polymorphic Evolutionary Sequence to a coevolutionary framework. We then assume that mutations have a small impact on phenotypes and consider the evolution of monomorphic prey and predator populations. The limit of small mutation steps leads to a system of two differential equations which is a version of the canonical equation of adaptive dynamics for the prey-predator coevolution. We illustrate these results with an example including different prey defense mechanisms. (10.1007/s00285-015-0895-y)
  DOI : 10.1007/s00285-015-0895-y
• Moutard type transform for matrix generalized analytic functions and gauge transforms
  
  • Novikov Roman
  • Taimanov Iskander
  Russian Mathematical Surveys, Turpion, 2016, 71 (5), pp.970-972. A Moutard type transform for matrix generalized analytic functions is derived. Relations between Moutard type transforms and gauge transforms are demonstrated.
• The influence of acquisition parameters on the metrics of the bi-exponential IVIM model
  
  • Fournet Gabrielle
  • Li Jing-Rebecca
  • Le Bihan Denis
  • Ciobanu Luisa
  , 2016. The IntraVoxel Incoherent Motion (IVIM) MRI signal, typically described as a mono-exponential decay, can sometimes be better modeled as a bi-exponential function accounting for two vascular pools, capillaries and medium-size vessels. The goal of this work is to define precisely in which conditions the IVIM signal shape becomes bi-exponential and to understand the evolution of the IVIM outputs with different acquisition parameters. Rats were scanned at 7T and 11.7T using diffusion-weighted pulsed-gradient spin-echo (SE) and stimulated-echo (STE) sequences with different repetition times (TR) and diffusion encoding times. The obtained IVIM signals were fit to the mono- and bi-exponential models and the output parameters compared. The bi-exponential and mono-exponential models converge at long diffusion encoding times and long TRs. The STE is less sensitive to inflow effects present at short TRs, leading to a smaller volume fraction for the fast pool when compared to the SE sequence. The two vascular components are more easily separated at short diffusion encoding times, short TRs and when using a SE sequence. The volume fractions of the two blood pools depend on the pulse sequence, TR and diffusion encoding times while the pseudo-diffusion coefficients are only affected by the diffusion encoding time.
• A volume integral method for solving scattering problems from locally perturbed infinite periodic layers
  
  • Haddar Houssem
  • Nguyen Thi Phong
  Applicable Analysis, Taylor & Francis, 2016, pp.29. We investigate the scattering problem for the case of locally perturbed periodic layers in $\R^d$, $d=2,3$. Using the Floquet-Bloch transform in the periodicity direction we reformulate this scattering problem as an equivalent system of coupled volume integral equations. We then apply a spectral method to discretize the obtained system after periodization in the direction orthogonal to the periodicity directions of the medium. The convergence of this method is established and validating numerical results are provided. (10.1080/00036811.2016.1221942)
  DOI : 10.1080/00036811.2016.1221942
• An Adaptive Multipreconditioned Conjugate Gradient Algorithm
  
  • Spillane Nicole
  , 2016. This article introduces and analyzes a new adaptive algorithm for solving symmetric positive definite linear systems in cases where several preconditioners are available or the usual preconditioner is a sum of contributions. A new theoretical result allows to select, at each iteration, whether a classical Preconditioned CG iteration is sufficient (i.e. the error decreases by a factor of at least some chosen ratio) or whether convergence needs to be accelerated by performing an iteration of Multi Preconditioned CG [4]. We first present this in an abstract framework with the one strong assumption being that a bound for the smallest eigenvalue of the preconditioned operator is available. Then, we apply the algorithm to the Balancing Domain Decomposition method and illustrate its behaviour numerically. In particular we observe that it is optimal in terms of local solves, both for well conditioned and ill conditioned test cases, which makes it a good candidate to be a default parallel linear solver.
• Mean-field inference of Hawkes point processes
  
  • Bacry Emmanuel
  • Gaïffas Stéphane
  • Mastromatteo Iacopo
  • Muzy Jean-François
  Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2016, 49 (17), pp.174006. We propose a fast and efficient estimation method that is able to accurately recover the parameters of a d -dimensional Hawkes point-process from a set of observations. We exploit a mean-field approximation that is valid when the fluctuations of the stochastic intensity are small. We show that this is notably the case in situations when interactions are sufficiently weak, when the dimension of the system is high or when the fluctuations are self-averaging due to the large number of past events they involve. In such a regime the estimation of a Hawkes process can be mapped on a least-squares problem for which we provide an analytic solution. Though this estimator is biased, we show that its precision can be comparable to the one of the maximum likelihood estimator while its computation speed is shown to be improved considerably. We give a theoretical control on the accuracy of our new approach and illustrate its efficiency using synthetic datasets, in order to assess the statistical estimation error of the parameters. (10.1088/1751-8113/49/17/174006)
  DOI : 10.1088/1751-8113/49/17/174006
• Nondestructive testing of the delaminated interface between two materials
  
  • Cakoni Fioralba
  • de Teresa Irene
  • Haddar Houssem
  • Monk Peter
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (6), pp.2306–2332. We consider the problem of detecting if two materials that should be in contact have separated or delaminated. The goal is to find an acoustic technique to detect the delamination. We model the delamination as a thin opening between two materials of different acoustic properties, and using asymptotic techniques we derive a asymptotic model where the delaminated region is replaced by jump conditions on the acoustic field and flux. The asymptotic model has potential singularities due to the edges of the delaminated region, and we show that the forward problem is well posed for a large class of possible delaminations. We then design a special Linear Sampling Method (LSM) for detecting the shape of the delamination assuming that the background, undamaged, state is known. Finally we show, by numerical experiments, that our LSM can indeed determine the shape of delaminated regions. (10.1137/16M1064167)
  DOI : 10.1137/16M1064167
• Generalized linear sampling method for elastic-wave sensing of heterogeneous fractures
  
  • Pourahmadian Fatemeh
  • Guzina Bojan B
  • Haddar Houssem
  Inverse Problems, IOP Publishing, 2016, pp.28. A theoretical foundation is developed for active seismic reconstruction of fractures endowed with spatially-varying interfacial condition (e.g. partially-closed fractures, hydraulic fractures). The proposed indicator functional carries a superior localization property with no significant sensitivity to the fracture's contact condition, measurement errors, and illumination frequency. This is accomplished through the paradigm of the F-factorization technique and the recently developed Generalized Linear Sampling Method (GLSM) applied to elastodynamics. The direct scattering problem is formulated in the frequency domain where the fracture surface is illuminated by a set of incident plane waves, while monitoring the induced scattered field in the form of (elastic) far-field patterns. The analysis of the well-posedness of the forward problem leads to an admissibility condition on the fracture's (linearized) contact parameters. This in turn contributes toward establishing the applicability of the F-factorization method, and consequently aids the formulation of a convex GLSM cost functional whose minimizer can be computed without iterations. Such minimizer is then used to construct a robust fracture indicator function, whose performance is illustrated through a set of numerical experiments. For completeness, the results of the GLSM reconstruction are compared to those obtained by the classical linear sampling method (LSM).
• A global Riemann-Hilbert problem for two-dimensional inverse scattering at fixed energy
  
  • Lakshtanov Evgeny
  • Novikov Roman
  • Vainberg Boris
  Rendiconti dell'Istituto di Matematica dell'Universita di Trieste: an International Journal of Mathematics, Università di Trieste, 2016, 48, pp.21-47. We develop the Riemann-Hilbert problem approach to inverse scattering for the two-dimensional Schrodinger equation at fixed energy. We obtain global or generic versions of the key results of this approach for the case of positive energy and compactly supported potentials. In particular, we do not assume that the potential is small or that Faddeev scattering solutions do not have singularities (i.e. we allow the Faddeev exceptional points to exist). Applications of these results to the Novikov-Veselov equation are also considered.
• Molding direction constraints in structural optimization via a level-set method
  
  • Allaire Grégoire
  • Jouve François
  • Michailidis Georgios
  , 2016, pp.1-39. In the framework of structural optimization via a level-set method, we develop an approach to handle the directional molding constraint for cast parts. A novel molding condition is formulated and a penalization method is used to enforce the constraint. A first advantage of our new approach is that it does not require to start from a feasible initialization, but it guarantees the convergence to a castable shape. A second advantage is that our approach can incorporate thickness constraints too. We do not adress the optimization of the casting system, which is considered a priori defined. We show several 3d examples of compliance minimization in linearized elasticity under molding and minimal or maximal thickness constraints. We also compare our results with formulations already existing in the literature. (10.1007/978-3-319-45680-5)
  DOI : 10.1007/978-3-319-45680-5
• Choice of measure source terms in interface coupling for a model problem in gas dynamics.
  
  • Coquel Frédéric
  • Godlewski Edwige
  • Haddaoui Khalil
  • Marmignon Claude
  • Renac Florent
  Mathematics of Computation, American Mathematical Society, 2016, 85, pp.2305-2339. This paper is devoted to the mathematical and numerical analysis of a coupling procedure for one-dimensional Euler systems. The two systems have different closure laws and are coupled through a thin fixed interface. Following the work of [5], we propose to couple these systems by a bounded vector-valued Dirac measure, concentrated at the coupling interface, which in the applications may have a physical meaning. We show that the proposed framework allows to control the coupling conditions and we propose an approximate Riemann solver based on a relaxation approach preserving equilibrium solutions of the coupled problem. Numerical experiments in constrained optimization problems are then presented to assess the performances of the present method. 1. Introduction The study of large-scale and complex problems exhibiting a wide range of physical space and time scales (see for instance [62, 35, 14]), usually requires separate solvers adapted to the resolution of specific scales. This is the case of many industrial flows. Let us quote, for example, the numerical simulation of two-phase flows applied to the burning liquid oxygen-hydrogen gas in rocket engines [58]. This kind of flow contains both separated and dispersed two-phase flows, due to atomization and evaporation phenomena. This requires appropriate models and solvers for separated and dispersed phases that have to be appropriately coupled. Another example concerns turbomachine flows which can be modeled by the Euler equations of gas dynamics with different closure laws between the stages of the turbine, where the conditions of temperature and pressure are strongly heterogeneous. The coupling of these different systems is thus necessary to give a complete description of the flow inside the whole turbine. The method of interface coupling allows to represent the evolution of such flows, where different models are separated by fixed interfaces. First, coupling conditions are specified at the interface to exchange information between the systems. The definition of transmission conditions generally results from physical consideration, e.g. the conservation or the continuity of given variables. Then, the transmission conditions are represented at the discrete level. The study of interface coupling for nonlinear hyperbolic systems has received attention for several years. In [43], the authors study the scalar case from both mathematical and numerical points of view. (10.1090/mcom%2F3063)
  DOI : 10.1090/mcom%2F3063
• Moutard transform for the generalized analytic functions
  
  • Grinevich Piotr
  • Novikov Roman
  The Journal of Geometric Analysis, Springer, 2016, 26 (4), pp.2984–2995. We construct a Moutard-type transform for the generalized analytic functions. The first theorems and the first explicit examples in this connection are given.
• Spatial Prediction Under Location Uncertainty in Cellular Networks
  
  • Braham Hajer
  • Jemaa Sana Ben
  • Fort Gersende
  • Moulines Éric
  • Sayrac Berna
  IEEE Transactions on Wireless Communications, Institute of Electrical and Electronics Engineers, 2016, 15, pp.7633 - 7643. Coverage optimization is an important process for the operator, as it is a crucial prerequisite toward offering a satisfactory quality of service to the end users. The first step of this process is coverage prediction, which can be performed by interpolating geo-located measurements reported to the network by mobile user's equipments. In the previous works, we proposed a low complexity coverage prediction algorithm based on the adaptation of the geo-statistics fixed rank kriging (FRK) algorithm. We supposed that the geo-location information reported with the radio measurements was perfect, which is not the case in reality. In this paper, we study the impact of location uncertainty on the coverage prediction accuracy and we extend the previously proposed algorithm to include geo-location error in the prediction model. We validate the proposed algorithm using both simulated and real-field measurements. The FRK is extended to take into account that the location uncertainty proves to enhance the prediction accuracy while keeping a reasonable computational complexity. (10.1109/TWC.2016.2605676)
  DOI : 10.1109/TWC.2016.2605676
• 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
• Existence and Uniqueness for a Crystalline Mean Curvature Flow
  
  • Chambolle Antonin
  • Morini Massimiliano
  • Ponsiglione Marcello
  Communications on Pure and Applied Mathematics, Wiley, 2016. An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The comparison principle is obtained by means of a suitable weak formulation of the flow, while the existence of a global-in-time solution follows via a minimizing movements approach. (10.1002/cpa.21668)
  DOI : 10.1002/cpa.21668
• Geometry, Analysis and Dynamics on sub-Riemannian Manifolds - Volume I
  
  • Barilari Davide
  • Boscain Ugo
  • Sigalotti Mario
  , 2016. (10.4171/162)
  DOI : 10.4171/162