Section: New Results

Inverse problems

Linear Sampling Method with realistic data in waveguides

Participants : Laurent Bourgeois, Arnaud Recoquillay.

Our activities in the field of inverse scattering in waveguides with the help of sampling methods has now a quite long history. Very recently, we have focused on elastodynamics and realistic data, that is surface data in the time domain. This has been the subject of the PhD of Arnaud Recoquillay. It was motivated by Non Destructive Testing activities for tubular structures and was the object of a partnership with CEA List (Vahan Baronian).

Our strategy consists in transforming the time domain problem into a multi-frequency problem by the Fourier transform. This allows us to take full advantage of the established efficiency of modal frequency-domain sampling methods. In particular, we have shown how to optimize the number of sources/receivers and the distance between them in order to obtain the best possible imaging results.

Our main achievement is an experimental validation of such approach in the presence of real data: the measurements were carried at CEA on steel plates with the help of piezoelectric sensors. The identification results are encouraging and pave the way of a future integration of sampling methods in real NDT activities.

The "exterior approach" to solve inverse obstacle problems

Participants : Laurent Bourgeois, Arnaud Recoquillay, Dmitry Ponomarev.

This work is done in collaboration with Jérémi Dardé (IMT Toulouse).

We consider some inverse obstacle problems in acoustics by using a single incident wave, either in the frequency or in the time domain. When so few data are available, a Linear Sampling type method cannot be applied. In order to solve those kinds of problem, we propose an "exterior approach", coupling a mixed formulation of quasi-reversibility and a simple level set method. In such iterative approach, for a given defect $D$, we update the solution $u$ with the help of a mixed formulation of quasi-reversibility while for a given solution $u$, we update the defect $D$ with the help of a level set method based on a Poisson problem. We have studied two cases. The first case concerns the waveguide geometry in the frequency domain. The second case concerns a bounded spatial set in the time domain when data are given in a finite time interval. This last case is challenging because it raises the (open) question of the minimal final time which is required to ensure uniqueness of the obstacle from the lateral Cauchy data.

Inverse acoustic scattering using high-order small-inclusion expansion of misfit function

Participant : Marc Bonnet.

This work concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function $J$ is expanded in powers of the characteristic radius $a$ of a single small inhomogeneity. The $O\left(a^6\right)$ approximation of $J$ is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape, and the generalization to multiple small obstacles is outlined. Simpler and more explicit expressions are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing the $O\left(a^6\right)$ approximation over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of $J$, is considered. measurements configuration situated far enough from the probing region.

Microstructural topological sensitivities of the second-order macroscopic model for waves in periodic media

Participant : Marc Bonnet.

This work is done in collaboration with Bojan Guzina (University of Minnesota, USA) and Rémi Cornaggia (IRMAR, Rennes).

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 relevant 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 counterparts. 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.

Analysis of topological derivative as a tool for qualitative identification

Participant : Marc Bonnet.

This work is a collaboration with Fioralba Cakoni (Rutgers University, USA).

The concept of topological derivative has proved effective as a qualitative inversion tool for a wave-based identification of finite-sized objects. Although for the most part, this approach remains based on a heuristic interpretation of the topological derivative, a first attempt toward its mathematical justification was done in Bellis et al. (Inverse Problems 29:075012, 2013) for the case of isotropic media with far field data and inhomogeneous refraction index. Our paper extends the analysis there to the case of anisotropic scatterers and background with near field data. Topological derivative-based imaging functional is analyzed using a suitable factorization of the near fields, which became achievable thanks to a new volume integral formulation recently obtained in Bonnet (J. Integral Equ. Appl. 29:271-295, 2017). Our results include justification of sign heuristics for the topological derivative in the isotropic case with jump in the main operator and for some cases of anisotropic media, as well as verifying its decaying property in the isotropic case with near field spherical measurements configuration situated far enough from the probing region.

Elasticity imaging by error in constitutive equation functionals

Participant : Marc Bonnet.

This work is done in collaboration with Wilkins Aquino (Duke University, USA).

We formulate the identification of heterogeneous linear elastic moduli in the context of time-harmonic elastodynamics as the minimization of the modified error in constitutive equation (MECE) functional. Our main goal is to develop theoretical foundations, in a continuous setting, allowing to explain and justify some known beneficial properties of this treatment. A specific feature of MECE formulations is that forward and adjoint solutions are governed by a fully coupled system, whose mathematical properties play a fundamental role in the qualitative and computational aspects of MECE minimization. We prove that this system has a unique and stable solution at any frequency, provided data is abundant enough (in a sense made precise), even though the relevant forward problem is not a priori clearly defined. This result has practical implications such as applicability of MECE to partial interior data (with important practical applications including ultrasound elastography), convergence of finite element discretizations and differentiability of the reduced MECE functional. In addition, we establish that usual least squares and pure ECE formulations are limiting cases of MECE formulations for small and large values of the weight of the data misfit component of the functional, respectively. For the latter case, we furthermore show that the reduced MECE Hessian is asymptotically positive for any parameter perturbation supported on the measurement region, thereby corroborating existing computational evidence on convexity improvement brought by MECE functionals. Finally, numerical studies including parameter reconstruction examples using interior data support our findings.

A continuation method for building large invisible obstacles in waveguides

Participants : Antoine Bera, Anne-Sophie Bonnet-Ben Dhia.

This work is done on collaboration with Lucas Chesnel (EPI DEFI).

We are interested in building invisible obstacles in waveguides, at a given frequency. The invisibility is characterized by the nullity of the scattering coefficients associated to propagating modes. In previous papers, a method has been proposed to prove the existence of invisible obstacles and to build them. But its main drawback was its limitation to small obstacles. In order to get larger invisible obstacles, we have developed a new approach which combines the previous idea with a continuation method: we are building a sequence of invisible obstacles, each of them being a small perturbation of the previous one. This algorithm is based, at each step, on the ontoness of an application and on the fixed-point theorem. We have implemented the method in the finite element library XLiFE++, in the case of penetrable obstacles of a two-dimensional acoustic waveguide, in multi-modal regime. A remarkable result is that the ontoness condition can be ensured in many cases, so that the algorithm can be iterated as long as required. Another interesting feature of our approach is that it allows to prescribe some properties of the obstacle (shape of the obstacle, piecewise constant index, ...), but a drawback is that the algorithm can produce non-realistic negative indices. This is a question that we are currently working on. Finally, let us emphasize that the formalism of the method is very general and flexible. In particular, it can be directly extended to 3D waveguides, or to the scattering in free space.