Publications

2022

• A mathematical study of a hyperbolic metamaterial in free space
  
  • Ciarlet Patrick
  • Kachanovska Maryna
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (2), pp.2216-2250. Wave propagation in hyperbolic metamaterials is described by the Maxwell equations with a frequency dependent tensor of dielectric permittivity, whose eigenvalues are of different signs. In this case the problem becomes hyperbolic (Klein-Gordon equation) for a certain range of frequencies. The principal theoretical and numerical difficulty comes from the fact that this hyperbolic equation is posed in a free space, without initial conditions provided. The subject of the work is the theoretical justification of this problem. In particular, this includes the construction of a radiation condition, a well-posedness result, a limiting absorption principle and regularity estimates on the solution. (10.1137/21M1404223)
  DOI : 10.1137/21M1404223
• The Complex-Scaled Half-Space Matching Method
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N.
  • Fliss Sonia
  • Hazard Christophe
  • Perfekt Karl-Mikael
  • Tjandrawidjaja Yohanes
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2022, 54 (1), pp.512-557. The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results. (10.1137/20M1387122)
  DOI : 10.1137/20M1387122
• On some path-dependent SDEs involving distributional drifts
  
  • Ohashi Alberto
  • Russo Francesco
  • Teixeira Alan
  Modern Stochastics: Theory and Applications, VTEX, 2022, 9 (1), pp.65-87. In this paper, we study (strong and weak) existence and uniqueness of a class of non-Markovian SDEs whose drift contains the derivative in the sense of distributions of a continuous function. (10.15559/21-VMSTA197)
  DOI : 10.15559/21-VMSTA197
• Fokker-Planck equations with terminal condition and related McKean probabilistic representation
  
  • Izydorczyk Lucas
  • Oudjane Nadia
  • Russo Francesco
  • Tessitore Gianmario
  Nonlinear Differential Equations and Applications, Springer Verlag, 2022, volume 29 (10). Usually Fokker-Planck type partial differential equations (PDEs) are well-posed if the initial condition is specified. In this paper, alternatively, we consider the inverse problem which consists in prescribing final data: in particular we give sufficient conditions for existence and uniqueness. In the second part of the paper we provide a probabilistic representation of those PDEs in the form a solution of a McKean type equation corresponding to the time-reversal dynamics of a diffusion process. (10.1007/s00030-021-00736-1)
  DOI : 10.1007/s00030-021-00736-1
• A Mayer optimal control problem on Wasserstein spaces over Riemannian manifolds
  
  • Jean Frédéric
  • Jerhaoui Othmane
  • Zidani Hasnaa
  , 2022, 55 (16), pp.44-49. (10.1016/j.ifacol.2022.08.079)
  DOI : 10.1016/j.ifacol.2022.08.079
• Local transparent boundary conditions for wave propagation in fractal trees (ii): error and complexity analysis
  
  • Joly Patrick
  • Kachanovska Maryna
  SIAM Journal on Numerical Analysis, Society for Industrial and Applied Mathematics, 2022, 60 (2). This work is dedicated to a refined error analysis of the high-order transparent boundary conditions introduced in the companion work [8] for the weighted wave equation on a fractal tree. The construction of such boundary conditions relies on truncating the meromorphic series that represents the symbol of the Dirichlet-to-Neumann operator. The error induced by the truncation depends on the behaviour of the eigenvalues and the eigenfunctions of the weighted Laplacian on a self-similar metric tree. In this work we quantify this error by computing asymptotics of the eigenvalues and bounds for Neumann traces of the eigenfunctions. We prove the sharpness of the obtained bounds for a class of self-similar trees. (10.1137/20M1357524)
  DOI : 10.1137/20M1357524
• Gâteaux type path-dependent PDEs and BSDEs with Gaussian forward processes
  
  • Barrasso Adrien
  • Russo Francesco
  Stochastics and Dynamics, World Scientific Publishing, 2022, 22, pp.2250007,. We are interested in path-dependent semilinear PDEs, where the derivatives are of Gâteaux type in specific directions k and b, being the kernel functions of a Volterra Gaussian process X. Under some conditions on k, b and the coefficients of the PDE, we prove existence and uniqueness of a decoupled mild solution, a notion introduced in a previous paper by the authors. We also show that the solution of the PDE can be represented through BSDEs where the forward (underlying) process is X. (10.1142/S0219493722500071)
  DOI : 10.1142/S0219493722500071
• Optimistic Planning Algorithms For State-Constrained Optimal Control Problems
  
  • Bokanowski Olivier
  • Gammoudi Nidhal
  • Zidani Hasnaa
  Computers & Mathematics with Applications, Elsevier, 2022, 109 (1), pp.158-179. In this work, we study optimistic planning methods to solve some state-constrained optimal control problems in finite horizon. While classical methods for calculating the value function are generally based on a discretization in the state space, optimistic planning algorithms have the advantage of using adaptive discretization in the control space. These approaches are therefore very suitable for control problems where the dimension of the control variable is low and allow to deal with problems where the dimension of the state space can be very high. Our algorithms also have the advantage of providing, for given computing resources, the best control strategy whose performance is as close as possible to optimality while its corresponding trajectory comply with the state constraints up to a given accuracy. (10.1016/j.camwa.2022.01.016)
  DOI : 10.1016/j.camwa.2022.01.016
• The Morozov's principle applied to data assimilation problems
  
  • Bourgeois Laurent
  • Dardé Jérémi
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2022. This paper is focused on the Morozov's principle applied to an abstract data assimilation framework, with particular attention to three simple examples: the data assimilation problem for the Laplace equation, the Cauchy problem for the Laplace equation and the data assimilation problem for the heat equation. Those ill-posed problems are regularized with the help of a mixed type formulation which is proved to be equivalent to a Tikhonov regularization applied to a well-chosen operator. The main issue is that such operator may not have a dense range, which makes it necessary to extend well-known results related to the Morozov's choice of the regularization parameter to that unusual situation. The solution which satisfies the Morozov's principle is computed with the help of the duality in optimization, possibly by forcing the solution to satisfy given a priori constraints. Some numerical results in two dimensions are proposed in the case of the data assimilation problem for the Laplace equation. (10.1051/m2an/2022061)
  DOI : 10.1051/m2an/2022061
• Structure of optimal control for planetary landing with control and state constraints
  
  • Leparoux Clara
  • Hérissé Bruno
  • Jean Frédéric
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28. This paper studies a vertical powered descent problem in the context of planetary landing, considering glide-slope and thrust pointing constraints and minimizing any final cost. In a first time, it proves the Max-Min-Max or Max-Singular-Max form of the optimal control using the Pontryagin Maximum Principle, and it extends this result to a problem formulation considering the effect of an atmosphere. It also shows that the singular structure does not appear in generic cases. In a second time, it theoretically analyzes the optimal trajectory for a more specific problem formulation to show that there can be at most one contact or boundary interval with the state constraint on each Max or Min arc. (10.1051/cocv/2022065)
  DOI : 10.1051/cocv/2022065
• An efficient numerical method for time domain electromagnetic wave propagation in co-axial cables
  
  • Beni Hamad Akram
  • Beck Geoffrey
  • Imperiale Sébastien
  • Joly Patrick
  Computational Methods in Applied Mathematics, De Gruyter, 2022, 22 (4). In this work we construct an efficient numerical method to solve 3D Maxwell's equations in coaxial cables. Our strategy is based upon an hybrid explicit-implicit time discretization combined with edge elements on prisms and numerical quadrature. One of the objective is to validate numerically generalized Telegrapher's models that are used to simplify the 3D Maxwell equations into a 1D problem. This is the object of the second part of the article. (10.1515/cmam-2021-0195)
  DOI : 10.1515/cmam-2021-0195
• On the approximation of electromagnetic fields by edge finite elements. Part 4: analysis of the model with one sign-changing coefficient
  
  • Ciarlet Patrick
  Numerische Mathematik, Springer Verlag, 2022, 152, pp.223-257. In electromagnetism, in the presence of a negative material surrounded by a classical material, the electric permittivity, and possibly the magnetic permeability, can exhibit a sign-change at the interface. In this setting, the study of electromagnetic phenomena is a challenging topic. We focus on the time-harmonic Maxwell equations in a bounded set $\Omega$ of ${\mathbb R}^3$, and more precisely on the numerical approximation of the electromagnetic fields by edge finite elements. Special attention is paid to low-regularity solutions, in terms of the Sobolev scale $({\boldsymbol{H}}^{\mathtt{s}}(\Omega))_{\mathtt{s}>0}$. With the help of T-coercivity, we address the case of one sign-changing coefficient, both for the model itself, and for its discrete version. Optimal a priori error estimates are derived. (10.1007/s00211-022-01315-x)
  DOI : 10.1007/s00211-022-01315-x
• Kernel representation of Kalman observer and associated H-matrix based discretization
  
  • Aussal Matthieu
  • Moireau Philippe
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2022, 28, pp.78. In deterministic estimation, applying a Kalman filter to a dynamical model based on partial differential equations is theoretically seducing but solving the associated Riccati equation leads to a so-called curse of dimensionality for its numerical implementation. In this work, we propose to entirely revisit the theory of Kalman filters for parabolic problems where additional regularity results proves that the Riccati equation solution belongs to the class of Hilbert-Schmidt operators. The regularity of the associated kernel then allows to proceed to the numerical analysis of the Kalman full space-time discretization in adapted norms, hence justifying the implementation of the related Kalman filter numerical algorithm with H-matrices typically developed for integral equations discretization. (10.1051/cocv/2022071)
  DOI : 10.1051/cocv/2022071