Publications

2022

• On the justification of topological derivative for wave-based qualitative imaging of finite-sized defects in bounded media
  
  • Bonnet Marc
  Engineering Computations, Emerald, 2022, 39 (1), pp.313-336. The concept of topological derivative (TD) is known to provide, through its heuristic interpretation involving its sign and its spatial decay away from the true anomaly, a basis for the qualitative imaging of finite-sized anomalies. The TD imaging heuristic is currently partially backed by conditional mathematical justifications. Continuing earlier efforts towards the justification of TD-based identification, this work investigates the acoustic wave-based imaging of finite-sized (i.e. not necessarily small) medium anomalies embedded in bounded domains and affecting the leading-order term of the acoustic field equation. Both the probing excitation and the measurement are assumed to take place on the domain boundary. We extend to this setting the analysis approach previously used for unbounded media with either refraction-index anomalies and far-field measurements (Bellis et al., \emph{Inverse Problems} \textbf{29}:075012, 2013) or mass-density anomalies and meaurements at finite distance (Bonnet, Cakoni, \emph{Inverse Problems} \textbf{35}:104007, 2019). Like in the latter work, TD-based imaging functionals are reformulated for analysis using a suitable factorization of the acoustic fields, facilitated by a volume integral formulation. Our results, which echo corresponding results of our earlier investigations, conditionally validate the TD imaging heuristic. Moreover, we show on a geometrically simple configuration that the spatial behavior of the TD associated with standard $L^2$ cost functionals is degraded by ``echoes'' of the true anomaly, an aspect specific to the present bounded-domain framework. This undesirable effect is removed by a combination of (i) post-processing the measurements by application of a suitable integral operator (a treatment introduced by Ammari et al., 2011, for the analysis of TD-based imaging involving true flaws modelled using small-anomaly asymptotics), and (ii) expressing the background field as an incoming single-layer potential defined in the full space (after an idea used in Bonnet, Cakoni, 2019). Finally, we also show that selecting eigenfunctions of the source-to-measurement operator as excitations enhances the spatial decay properties of the TD functionals (10.1108/EC-08-2021-0471)
  DOI : 10.1108/EC-08-2021-0471
• Backward Stochastic Differential Equations with no driving martingale, Markov processes and associated Pseudo Partial Differential Equations
  
  • Barrasso Adrien
  • Russo Francesco
  Journal of Stochastic Analysis, Louisiana State University, 2022, 3 (1). We discuss a class of Backward Stochastic Differential Equations (BSDEs) with no driving martingale. When the randomness of the driver depends on a general Markov process $X$, those BSDEs are denominated Markovian BSDEs and can be associated to a deterministic problem, called Pseudo-PDE which constitute the natural generalization of a parabolic semilinear PDE which naturally appears when the underlying filtration is Brownian. We consider two aspects of well-posedness for the Pseudo-PDEs: "classical" and "martingale" solutions. (10.31390/josa.3.1.03)
  DOI : 10.31390/josa.3.1.03
• Limiting amplitude principle and resonances in plasmonic structures with corners: numerical investigation
  
  • Carvalho Camille
  • Ciarlet Patrick
  • Scheid Claire
  Computer Methods in Applied Mechanics and Engineering, Elsevier, 2022, 388, pp.114207. The limiting amplitude principle states that the response of a scatterer to a harmonic light excitation is asymptotically harmonic with the same pulsation. Depending on the geometry and nature of the scatterer, there might or might not be an established theoretical proof validating this principle. In this paper, we investigate a case where the theory is missing: we consider a two-dimensional dispersive Drude structure with corners. In the non lossy case, it is well known that looking for harmonic solutions leads to an ill-posed problem for a specific range of critical pulsations, characterized by the metal’s properties and the aperture of the corners. Ill-posedness is then due to highly oscillatory resonances at the corners called black-hole waves. However, a time-domain formulation with a harmonic excitation is always mathematically valid. Based on this observation, we conjecture that the limiting amplitude principle might not hold for all pulsations. Using a time-domain setting, we propose a systematic numerical approach that allows to give numerical evidences of the latter conjecture, and find clear signature of the critical pulsa- tions. Furthermore, we connect our results to the underlying physical plasmonic resonances that occur in the lossy physical metallic case. (10.1016/j.cma.2021.114207)
  DOI : 10.1016/j.cma.2021.114207
• An extension of the proximal point algorithm beyond convexity
  
  • Grad Sorin-Mihai
  • Lara Felipe
  Journal of Global Optimization, Springer Verlag, 2022, 82, pp.313–329. Abstract We introduce and investigate a new generalized convexity notion for functions called prox-convexity. The proximity operator of such a function is single-valued and firmly nonexpansive. We provide examples of (strongly) quasiconvex, weakly convex, and DC (difference of convex) functions that are prox-convex, however none of these classes fully contains the one of prox-convex functions or is included into it. We show that the classical proximal point algorithm remains convergent when the convexity of the proper lower semicontinuous function to be minimized is relaxed to prox-convexity. (10.1007/s10898-021-01081-4)
  DOI : 10.1007/s10898-021-01081-4
• 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 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
• On the Half-Space Matching Method for Real Wavenumber
  
  • Bonnet-Ben Dhia Anne-Sophie
  • Chandler-Wilde Simon N
  • Fliss Sonia
  SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2022, 82 (4). 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 coupling (1) a standard finite element discretisation localised around the scatterer and (2) integral equations whose unknowns are traces of the solution on the boundaries of a finite number of overlapping half-planes contained in the domain. While satisfactory numerical results have been obtained for real wavenumbers, well-posedness and equivalence of this HSM formulation to the original scattering problem have been established only for complex wavenumbers. In the present paper we show, in the case of a homogeneous background, that the HSM formulation is equivalent to the original scattering problem also for real wavenumbers, and so is well-posed, provided the traces satisfy radiation conditions at infinity analogous to the standard Sommerfeld radiation condition. As a key component of our argument we show that, if the trace on the boundary of a half-plane satisfies our new radiation condition, then the corresponding solution to the half-plane Dirichlet problem satisfies the Sommerfeld radiation condition in a slightly smaller half-plane. We expect that this last result will be of independent interest, in particular in studies of rough surface scattering. (10.1137/21M1459216)
  DOI : 10.1137/21M1459216
• 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