Publications

2018

• Solving chance constrained optimal control problems in aerospace via Kernel Density Estimation
  
  • Caillau Jean-Baptiste
  • Cerf Max
  • Sassi Achille
  • Trélat Emmanuel
  • Zidani Hasnaa
  Optimal Control Applications and Methods, Wiley, 2018, 39 (5), pp.1833-1858. The goal of this paper is to show how non-parametric statistics can be used to solve some chance constrained optimization and optimal control problems. We use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution , from a relatively small sample. We then show how this technique can be applied and implemented for a class of problems including the God-dard problem and the trajectory optimization of an Ariane 5-like launcher. (10.1002/oca.2445)
  DOI : 10.1002/oca.2445
• Sub-wavelength sensing of bi-periodic materials using topological sensitivity of second-order homogenized model
  
  • Bonnet Marc
  • Cornaggia Rémi
  • Guzina Bojan B
  Journal of Physics: Conference Series, IOP Science, 2018, 1131, pp.012008. We aim to detect defects or perturbations of periodic media, e.g. due to a defective manufacturing process. To this end, we consider scalar waves in such media through the lens of a second-order macroscopic description, and we compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. Then, we present a method that permits sub-wavelength sensing of periodic media, given the (anisotropic) phase velocity of plane waves illuminating the considered medium for several angles and wavenumbers. (10.1088/1742-6596/1131/1/012008)
  DOI : 10.1088/1742-6596/1131/1/012008
• Infinite Horizon Stochastic Optimal Control Problems with Running Maximum Cost
  
  • Kröner Axel
  • Picarelli Athena
  • Zidani Hasnaa
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2018, 56 (5), pp.3296-3319. An infinite horizon stochastic optimal control problem with running maximum cost is considered. The value function is characterized as the viscosity solution of a second-order Hamilton-Jacobi-Bellman (HJB) equation with mixed boundary condition. A general numerical scheme is proposed and convergence is established under the assumptions of consistency, monotonicity and stability of the scheme. These properties are verified for a specific semi-Lagrangian scheme. (10.1137/17M115253X)
  DOI : 10.1137/17M115253X
• Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions
  
  • Ghilli Daria
  • Rao Zhiping
  • Zidani Hasnaa
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2018. This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results in [26] in a more general framework with switching running costs and weaker controllability assumptions. The comparison principle has been established to guarantee the uniqueness and the stability results for the HJB system on such multi-domains. In the lower semi-continuous case, we characterize the value function as the unique lower semi-continuous viscosity solution of the HJB system, under a local controllability assumption. (10.1051/cocv/2018072)
  DOI : 10.1051/cocv/2018072
• Monte-Carlo Algorithms for Forward Feynman-Kac type representation for semilinear nonconservative Partial Differential Equations
  
  • Lecavil Anthony
  • Oudjane Nadia
  • Russo Francesco
  Monte Carlo Methods and Applications, De Gruyter, 2018, 24 (1), pp.55-70. The paper is devoted to the construction of a probabilistic particle algorithm. This is related to nonlin-ear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). Illustrations of the efficiency of the algorithm are provided by numerical experiments. (10.1515/mcma-2018-0005)
  DOI : 10.1515/mcma-2018-0005
• Error estimates for numerical approximation of Hamilton-Jacobi equations related to hybrid control systems
  
  • Ferretti Roberto
  • Sassi Achille
  • Zidani Hasnaa
  Applied Mathematics and Optimization, Springer Verlag (Germany), 2018. Hybrid control systems are dynamical systems that can be controlled by a combination of both continuous and discrete actions. In this paper we study the approximation of optimal control problems associated to this kind of systems, and in particular of the Quasi-Variational Inequality which characterizes the value function. Our main result features the error estimates between the value function of the problem and its approximation. We also focus on the hypotheses describing the mathematical model and the properties defining the class of numerical scheme for which the result holds true. (10.1007/s00245-018-9515-8)
  DOI : 10.1007/s00245-018-9515-8
• Minimal graphs for matching extension
  
  • Costa Marie-Christine
  • de Werra Dominique
  • Picouleau Christophe
  Discrete Applied Mathematics, Elsevier, 2018, 234, pp.47-55. Let G = (V, E) be a simple loopless finite undirected graph. We say that G is (2-factor) expandable if for any non-edge uv, G + uv has a 2-factor F that contains uv. We are interested in the following: Given a positive integer n = |V |, what is the minimum cardinality of E such that there exists G = (V, E) which is 2-factor expandable? This minimum number is denoted by Exp 2 (n). We give an explicit formula for Exp 2 (n) and provide 2-factor expandable graphs of minimum size Exp 2 (n). (10.1016/j.dam.2015.11.007)
  DOI : 10.1016/j.dam.2015.11.007
• A Family of Crouzeix-Raviart Finite Elements in 3D
  
  • Ciarlet Patrick
  • Dunkl Charles F
  • Sauter Stefan A
  Analysis and Applications, World Scientific Publishing, 2018. In this paper we will develop a family of non-conforming " Crouzeix-Raviart " type finite elements in three dimensions. They consist of local polynomials of maximal degree p ∈ N on simplicial finite element meshes while certain jump conditions are imposed across adjacent simplices. We will prove optimal a priori estimates for these finite elements. The characterization of this space via jump conditions is implicit and the derivation of a local basis requires some deeper theoretical tools from orthogonal polynomials on triangles and their representation. We will derive these tools for this purpose. These results allow us to give explicit representations of the local basis functions. Finally we will analyze the linear independence of these sets of functions and discuss the question whether they span the whole non-conforming space. (10.1142/S0219530518500070)
  DOI : 10.1142/S0219530518500070
• Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media
  
  • Bonnet Marc
  • Cornaggia Rémi
  • Guzina Bojan B
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.2057-2082. We consider scalar waves in periodic media through the lens of a second-order effective i.e. macroscopic description, and we aim to compute the sensitivities of the germane effective parameters due to topological perturbations of a microscopic unit cell. Specifically, our analysis focuses on the tensorial coefficients in the governing mean-field equation – including both the leading order (i.e. quasi-static) terms, and their second-order companions bearing the effects of incipient wave dispersion. The results demonstrate that the sought sensitivities are computable in terms of (i) three unit-cell solutions used to formulate the unperturbed macroscopic model; (ii) two adoint-field solutions driven by the mass density variation inside the unperturbed unit cell; and (iii) the usual polarization tensor, appearing in the related studies of non-periodic media, that synthesizes the geometric and constitutive features of a point-like perturbation. The proposed developments may be useful toward (a) the design of periodic media to manipulate macroscopic waves via the microstructure-generated effects of dispersion and anisotropy, and (b) sub-wavelength sensing of periodic defects or perturbations. (10.1137/17M1149018)
  DOI : 10.1137/17M1149018
• A mixed formulation of the Tikhonov regularization and its application to inverse PDE problems
  
  • Bourgeois Laurent
  • Recoquillay Arnaud
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (1), pp.123-145. This paper is dedicated to a new way of presenting the Tikhonov regularization in the form of a mixed formulation. Such formulation is well adapted to the regularization of linear ill-posed partial differential equations because when it comes to discretization, the mixed formulation enables us to use some standard finite elements. As an application of our theory, we consider an inverse obstacle problem in an acoustic waveguide. In order to solve it we use the so-called “exterior approach”, which couples the mixed formulation of Tikhonov regularization and a level set method. Some 2d numerical experiments show the feasibility of our approach. (10.1051/m2an/2018008)
  DOI : 10.1051/m2an/2018008
• Optimal control of normalized SIMR models with vaccination and treatment
  
  • Pinho Maria Do Rosário De
  • Maurer Helmut
  • Zidani Hasnaa
  Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2018, 23 (1), pp.79 - 99. (10.3934/dcdsb.2018006)
  DOI : 10.3934/dcdsb.2018006
• On measures in sub-Riemannian geometry
  
  • Ghezzi Roberta
  • Jean Frédéric
  Séminaire de Théorie Spectrale et Géométrie, Grenoble : Université de Grenoble 1, Institut Fourier, 1983-, 2018, 33 (2015-2016). In \cite{gjha} we give a detailed analysis of spherical Hausdorff measures on sub-Riemannian manifolds in a general framework, that is, without the assumption of equiregularity. The present paper is devised as a complement of this analysis, with both new results and open questions. The first aim is to extend the study to other kinds of intrinsic measures on sub-Riemannian manifolds, namely Popp's measure and general (i.e., non spherical) Hausdorff measures. The second is to explore some consequences of \cite{gjha} on metric measure spaces based on sub-Riemannian manifolds. (10.5802/tsg.312)
  DOI : 10.5802/tsg.312
• Gas storage valuation and hedging. A quantification of the model risk.
  
  • Henaff Patrick
  • Laachir Ismail
  • Russo Francesco
  International Journal of Financial Studies, MDPI, 2018, 6 (1 (27)). This paper focuses on the valuation and hedging of gas storage facilities, using a spot-based valuation framework coupled with a financial hedging strategy implemented with futures contracts. The first novelty consist in proposing a model that unifies the dynamics of the futures curve and the spot price, which accounts for the main stylized facts of the US natural gas market, such as seasonality and presence of price spikes. The second aspect of the paper is related to the quantification of model uncertainty related to the spot dynamics. (10.3390/ijfs6010027)
  DOI : 10.3390/ijfs6010027