Publications

2020

• A unifying vision of Particle Filtering and Explicit Dual Control
  
  • Flayac Emilien
  • Dahia Karim
  • Hérissé Bruno
  • Jean Frédéric
  , 2020. This paper presents a joint optimisation framework for optimal estimation and stochastic optimal control with imperfect information. It provides a estimation and control scheme that can be decomposed into a classical optimal estimation step and an optimal control step where a new term coming from optimal estimation is added to the cost. It is shown that a specific particle filter algorithm allows one to solve the first step approximately in the case of Mean Square Error minimisation and under suitable assumptions on the model. Then, it is shown that the estimation-based control step can justify formally the use of Explicit dual controllers which are most of the time derived from empirical matters. Finally, a relevant example from Aerospace engineering is presented.
• Exponentially convergent non overlapping domain decomposition methods for the Helmholtz equation
  
  • Collino Francis
  • Joly Patrick
  • Lecouvez Matthieu
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020, 54 (3), pp.775-810. In this paper, we develop in a general framework a non overlapping Domain Decomposition Method that is proven to be well-posed and converges exponentially fast, provided that specific transmission operators are used. These operators are necessarily non local and we provide a class of such operators in the form of integral operators. To reduce the numerical cost of these integral operators, we show that a truncation process can be applied that preserves all the properties leading to an exponentially fast convergent method. A modal analysis is performed on a separable geometry to illustrate the theoretical properties of the method and we exhibit an optimization process to further reduce the convergence rate of the algorithm. (10.1051/m2an/2019050)
  DOI : 10.1051/m2an/2019050
• Exact converging bounds for Stochastic Dual Dynamic Programming via Fenchel duality
  
  • Leclère Vincent
  • Carpentier Pierre
  • Chancelier Jean-Philippe
  • Lenoir Arnaud
  • Pacaud François
  SIAM Journal on Optimization, Society for Industrial and Applied Mathematics, 2020, 30 (2), pp.1223–1250. The Stochastic Dual Dynamic Programming (SDDP) algorithm has become one of the main tools to address convex multistage stochastic optimal control problem. Recently a large amount of work has been devoted to improve the convergence speed of the algorithm through cut-selection and regularization, or to extend the field of applications to non-linear, integer or risk-averse problems. However one of the main downside of the algorithm remains the difficulty to give an upper bound of the optimal value, usually estimated through Monte Carlo methods and therefore difficult to use in the algorithm stopping criterion. In this paper we present a dual SDDP algorithm that yields a converging exact upper bound for the optimal value of the optimization problem. Incidently we show how to compute an alternative control policy based on an inner approximation of Bellman value functions instead of the outer approximation given by the standard SDDP algorithm. We illustrate the approach on an energy production problem involving zones of production and transportation links between the zones. The numerical experiments we carry out on this example show the effectiveness of the method. (10.1137/19M1258876)
  DOI : 10.1137/19M1258876
• The Geometry of Isochrone Orbits from Archimedes' parabolae to Kepler's third law
  
  • Ramond Paul
  • Perez Jérôme
  celestial Mechanics & Dynamical Astronomy, 2020, 132 (4), pp.1-47. The Kepler potential ∝ −1/r and the Harmonic potential ∝ r 2 share the following remarkable property: in either of these potentials, a bound test particle orbits with a radial period that is independent of its angular momentum. For this reason, the Kepler and Harmonic potentials are called isochrone. In this paper, we solve the following general problem: are there any other isochrone potentials, and if so, what kind of orbits do they contain? To answer these questions, we adopt a geometrical point of view initiated by Michel Hénon in 1959, in order to explore and classify exhaustively the set of isochrone potentials and isochrone orbits. In particular, we provide a geometric generalization of Kepler's third law, and give a similar law for the apsidal angle, of any isochrone orbit. We also relate the set of isochrone orbits to the set of parabolae in the plane under linear transformations, and use this to derive an analytical parameterization of any isochrone orbit. Along the way we compare our results to known ones, pinpoint some interesting details of this mathematical physics problem, and argue that our geometrical methods can be exported to more generic orbits in potential theory.
• The Linear Sampling Method for Kirchhoff-Love Infinite Plates
  
  • Bourgeois Laurent
  • Recoquillay Arnaud
  Inverse Problems and Imaging, AIMS American Institute of Mathematical Sciences, 2020, 14 (2), pp.363-384. This paper addresses the problem of identifying impenetrable obstacles in a Kirchhoff-Love infinite plate from multistatic near-field data. The Linear Sampling Method is introduced in this context. We firstly prove a uniqueness result for such an inverse problem. We secondly provide the classical theoretical foundation of the Linear Sampling Method. We lastly show the feasibility of the method with the help of numerical experiments. (10.3934/ipi.2020016)
  DOI : 10.3934/ipi.2020016
• Adaptive solution of the neutron diffusion equation with heterogeneous coefficients using the mixed finite element method on structured meshes
  
  • Do M.-H.
  • Madiot F.
  • Ciarlet Patrick
  , 2020.
• On quasi-reversibility solutions to the Cauchy problem for the Laplace equation: regularity and error estimates
  
  • Bourgeois Laurent
  • Chesnel Lucas
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2020. We are interested in the classical ill-posed Cauchy problem for the Laplace equation. One method to approximate the solution associated with compatible data consists in considering a family of regularized well-posed problems depending on a small parameter ε > 0. In this context, in order to prove convergence of finite elements methods, it is necessary to get regularity results of the solutions to these regularized problems which hold uniformly in ε. In the present work, we obtain these results in smooth domains and in 2D polygonal geometries. In presence of corners, due to the particular structure of the regularized problems, classical techniques à la Grisvard do not work and instead, we apply the Kondratiev approach. We describe the procedure in detail to keep track of the dependence in ε in all the estimates. The main originality of this study lies in the fact that the limit problem is ill-posed in any framework. (10.1051/m2an/2019073)
  DOI : 10.1051/m2an/2019073
• Perfect Brewster transmission through ultrathin perforated films
  
  • Pham Kim
  • Maurel Agnès
  • Mercier Jean-François
  • Félix Simon
  • Cordero Maria Luisa
  • Horvath Camila
  Wave Motion, Elsevier, 2020, 93, pp.102485. We address the perfect transmission of a plane acoustic wave at oblique incidence on a perforated, sound penetrable or rigid, film in two-dimensions. It is shown that the Brewster incidence θ * realizing so-called extraordinary transmission due to matched impedances varies significantly when the thickness e of the film decreases. For thick films, i.e. ke ≫ 1 with k the incident wavenumber, the classical effective medium model provides an accurate prediction of the Brewster angle independent of e (this Brewster angle is denoted θ B). However, for thinner films with ke < 1, θ * becomes dependent of e and it deviates from θ B. To properly describe this shift, an interface model is used (Marigo et al., 2017) which accurately reproduces the spectra of ultrathin to relatively thick perforated films. Depending on the contrasts in the material properties of the film and of the surrounding matrix, decreasing the film thickness can produce an increase or a decrease of θ * ; it can also produce the disappearance of a perfect transmission or to the contrary its appearance. (10.1016/j.wavemoti.2019.102485)
  DOI : 10.1016/j.wavemoti.2019.102485
• Problème de Correlation Clustering avec Médiateurs
  
  • Alès Zacharie
  • Engelbeen Céline
  • Figueiredo Rosa
  , 2020.
• Degenerate elliptic equations for resonant wave problems
  
  • Nicolopoulos Anouk
  • Campos Pinto Martin
  • Després Bruno
  • Ciarlet Patrick
  IMA Journal of Applied Mathematics, Oxford University Press (OUP), 2020, 85 (1), pp.132-159. The modeling of resonant waves in 2D plasma leads to the coupling of two degenerate elliptic equations with a smooth coeffcient alpha and compact terms. The coeffcient alpha changes sign. The region where alpha is positive is propagative, and the region where alpha is negative is non propagative and elliptic. The two models are coupled through the line Sigma, corresponding to alpha equal to zero. Generically, it is an ill-posed problem, and additional information must be introduced to get a satisfactory treatment at Sigma. In this work we define the solution by relying on the limit absorption principle (alpha is replaced by alpha + i0^+) in an adapted functional setting. This setting lies on the decomposition of the solution in a regular part and a singular part, which originates at Sigma, and on quasi-solutions. It leads to a new well-posed mixed variational formulation with coupling. As we design explicit quasi-solutions, numerical experiments can be carried out, which illustrate the good properties of this new tool for numerical computation. (10.1093/imamat/hxaa001)
  DOI : 10.1093/imamat/hxaa001
• The identification problem for BSDEs driven by possibly non quasi-left-continuous random measures
  
  • Bandini Elena
  • Russo Francesco
  , 2020. In this paper we focus on the so called identification problem for a backward SDE driven by a continuous local martingale and a possibly non quasi-left-continuous random measure. Supposing that a solution (Y, Z, U) of a backward SDE is such that $Y(t) = v(t, X(t))$ where X is an underlying process and v is a deterministic function, solving the identification problem consists in determining Z and U in term of v. We study the over-mentioned identification problem under various sets of assumptions and we provide a family of examples including the case when X is a non-semimartingale jump process solution of an SDE with singular coefficients.
• Efficient computation of optimal open-loop controls for stochastic systems
  
  • Berret Bastien
  • Jean Frédéric
  Automatica, Elsevier, 2020, 115. (10.1016/j.automatica.2020.108874)
  DOI : 10.1016/j.automatica.2020.108874
• A fast boundary element method using the Z-transform and high-frequency approximations for large-scale 3D transient wave problems
  
  • Mavaleix-Marchessoux Damien
  • Bonnet Marc
  • Chaillat Stéphanie
  • Leblé Bruno
  International Journal for Numerical Methods in Engineering, Wiley, 2020, 121, pp.4734-4767. 3D rapid transient acoustic problems are difficult to solve numerically when dealing with large geometries, because numerical methods based on geometry discretisation (mesh), such as the boundary element method (BEM) or the finite element method (FEM), often require to solve a linear system (from the spacial discretisation) for each time step. We propose a numerical method to efficiently deal with 3D rapid transient acoustic problems set in large exterior domains. Using the Z-transform and the convolution quadrature method (CQM), we first present a straightforward way to reframe the problem to the solving of a large amount (the number of time steps, M) of frequency-domain BEMs. Then, taking advantage of a well-designed high-frequency approximation (HFA), we drastically reduce the number of frequency-domain BEMs to be solved, with little loss of accuracy. The complexity of the resulting numerical procedure turns out to be O(1) in regards to the time discretisation and O(N log N) for the spacial discretisation, the latter being prescribed by the complexity of the used fast BEM solver. Examples of applications are proposed to illustrate the efficiency of the procedure in the case of fluid-structure interaction: the radiation of an acoustic wave into a fluid by a deformable structure with prescribed velocity, and the scattering of an abrupt wave by simple and realistic geometries. (10.1002/nme.6488)
  DOI : 10.1002/nme.6488
• Modified forward and inverse Born series for the Calderon and diffuse-wave problems
  
  • Abhishek Anuj
  • Bonnet Marc
  • Moskow Shari
  Inverse Problems, IOP Publishing, 2020, 36, pp.114001. We propose a new direct reconstruction method based on series inversion for Electrical Impedance Tomography (EIT) and the inverse scattering problem for diffuse waves. The standard Born series for the forward problem has the limitation that the series requires that the contrast lies within a certain radius for convergence. Here, we instead propose a modified Born series which converges for the forward problem unconditionally. We then invert this modified Born series and compare reconstructions with the usual inverse Born series. We also show that the modified inverse Born series has a larger radius of convergence. (10.1088/1361-6420/abae11)
  DOI : 10.1088/1361-6420/abae11
• On well-posedness of scattering problems in a Kirchhoff-Love infinite plate
  
  • Bourgeois Laurent
  • Hazard Christophe
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2020, 80 (3), pp.1546-1566. We address scattering problems for impenetrable obstacles in an infinite elastic Kirchhoff-Love two-dimensional plate. The analysis is restricted to the purely bending case and the time-harmonic regime. Considering four types of boundary conditions on the obstacle, well-posedness for those problems is proved with the help of a variational approach: (i) for any wave number k when the plate is clamped, simply supported or roller supported; (ii) for any k except a discrete set when the plate is free (this set is finite for convex obstacles).
• On the regularity of abnormal minimizers for rank 2 sub-Riemannian structures
  
  • Barilari Davide
  • Chitour Yacine
  • Jean Frédéric
  • Prandi Dario
  • Sigalotti Mario
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2020, 133, pp.118-138. We prove the C^1 regularity for a class of abnormal length-minimizers in rank 2 sub-Riemannian structures. As a consequence of our result, all length-minimizers for rank 2 sub-Riemannian structures of step up to 4 are of class C^1 (10.1016/j.matpur.2019.04.008)
  DOI : 10.1016/j.matpur.2019.04.008
• A Feynman-Kac result via Markov BSDEs with generalized driver
  
  • Issoglio Elena
  • Russo Francesco
  Bernoulli, Bernoulli Society for Mathematical Statistics and Probability, 2020, 26, pp.728-766. In this paper we investigate BSDEs where the driver contains a distributional term (in the sense of generalised functions) and derive general Feynman-Kac formulae related to these BSDEs. We introduce an integral operator to give sense to the equation and then we show the existence of a strong solution employing results on a related PDE. Due to the irregularity of the driver, the $Y$-component of a couple $(Y,Z)$ solving the BSDE is not necessarily a semimartingale but a weak Dirichlet process. (10.3150/19-BEJ1150)
  DOI : 10.3150/19-BEJ1150
• Periodical body's deformations are optimal strategies for locomotion
  
  • Giraldi Laetitia
  • Jean Frédéric
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2020, 58 (3), pp.1700–1714. A periodical cycle of body's deformation is a common strategy for locomotion (see for instance birds, fishes, humans). The aim of this paper is to establish that the auto-propulsion of deformable object is optimally achieved using periodic strategies of body's deformations. This property is proved for a simple model using optimal control theory framework. (10.1137/19M1280120)
  DOI : 10.1137/19M1280120
• Demand response versus storage flexibility in energy: multi-objective programming considerations
  
  • van Ackooij Wim
  • Paula Chorobura Ana
  • Sagastizábal Claudia
  • Zidani Hasnaa
  Optimization, Taylor & Francis, 2020, pp.1-28. (10.1080/02331934.2020.1732373)
  DOI : 10.1080/02331934.2020.1732373
• Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering
  
  • Modave Axel
  • Geuzaine Christophe
  • Antoine Xavier
  Journal of Computational Physics, Elsevier, 2020, 401, pp.109029. This paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal and polyhedral computational domains for the finite element solution of high-frequency acoustic scattering problems. While high-order absorbing boundary conditions (HABCs) are accurate for smooth fictitious boundaries, the precision of the solution drops in the presence of corners if no specific treatment is applied. We present and analyze two strategies to preserve the accuracy of Padé-type HABCs at corners: first by using compatibility relations (derived for right angle corners) and second by regularizing the boundary at the corner. Exhaustive numerical results for two- and three-dimensional problems are reported in the paper. They show that using the compatibility relations is optimal for domains with right angles. For the other cases, the error still remains acceptable, but depends on the choice of the corner treatment according to the angle. (10.1016/j.jcp.2019.109029)
  DOI : 10.1016/j.jcp.2019.109029
• About classical solutions of the path-dependent heat equation
  
  • Di Girolami Cristina
  • Russo Francesco
  Random Operators and Stochastic Equations, De Gruyter, 2020, 1, pp.35-62. This paper investigates two existence theorems for the path-dependent heat equation, which is the Kolmogorov equation related to the window Brownian motion, considered as a C([−T, 0])-valued process. We concentrate on two general existence results of its classical solutions related to different classes of final conditions: the first one is given by a cylindrical non necessarily smooth r.v., the second one is a smooth generic functional. (10.1515/rose-2020-2028)
  DOI : 10.1515/rose-2020-2028
• Stability and Reachability analysis for a controlled heterogeneous population of cells
  
  • Carrère Cécile
  • Zidani Hasnaa
  Optimal Control Applications and Methods, Wiley, 2020, 41, pp.1678–1704. This paper is devoted to the study of a controlled population of cells. The modelling of the problem leads to a mathematical formulation of stability and reachability properties of some controlled systems under uncertainties. We use the Hamilton-Jacobi (HJ) approach to address theses problems and to design a numerical method that we analyse on several numerical simulations.
• On the approximation of electromagnetic fields by edge finite elements. Part 3: sensitivity to coefficients
  
  • Ciarlet Patrick
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2020. In bounded domains, the regularity of the solutions to boundary value problems depends on the gometry, and on the coefficients that enter into the definition of the model. This is in particular the case for the time-harmonic Maxwell equations, whose solutions are the electromagnetic fields. In this paper, emphasis is put on the electric field. We study the regularity in terms of the fractional order Sobolev spaces $H^s$, $s\in[0,1]$. Precisely, our first goal is to determine the regularity of the electric field and of its curl, that is to find some regularity exponent $\tau\in(0,1)$, such that they both belong to $H^s$, for all $s\in[0,\tau)$. After that, one can derive error estimates. Here, the error is defined as the difference between the exact field and its approximation, where the latter is built with N\'ed\'elec's first family of finite elements. In addition to the regularity exponent, one needs to derive a stability constant that relates the norm of the error to the norm of the data: this is our second goal. We provide explicit expressions for both the regularity exponent and the stability constant with respect to the coefficients. We also discuss the accuracy of these expressions, and we provide some numerical illustrations. (10.1137/19M1275383)
  DOI : 10.1137/19M1275383
• Shape optimization of Stokesian peristaltic pumps using boundary integral methods
  
  • Bonnet Marc
  • Liu Ruowen
  • Veerapaneni Shravan
  Journal of Computational and Applied Mathematics, Elsevier, 2020, 46, pp.18. This article presents a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only.By employing these formulas in conjunction with a boundary integral approach for solving forward and adjoint problems, we completely avoid the issue of volume remeshing when updating the pump shape as the optimization proceeds. This leads to significant cost savings and we demonstrate the performance on several numerical examples (10.1007/s10444-020-09761-7)
  DOI : 10.1007/s10444-020-09761-7
• Decoupled mild solutions of path-dependent PDEs and IPDEs represented by BSDEs driven by cadlag martingales.
  
  • Barrasso Adrien
  • Russo Francesco
  Potential Analysis, Springer Verlag, 2020, 53, pp.449-481. We focus on a class of path-dependent problems which include path-dependent PDEs and Integro PDEs (in short IPDEs), and their representation via BSDEs driven by a cadlag martingale. For those equations we introduce the notion of {\it decoupled mild solution} for which, under general assumptions, we study existence and uniqueness and its representation via the aforementioned BSDEs. This concept generalizes a similar notion introduced by the authors in recent papers in the framework of classical PDEs and IPDEs. For every initial condition $(s,\eta)$, where $s$ is an initial time and $\eta$ an initial path, the solution of such BSDE produces a couple of processes $(Y^{s,\eta},Z^{s,\eta})$. In the classical (Markovian or not) literature the function $u(s,\eta):= Y^{s,\eta}_s$ constitutes a viscosity type solution of an associated PDE (resp. IPDE); our approach allows not only to identify $u$ as the unique decoupled mild solution, but also to solve quite generally the so called {\it identification problem}, i.e. to also characterize the $(Z^{s,\eta})_{s,\eta}$ processes in term of a deterministic function $v$ associated to the (above decoupled mild) solution $u$. (10.1007/s11118-019-09775-x)
  DOI : 10.1007/s11118-019-09775-x