Section: New Results

Inverse problems

Quasi-Reversibility method and exterior approach for evolution problems

Participant : Laurent Bourgeois.

This work is a collaboration with Jérémi Dardé from Toulouse University. We address some linear ill-posed problems involving the heat or the wave equation, in particular the heat/wave equation with lateral Cauchy data. We have introduced several kinds of variational mixed formulations of quasi-reversibility which enable us to solve these ill-posed problems by using classical Lagrange finite elements. We have also designed a new approach called the “exterior approach” to solve inverse obstacle problems with initial condition and lateral Cauchy data for the heat/wave equation. It is based on a combination of an elementary level set method and the quasi-reversibility methods we have just mentioned. Some numerical experiments have proved the feasibility of our strategy to identify obstacles from lateral Cauchy data for the heat equation in 2D and for the wave equation in 1D. Our objective is now to focus on the wave equation in 2D. Firstly we wish to obtain a minimal value of the final time in order to ensure uniqueness of the obstacle from the lateral Cauchy data. Secondly we want to test our exterior approach numerically. We expect better results than with the heat equation.

Higher-order expansion of misfit functional for defect identification in elastic solids

Participants : Marc Bonnet, Rémi Cornaggia.

This work, done in the context of the PhD of Rémi Cornaggia, concerns the identification of scatterers of moderate size, modelled as elastic inhomogeneities embedded in an homogeneous elastic background medium, by time-harmonic elastodynamic measurements. Least-squares functionals, commonly used for defect identification, are expanded in powers of the small characteristic radius $a$ of a trial inhomogeneity. This entails the expansion of the elastodynamic scattering problem, which is needed only on the support of the trial inhomogeneity and is established by means of a Lippmann-Schwinger volume integral equation. This approach generalizes, to higher orders in $a$, the well-known concept of topological derivative. Such expansion, whose derivation and evaluation are facilitated by using an adjoint state, provides a basis for the quantitative estimation of flaws whereby a region of interest may be exhaustively probed at reasonable computational cost. So far, the higher-order expansion has been derived under fairly general conditions, mathematically justified, and demonstrated on simple numerical examples involving the identification of a spherical inhomogeneity in an unbounded 3D medium.

Complete transmission invisibility in waveguides

Participant : Anne-Sophie Bonnet-Ben Dhia.

In collaboration with Lucas Chesnel (Inria, Defi) and Sergei Nazarov (Saint-Petersburg University), we consider time harmonic acoustic problems in waveguides. We are interested in finding localized perturbations of a straight waveguide which are not detectable in the far field, as they produce neither reflection nor conversion of propagative modes. In other words, such invisible perturbation produces a scattered field which is exponentially decaying at infinity in the two infinite outlets of the waveguide.

In our previous contributions, we found a way to build smooth and small perturbations of the boundary which were almost invisible, in the sense that they were producing no reflexions but maybe a phase shift in transmission.

During the visit of Sergei Nazarov, we found a new approach which allows to build completely invisible perturbations in the mono-mode regime (i.e. when the frequency is chosen below the first cut-off frequency) with no phase shift in transmission. These perturbations include some kinds of thin resonators whose height is adapted to the frequency.

All our results mainly rely on asymptotic theory.

Energy-based cost functional for three-dimensional transient elastodynamic imaging

Participant : Marc Bonnet.

This work is a continuing collaboration with Wilkins Aquino (Duke University, USA). It is concerned with three-dimensional elastodynamic imaging by means of the modified error in constitutive relation (MECR), combining the energy norm of the constitutive residual and a more-classical $L^2$ norm on the measurement residuals.

We have in particular considered the case of imaging using interior data. The stationarity equations associated with the minimization of a MECR objective function, subject to the conservation of linear momentum, yields a well-posed problem coupling two elastodynamic fields, even in cases where boundary conditions are initially underspecified (making it difficult to define a priori a forward problem). Numerical results demonstrate the robust performance of the method in situations where the available measurement data is incomplete and corrupted by noise of varying levels.

In a separate study, elastodynamic imaging using transient data and based on time-domain solvers has been investigated. In this context, each evaluation of a time-domain MECR cost functional entails solving two elastodynamic problems (one forward, one backward), which moreover are coupled (unlike the case of $L^2$ misfit functionals). This coupling creates a major computational bottleneck, making MECR-based inversion difficult for spatially 2D or 3D configurations. To overcome this obstacle, we propose to (a) set the entire computational procedure in a consistent time-discrete framework that incorporates the chosen time-stepping algorithm, and (b) use an iterative successive over-relaxation-like method for the resulting stationarity equations. The resulting MECR-based inversion algorithm is formulated under quite general conditions, allowing for 3D transient elastodynamics, straightforward use of available parallel solvers, a wide array of time-stepping algorithms commonly used for transient structural dynamics, and flexible boundary conditions and measurement settings. The proposed MECR algorithm is then demonstrated on computational experiments involving 2D and 3D transient elastodynamics and up to over 500 000 unknown elastic moduli.

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. We now intend to apply these methods in the case of realistic data, that is surface data in the time domain. This is the subject of the PhD of Arnaud Recoquillay. It is motivated by Non Destructive Testing activities for tubular structures and is 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. We have already proved the feasibility of our approach in the 2D acoustic case. 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 identification result. The next steps consist in extending such an approach to the elastic case and trying it experimentally, that is with real data. Experiments will be carried in CEA.