Publications

2018

• 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
• 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
• 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
• Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem.
  
  • Assellaou Mohamed
  • Bokanowski Olivier
  • Desilles Anna
  • Zidani Hasnaa
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (1), pp.305--335. The aim of this article is to study the Hamilton Jacobi Bellman (HJB) approach for state-constrained control problems with maximum cost. In particular, we are interested in the characterization of the value functions of such problems and the analysis of the associated optimal trajectories, without assuming any controllability assumption. The rigorous theoretical results lead to several trajectory reconstruction procedures for which convergence results are also investigated. An application to a five-state aircraft abort landing problem is then considered, for which several numerical simulations are performed to analyse the relevance of the theoretical approach. (10.1051/m2an/2017064)
  DOI : 10.1051/m2an/2017064
• Infinite-dimensional calculus under weak spatial regularity of the processes.
  
  • Flandoli Franco
  • Russo Francesco
  • Zanco Giovanni
  Journal of Theoretical Probability, Springer, 2018, 31, pp.789–826. Two generalizations of Itô formula to infinite-dimensional spaces are given. The first one, in Hilbert spaces, extends the classical one by taking advantage of cancellations, when they occur in examples and it is applied to the case of a group generator. The second one, based on the previous one and a limit procedure, is an Itô formula in a special class of Banach spaces, having a product structure with the noise in a Hilbertian component; again the key point is the extension due to a cancellation. This extension to Banach spaces and in particular the specific cancellation are motivated by path-dependent Itô calculus. (10.1007/s10959-016-0724-2)
  DOI : 10.1007/s10959-016-0724-2
• Accuracy of a Low Mach Number Model for Time-Harmonic Acoustics
  
  • Mercier Jean-François
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2018, 78 (4), pp.1891-1912. We study the time-harmonic acoustic radiation in a fluid in flow. To go beyond the convected Helmholtz equation adapted only to potential flows, starting from the Goldstein equations, coupling exactly the acoustic waves to the hydrodynamic field, we develop a new model in which the description of the hydrodynamic phenomena is simplified. This model, initially developed for a carrier flow of low Mach number M , is proved theoretically to be accurate, associated to a low error bounded by M 2 . Numerical experiments confirm the M 2 law and show that the model remains of very good quality for flow of moderate Mach numbers. (10.1137/17M113976X)
  DOI : 10.1137/17M113976X
• Asymptotic method for estimating magnetic moments from field measurements on a planar grid
  
  • Baratchart Laurent
  • Chevillard Sylvain
  • Leblond Juliette
  • Lima Eduardo Andrade
  • Ponomarev Dmitry
  , 2018. Scanning magnetic microscopes typically measure the vertical component B_3 of the magnetic field on a horizontal rectangular grid at close proximity to the sample. This feature makes them valuable instruments for analyzing magnetized materials at fine spatial scales, e.g., in geosciences to access ancient magnetic records that might be preserved in rocks by mapping the external magnetic field generated by the magnetization within a rock sample. Recovering basic characteristics of the magnetization (such as its net moment, i.e., the integral of the magnetization over the sample's volume) is an important problem, specifically when the field is too weak or the magnetization too complex to be reliably measured by standard bulk moment magnetometers. In this paper, we establish formulas, asymptotically exact when R goes large, linking the integral of x_1 B_3, x_2 B_3, and B_3 over a square region of size R to the first, second, and third component of the net moment (and higher moments), respectively, of the magnetization generating B_3. The considered square regions are centered at the origin and have sides either parallel to the axes or making a 45-degree angle with them. Differences between the exact integrals and their approximations by these asymptotic formulas are explicitly estimated, allowing one to establish rigorous bounds on the errors. We show how the formulas can be used for numerically estimating the net moment, so as to effectively use scanning magnetic microscopes as moment magnetometers. Illustrations of the method are provided using synthetic examples.
• Numerical analysis of the mixed finite element method for the neutron diffusion eigenproblem with heterogeneous coefficients
  
  • Ciarlet Patrick
  • Giret Léandre
  • Jamelot Erell
  • Kpadonou Félix D.
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2018, 52 (5), pp.2003-2035. We study first the convergence of the finite element approximation of the mixed diffusion equations with a source term, in the case where the solution is of low regularity. Such a situation commonly arises in the presence of three or more intersecting material components with different characteristics. Then we focus on the approximation of the associated eigenvalue problem. We prove spectral correctness for this problem in the mixed setting. These studies are carried out without, and then with a domain decomposition method. The domain decomposition method can be non-matching in the sense that the traces of the finite element spaces may not fit at the interface between subdomains. Finally, numerical experiments illustrate the accuracy of the method. (10.1051/m2an/2018011)
  DOI : 10.1051/m2an/2018011
• Special weak Dirichlet processes and BSDEs driven by a random measure
  
  • Bandini Elena
  • Russo Francesco
  Bernoulli, 2018, 24 (4A), pp.2569-2609. This paper considers a forward BSDE driven by a random measure, when the underlying forward process X is special semimartingale, or even more generally, a special weak Dirichlet process. Given a solution (Y, Z, U), generally Y appears to be of the type u(t, X_t) where u is a deterministic function. In this paper we identify Z and U in terms of u applying stochastic calculus with respect to weak Dirichlet processes. (10.3150/17-BEJ937)
  DOI : 10.3150/17-BEJ937