Publications

2017

• Domain decomposition methods for the diffusion equation with low-regularity solution
  
  • Ciarlet Patrick
  • Jamelot Erell
  • Kpadonou Félix D.
  Computers & Mathematics with Applications, Elsevier, 2017. We analyze matching and non-matching domain decomposition methods for the numerical approximation of the mixed diffusion equations. Special attention is paid to 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. The domain decomposition method can be non-matching in the sense that the traces of the finite elements spaces may not fit at the interface between subdomains. We prove well-posedness of the discrete problem, that is solvability of the corresponding linear system, provided two algebraic conditions are fulfilled. If moreover the conditions hold independently of the discretization, convergence is ensured. (10.1016/j.camwa.2017.07.017)
  DOI : 10.1016/j.camwa.2017.07.017
• On some extremal problems for analytic functions with constraints on real or imaginary parts
  
  • Leblond Juliette
  • Ponomarev Dmitry
  , 2017, pp.219-236. We study some approximation problems by functions in the Hardy space H 2 of the upper half-plane or by their real or imaginary parts, with constraint on their real or imaginary parts on the boundary. Situations where the criterion acts on subsets of the boundary or of horizontal lines inside the half-plane are considered. Existence and uniqueness results are established, together with novel solution formulas and techniques. As a by-product, we devise a regularized inversion scheme for Poisson and conjugate Poisson integral transforms. (10.1007/978-3-319-62362-7_8)
  DOI : 10.1007/978-3-319-62362-7_8
• Finite Element and Discontinuous Galerkin Methods for Transient Wave Equations
  
  • Cohen Gary
  • Pernet Sebastien
  , 2017. This monograph presents numerical methods for solving transient wave equations (i.e. in time domain). More precisely, it provides an overview of continuous and discontinuous finite element methods for these equations, including their implementation in physical models, an extensive description of 2D and 3D elements with different shapes, such as prisms or pyramids, an analysis of the accuracy of the methods and the study of the Maxwell’s system and the important problem of its spurious free approximations. After recalling the classical models, i.e. acoustics, linear elastodynamics and electromagnetism and their variational formulations, the authors present a wide variety of finite elements of different shapes useful for the numerical resolution of wave equations. Then, they focus on the construction of efficient continuous and discontinuous Galerkin methods and study their accuracy by plane wave techniques and a priori error estimates. A chapter is devoted to the Maxwell’s system and the important problem of its spurious-free approximations. Treatment of unbounded domains by Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML) is described and analyzed in a separate chapter. The two last chapters deal with time approximation including local time-stepping and with the study of some complex models, i.e. acoustics in flow, gravity waves and vibrating thin plates. Throughout, emphasis is put on the accuracy and computational efficiency of the methods, with attention brought to their practical aspects.This monograph also covers in details the theoretical foundations and numerical analysis of these methods. As a result, this monograph will be of interest to practitioners, researchers, engineers and graduate students involved in the numerical simulationof waves. (10.1007/978-94-017-7761-2)
  DOI : 10.1007/978-94-017-7761-2
• Uniqueness for a class of stochastic Fokker-Planck and porous media equations
  
  • Röckner Michael
  • Russo Francesco
  Journal of Evolution Equations, Springer Verlag, 2017, 17 (3), pp.1049-1062. The purpose of the present note consists of first showing a uniqueness result for a stochastic Fokker-Planck equation under very general assumptions. In particular, the second order coefficients may be just measurable and degenerate. We also provide a proof for uniqueness of a stochastic porous media equation in a fairly large space. (10.1007/s00028-016-0372-0)
  DOI : 10.1007/s00028-016-0372-0
• Manipulating light at subwavelength scale by exploiting defect-guided spoof plasmon modes
  
  • Ourir Abdelwaheb
  • Maurel Agnes
  • Félix Simon
  • Mercier Jean-François
  • Fink Mathias
  Physical Review B: Condensed Matter and Materials Physics (1998-2015), American Physical Society, 2017, 96 (12). We study the defect-guided modes supported by a set of metallic rods structured at the subwavelength scale. Following the idea of photonic crystal waveguide, we show that spoof plasmon surface waves can be manipulated at subwavelength scale. We demonstrate that these waves can propagate without leakage along a row of rods having a different length than the surrounding medium and we provide the corresponding dispersion relation. The principle of this subwavelength colored guide is validated experimentally. This allows us to propose the design of a wavelength demultiplexer whose efficiency is illustrated in the microwave regime. (10.1103/PhysRevB.96.125133)
  DOI : 10.1103/PhysRevB.96.125133
• Inverse Optimal Control Problem: the Sub-Riemannian Case
  
  • Jean Frédéric
  • Maslovskaya Sofya
  • Zelenko Igor
  IFAC-PapersOnLine, Elsevier, 2017, 50 (1). The object of this paper is to study the uniqueness of solutions of inverse control problems in the case where the dynamics is given by a control-affine system without drift and the costs are length and energy functionals. (10.1016/j.ifacol.2017.08.105)
  DOI : 10.1016/j.ifacol.2017.08.105
• Hamilton–Jacobi–Bellman Equations
  
  • Festa Adriano
  • Guglielmi Roberto
  • Hermosilla Cristopher
  • Picarelli Athena
  • Sahu Smita
  • Sassi Achille
  • Silva Francisco José
  , 2017, Optimal Control: Novel Directions and Applications (2180), pp.127-261. In this chapter we present recent developments in the theory of Hamilton–Jacobi–Bellman (HJB) equations as well as applications. The intention of this chapter is to exhibit novel methods and techniques introduced few years ago in order to solve long-standing questions in nonlinear optimal control theory of Ordinary Differential Equations (ODEs). (10.1007/978-3-319-60771-9)
  DOI : 10.1007/978-3-319-60771-9
• Doubly probabilistic representation for the stochastic porous media type equation.
  
  • Barbu Viorel
  • Röckner Michael
  • Russo Francesco
  Annales de l'Institut Henri Poincaré, Presses universitaires de France — PUF, 2017. The purpose of the present paper consists in proposing and discussing a doubly probabilistic representation for a stochastic porous media equation in the whole space R^1 perturbed by a multiplicative coloured noise. For almost all random realizations ω, one associates a stochastic differential equation in law with random coefficients, driven by an independent Brownian motion. (10.1214/16-AIHP783)
  DOI : 10.1214/16-AIHP783
• Mathematical models for dispersive electromagnetic waves: An overview
  
  • Cassier Maxence
  • Joly Patrick
  • Kachanovska Maryna
  Computers & Mathematics with Applications, Elsevier, 2017, 74 (11), pp.2792-2830. In this work, we investigate mathematical models for electromagnetic wave propagation in dispersive isotropic media. We emphasize the link between physical requirements and mathematical properties of the models. A particular attention is devoted to the notion of non-dissipativity and passivity. We consider successively the case of so-called local media and general passive media. The models are studied through energy techniques, spectral theory and dispersion analysis of plane waves. For making the article self-contained, we provide in appendix some useful mathematical background. (10.1016/j.camwa.2017.07.025)
  DOI : 10.1016/j.camwa.2017.07.025
• Weak Dirichlet processes with jumps
  
  • Bandini Elena
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2017, 12, pp.4139-4189. This paper develops systematically stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N, A] = 0, for any continuous local martingale N. In particular, given a function u : [0, T ] × R → R, which is of class C^{0,1} (or sometimes less), we provide a chain rule type expansion for X_t = u(t, X_t) which stands in applications for a chain Itô type rule. (10.1016/j.spa.2017.04.001)
  DOI : 10.1016/j.spa.2017.04.001
• Décomposition-coordination en optimisation déterministe et stochastique
  
  • Carpentier Pierre
  • Cohen Guy
  , 2017, 81. (10.1007/978-3-662-55428-9)
  DOI : 10.1007/978-3-662-55428-9
• A note on time-dependent additive functionals
  
  • Barrasso Adrien
  • Russo Francesco
  Communications on Stochastic Analysis, Serials Publications, 2017, 11 (3), pp.313-334. This note develops shortly the theory of time-inhomogeneous additive functionals and is a useful support for the analysis of time-dependent Markov processes and related topics. It is a significant tool for the analysis of BSDEs in law. In particular we extend to a non-homogeneous setup some results concerning the quadratic variation and the angular bracket of Martin-gale Additive Functionals (in short MAF) associated to a homogeneous Markov processes. (10.31390/cosa.11.3.04)
  DOI : 10.31390/cosa.11.3.04
• Infinite Dimensional Weak Dirichlet Processes and Convolution Type Processes
  
  • Fabbri Giorgio
  • Russo Francesco
  Stochastic Processes and their Applications, Elsevier, 2017, 127 (1), pp.325-357. The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of weak Dirichlet process in this context. Such a process X, taking values in a Banach space H, is the sum of a local martingale and a suitable orthogonal process. The concept of weak Dirichlet process fits the notion of convolution type processes, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an Itô’s type formula applied to to f(t, X(t)) where f : [0, T ] × H → R is a C0,1 function and X a convolution type processes. (10.1016/j.spa.2016.06.010)
  DOI : 10.1016/j.spa.2016.06.010
• Multidimensional stochastic differential equations with distributional drift
  
  • Flandoli Franco
  • Issoglio Elena
  • Russo Francesco
  Transactions of the American Mathematical Society, Series B, American Mathematical Society, 2017, 369 (3), pp.1655-1688. This paper investigates a time-dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of Sobolev spaces with negative derivation order. This is done through a careful analysis of the corresponding Kolmogorov equation whose coefficient is a distribution. (10.1090/tran/6729)
  DOI : 10.1090/tran/6729
• Multi-usage hydropower single dam management: chance-constrained optimization and stochastic viability
  
  • Carpentier Pierre
  • Alais Jean-Christophe
  • de Lara Michel
  Energy Systems, Springer, 2017, 8 (1), pp.7–30. We consider the management of a single hydroelectric dam, subject to uncertain inflows and electricity prices and to a so-called “tourism constraint”: the water storage level must be high enough during the tourist season with high enough probability. We cast the problem in the stochastic optimal control framework: we search at each time t the optimal control as a function of the available information at t. We lay out two approaches. First, we formulate a chance-constrained stochastic optimal control problem: we maximize the expected gain while guaranteeing a minimum storage level with a minimal prescribed probability level. Dualizing the chance constraint by a multiplier, we propose an iterative algorithm alternating additive dynamic programming and update of the multiplier value “à la Uzawa”. Our numerical results reveal that the random gain is very dispersed around its expected value; in particular, low gain values have a relatively high probability to materialize. This is why, to put emphasis on these low values, we outline a second approach. We propose a so-called stochastic viability approach that focuses on jointly guaranteeing a minimum gain and a minimum storage level during the tourist season. We solve the corresponding problem by multiplicative dynamic programming. To conclude, we discuss and compare the two approaches. Keywords (10.1007/s12667-015-0174-4)
  DOI : 10.1007/s12667-015-0174-4
• A modified volume integral equation for anisotropic elastic or conducting inhomogeneities. Unconditional solvability by Neumann series
  
  • Bonnet Marc
  Journal of Integral Equations and Applications, Rocky Mountain Mathematics Consortium, 2017, 29, pp.271-295. This work addresses the solvability and solution of volume integrodifferential equations (VIEs) associated with 3D free-space transmission problems (FSTPs) involving elastic or conductive inhomogeneities. A modified version of the singular volume integral equation (SVIE) associated with the VIE is introduced and shown to be of second kind involving a contraction operator, i.e. solvable by Neumann series, implying the well-posedness of the initial VIE. Then, the solvability of VIEs for frequency-domain FSTPs (modelling the scattering of waves by compactly-supported inhomogeneities) follows by a compact perturbation argument. This approach extends work by Potthast (1999) on 2D electromagnetic problems (transverse-electric polarization conditions) involving orthotropic inhomogeneities in a isotropic background, and contains recent results on the solvability of Eshelby's equivalent inclusion problem as special cases. The proposed modified SVIE is also useful for iterative solution methods, as Neumannn series converge (i) unconditionally for static problems and (ii) on some inhomogeneity configurations for which divergence occurs with the usual SVIE for wave scattering problems.
• On the duration of human movement: from self-paced to slow/fast reaches up to Fitts's law
  
  • Jean Frédéric
  • Berret Bastien
  , 2017, 117. In this chapter, we present a mathematical theory of human movement vigor. At the core of the theory is the concept of the cost of time. According to it, natural movement cannot be too slow because the passage of time entails a cost which makes slow moves undesirable. Within this framework, an inverse methodology is available to reliably and robustly characterize how the brain penalizes time from experimental motion data. Yet, a general theory of human movement pace should not only account for the self-selected speed but should also include situations where slow or fast speed instructions are given by an experimenter or required by a task. In particular, the limit case of a " maximal speed " instruction is linked to Fitts's law, i.e. the speed/accuracy trade-off. This chapter first summarizes the cost of time theory and the procedure used for its accurate identification. Then, the case of slow/fast movements is investigated but changing the duration of goal-directed movements can be done in various ways in this framework. Here we show that only one strategy seems plausible to account for both slow/fast and self-paced reaching movements. By relying upon a free-time optimal control formulation of the motor planning problem, this chapter provides a comprehensive treatment of the linear-quadratic
• Discontinuous solutions of Hamilton-Jacobi equations on networks *
  
  • Graber Philip Jameson
  • Hermosilla Cristopher
  • Zidani Hasnaa
  Journal of Differential Equations, Elsevier, 2017, 263 (12), pp.8418-8466. This paper studies optimal control problems on networks without controllability assumptions at the junctions. The Value Function associated with the control problem is characterized as solution to a system of Hamilton-Jacobi equations with appropriate junction conditions. The novel feature of the result lies in that the controllability conditions are not needed and the characterization remains valid even when the Value Function is not continuous. (10.1016/j.jde.2017.08.040)
  DOI : 10.1016/j.jde.2017.08.040
• Fast iterative boundary element methods for high-frequency scattering problems in 3D elastodynamics
  
  • Chaillat Stéphanie
  • Darbas Marion
  • Le Louër Frédérique
  Journal of Computational Physics, Elsevier, 2017, 341, pp.429 - 446. The fast multipole method is an efficient technique to accelerate the solution of large scale 3D scattering problems with boundary integral equations. However, the fast multipole accelerated boundary element method (FM-BEM) is intrinsically based on an iterative solver. It has been shown that the number of iterations can significantly hinder the overall efficiency of the FM-BEM. The derivation of robust preconditioners for FM-BEM is now inevitable to increase the size of the problems that can be considered. The main constraint in the context of the FM-BEM is that the complete system is not assembled to reduce computational times and memory requirements. Analytic preconditioners offer a very interesting strategy by improving the spectral properties of the boundary integral equations ahead from the discretization. The main contribution of this paper is to combine an approximate adjoint Dirichlet to Neumann (DtN) map as an analytic preconditioner with a FM-BEM solver to treat Dirichlet exterior scattering problems in 3D elasticity. The approximations of the adjoint DtN map are derived using tools proposed in [40]. The resulting boundary integral equations are preconditioned Combined Field Integral Equations (CFIEs). We provide various numerical illustrations of the efficiency of the method for different smooth and non smooth geometries. In particular, the number of iterations is shown to be completely independent of the number of degrees of freedom and of the frequency for convex obstacles. (10.1016/j.jcp.2017.04.020)
  DOI : 10.1016/j.jcp.2017.04.020
• Application of asymptotic analysis to the two-scale modeling of small defects in mechanical structures
  
  • Marenić Eduard
  • Brancherie Delphine
  • Bonnet Marc
  International Journal of Solids and Structures, Elsevier, 2017, 128, pp.199-209. This work aims at designing a numerical strategy towards assessing the nocivity of a small defect in terms of its size and position in a structure, at low computational cost, using only a mesh of the defect-free reference structure. The modification of the fields induced by the presence of a small defect is taken into account by using asymptotic corrections of displacements or stresses. This approach helps determining the potential criticality of defects by considering trial micro-defects with varying positions, sizes and mechanical properties, taking advantage of the fact that parametric studies on defect characteristics become feasible at virtually no extra computational cost. The proposed treatment is validated and demonstrated on two numerical examples involving 2D elastic configurations. (10.1016/j.ijsolstr.2017.08.029)
  DOI : 10.1016/j.ijsolstr.2017.08.029
• Stable perfectly matched layers for a class of anisotropic dispersive models. Part I: Necessary and sufficient conditions of stability
  
  • Bécache Eliane
  • Kachanovska Maryna
  ESAIM: Mathematical Modelling and Numerical Analysis, Société de Mathématiques Appliquées et Industrielles (SMAI) / EDP, 2017, 51 (6), pp.2399-2434. In this work we consider a problem of modelling of 2D anisotropic dispersive wave propagation in unbounded domains with the help of perfectly matched layers (PML). We study the Maxwell equations in passive media with a frequency-dependent diagonal tensor of dielectric permittivity and magnetic permeability. An application of the traditional PMLs to this kind of problems often results in instabilities. We provide a recipe for the construction of new, stable PMLs. For a particular case of non-dissipative materials, we show that a known necessary stability condition of the perfectly matched layers is also sufficient. We illustrate our statements with theoretical and numerical arguments. (10.1051/m2an/2017019)
  DOI : 10.1051/m2an/2017019
• Hamilton-Jacobi-Bellman equations for optimal control processes with convex state constraints
  
  • Hermosilla Cristopher
  • Vinter Richard
  • Zidani Hasnaa
  Systems and Control Letters, Elsevier, 2017, 109, pp.30–36.
• Fonctions d'une variable complexe (avec des applications et de nombreux dessins)
  
  • Lenoir Marc
  , 2017.
• The Mayer and Minimum Time Problems with Stratified State Constraints *
  
  • Hermosilla C.
  • Wolenski P.R.
  • Zidani Hasnaa
  Set-Valued and Variational Analysis, Springer, 2017. This paper studies optimal control problems with state constraints by imposing structural assumptions on the constraint domain coupled with a tangential restriction with the dynamics. These assumptions replace pointing or controllability assumptions that are common in the literature, and provide a framework under which feasible boundary trajectories can be analyzed directly. The value functions associated with the state constrained Mayer and minimal time problems are characterized as solutions to a pair of Hamilton-Jacobi inequalities with appropriate boundary conditions. The novel feature of these inequalities lies in the choice of the Hamiltonian. (10.1007/s11228-017-0413-z)
  DOI : 10.1007/s11228-017-0413-z
• Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time
  
  • Russo Francesco
  • Wurzer Lukas
  Stochastics and Dynamics, World Scientific Publishing, 2017, 17, pp.1750030. We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of a BSDE with random terminal time when the driving process is a general càdlàg martingale. (10.1142/S0219493717500307)
  DOI : 10.1142/S0219493717500307