## Section: New Results

### Seismic Imaging and Inverse Problems

#### Mathematical determination of the Fréchet derivative with respect to the domain for a fluid-structure scattering problem. Case of polygonal-shaped domains.

Participants : Hélène Barucq, Elodie Estecahandy.

The characterization of the Fréchet derivative of the elasto-acoustic scattered field with respect to Lipschitz continuous polygonal domains is established. The considered class of do- mains is of practical interest since two-dimensional scatterers are always transformed into polygonal- shaped domains when employing finite element methods for solving direct and inverse scattering problems. The obtained result indicates that the Fr´echet derivative with respect to the scatterer of the scattered field is the solution of the same elasto-acoustic scattering problem but with addi- tional right-hand side terms in the transmission conditions across the fluid-structure interface. This characterization has the potential to advance the state-of-the-art of the solution of inverse obstacle problems.

This work has been done in collaboration with Prof. Rabia Djellouli (California State University at Northridge) and has been accepted for publication in Siam Journal of Applied Mathematics [16].

#### Shape-reconstruction and parameter identification of an elastic object immersed in a fluid

Participants : Izar Azpiroz Iragorri, Hélène Barucq, Julien Diaz, Kevin Lagnoux.

We have developed a procedure to reconstruct the shape and material parameters of an elastic obstacle immersed in a fluid medium from some external measurements given by the so called far-field pattern. It is a nonlinear and ill-posed problem which is solved by applying a Newton-like iterative method involving the Fréchet derivatives of the scattered field. These derivatives express the sensitivity of the scattered field with respect to the parameters of interest. They are defined as the solution of boundary value problems which differ from the direct one only at the right-hand sides level. We have been able to establish the well-posedness of each problem in the case of a regular obstacle and it would be interesting in the near future to extend those results to the case of scatterers with polygonal boundaries. It requires to work with less regular Sobolev spaces for which the definition of traces is not obvious. We have also provided an analytical representation of the Fréchet derivatives in the case of a circle. This provide a way of validating the numerical experiments and it would be interesting to obtain their expression in the case of elliptical scatterers or spherical ones. It is worth mentioning that this work has been done only in the case of isotropic media and it would be interesting to extend it to anisotropic media as well. It requires to establish analytic representations of the scattered field in anisotropic media which is more difficult because it involves more parameters.

We have studied the response of the data to the different parameters. It turns out that the sensitivity of the far field pattern is very different regarding the shape or the material parameters. We have delivered a sensitivity analysis which has been essential for understanding that the reconstruction of the material parameters is conditioned by the recovering of the shape parameters. This makes the full reconstruction very difficult and sometimes unstable. In particular, in the case of a disk-shaped obstacle, when addressing the role of the frequency in the reconstruction, we have been faced to the issue of the existence of Jones modes which had been already observed by Elodie Estecahandy in her PhD thesis. Next, we have introduced a series of numerical experiments that have been performed by applying two algorithms which propose two strategies of full reconstruction regarding the material parameters are retrieved simultaneously with the shape or not. It turns out that both work similarly delivering the same level of accuracy but the simultaneous reconstruction requires less iterations. We have thus opted for retrieving all the parameters simultaneously. Since realistic configurations include noisy data, we have performed some simulations for the reconstruction of the shape along with the Lamé coefficients for different noise levels. Other interesting experiments have been carried out using a multistage procedure where the parameters of interest are the density of the solid interior, the shape of the obstacle and its position. We have considered the case of Limited Aperture Data in back-scattering configurations, using multiple incident plane waves, mimicing a physical disposal of non-destructive testing. This is an encouraging ongoing work which deserves to be completed by considering a wide range of examples including more general geometries of the scatterer. It should also be extended by dealing with limited aperture data using only one incident wave (which will probably require multiple frequency data).

These results have been obtained in collaboration with Rabia Djellouli (California State University at Northridge, USA) and were presented to the Waves 2017 conference.

#### Shape-reconstruction and parameter identification of an anisotropic elastic object immersed in a fluid

Participants : Izar Azpiroz Iragorri, Hélène Barucq, Julien Diaz.

We extended the solution methodology for reconstructing the shape and material parameters of an elastic obstacle (see 6.1.2) to the case of anisotropic media. This is a very challenging case which still deserves further works. We have obtained some results but since the impact of some of the anisotropic parameters on the FFP is even weaker than the Lamé coefficients, the reconstruction of these parameters together with the shape parameters requires several frequencies and carefully adapted regularization parameters. It is in particular difficult to retrieve the Thomsen parameters $\u03f5$ and $\delta $ because their reconstruction requires to have an accurate adjustment on the rest of material and shape parameters. The recovery process is thus computationally intensive and some efforts should be done in the near future to decrease the computational costs. We were able to recover all the anisotropic parameters when the shape were assumed to be known. However, when trying to recover both shape and material parameters, we could only recover the shape and some of the physical parameters (namely the three most important ones : the density and the two velocities ${V}_{p}$ and ${V}_{s}$ ). We should now find a way to determine all the Thomsen parameters together with the shape. Then, we will have to deal with more complex media such as TTI media (this will add the angle of anisotropy as additional parameter). The last step will be to consider general anisotropy, which could be done by recovering each element of the elastic stiffness tensor. This is simple to implement, since the derivative of the stiffness tensor with respect to one of its component is easily computable (it is a tensor composed of zeroes and ones). However, the stability of the reconstruction is not guaranteed, since we will strongly increase the number of components to be retrieved.

These results have been obtained in collaboration with Rabia Djellouli (California State University at Northridge, USA).

#### Mathematical analysis and solution methodology for a class of inverse spectral problems arising in the design of optical waveguides

Participant : Hélène Barucq.

We analyze mathematically the problem of determining refractive index profiles from some desired/measured guided waves propagating in optical fibers. We establish the uniqueness of the solution of this inverse spectral problem assuming that only one guided mode is known. Then, we propose an iterative computational procedure for solving numerically the considered inverse spectral problem. Numerical results are presented to illustrate the potential of the proposed regularized Newton algorithm to efficiently and accurately retrieve the refractive index profiles even when the guided mode measurements are highly noisy.

This work has been submitted for publication in a peer-reviewed journal. It has been done in collaboration with Rabia Djellouli (California State University at Northridge, USA) and Chokri Bekkey (University of Monastir, Tunisia)

#### Time-harmonic seismic inverse problem with Cauchy data

Participant : Florian Faucher.

*This work is a collaboration with
Giovanni Alessandrini (Università di Trieste),
Maarten V. de Hoop (Rice University),
Romina Gaburro (University of Limerick) and
Eva Sincich (Università di Trieste).*

We study the performance of Full Waveform Inversion (FWI) from time-harmonic Cauchy data via conditional well-posedness driven iterative regularization. The Cauchy data can be obtained with dual sensors measuring the pressure and the normal velocity. We define a novel misfit functional which, adapted to the Cauchy data, allows the independent location of experimental and computational sources. The conditional well-posedness is obtained for a hierarchy of subspaces in which the inverse problem with partial data is Lipschitz stable. Here, these subspaces yield piecewise linear representations of the wave speed on given domain partitions. Domain partitions can be adaptively obtained through segmentation of the gradient. The domain partitions can be taken as a coarsening of an unstructured tetrahedral mesh associated with a finite element discretization of the Helmholtz equation. We illustrate the effectiveness of the iterative regularization through computational experiments with data in dimension three. In comparison with earlier work, the Cauchy data do not suffer from eigenfrequencies in the configurations.

The resulting paper is [47] and is also connected to the following conference presentations, [36], [27].

#### Quantitative Convergence of Full Waveform Inversion in the Frequency Domain

Participants : Hélène Barucq, Florian Faucher.

*This work is a collaboration with Guy Chavent (Inria Rocquencourt).*

We study the convergence of the inverse problem associated with the frequency domain wave equations for the recovery of subsurface parameters. The numerical method selected for the resolution is the Full Waveform Inversion (FWI), which designs an iterative minimization algorithm. We study the convergence of the scheme in the context of least squares minimization. We establish numerical estimates based on the Fréchet derivatives for the radius of curvature and the deflection. We quantify the (complex) frequency progression to select to foster the convergence, and illustrate the effect of the subsurface geometry. From the curvature estimates, we also provide an insight of the robustness with noise depending on the situation. We supplement the numerical analysis with numerical experiments to demonstrate the results.

The results have been presented in the following conference, [36], [27], [26], [25].

#### Contributions to seismic full waveform inversion for time harmonic wave equations: stability estimates, convergence analysis, numerical experiments involving large scale optimization algorithms

Participants : Hélène Barucq, Florian Faucher.

In this project, we investigate the recovery of subsurface Earth parameters. We consider the seismic imaging as a large scale iterative minimization problem, and deploy the Full Waveform Inversion (FWI) method. The reconstruction is based on the wave equations because the characteristics of the measurements indicate the nature of the medium in which the waves propagate. First, the natural heterogeneity and anisotropy of the Earth require numerical methods that are adapted and efficient to solve the wave propagation problem. In this study, we have decided to work with the harmonic formulation, i.e., in the frequency domain.

The inverse problem is then established in order to frame the seismic imaging. It is a nonlinear and ill-posed inverse problem by nature, due to the limited available data, and the complexity of the subsurface characterization. However, we obtain a conditional Lipschitz-type stability in the case of piecewise constant model representation. We derive the lower and upper bound for the underlying stability constant, which allows us to quantify the stability with frequency and scale. It is of great use for the underlying optimization algorithm involved to solve the seismic problem. We review the foundations of iterative optimization techniques and provide the different methods that we have used in this project. The Newton method, due to the numerical cost of inverting the Hessian, may not always be accessible. We propose some comparisons to identify the benefits of using the Hessian, in order to study what would be an appropriate procedure regarding the accuracy and time. We study the convergence of the iterative minimization method, depending on different aspects such as the geometry of the subsurface, the frequency, and the parametrization. In particular, we quantify the frequency progression, from the point of view of optimization, by showing how the size of the basin of attraction evolves with frequency.

Following the convergence and stability analysis of the problem, the iterative minimization algorithm is conducted via a multi-level scheme where frequency and scale progress simultaneously. We perform a collection of experiments, including acoustic and elastic media, in two and three dimensions. The perspectives of attenuation and anisotropic reconstructions are also introduced.

#### Quantitative localization of small obstacles with single-layer potential fast solvers

Participants : Hélène Barucq, Florian Faucher, Ha Pham.

In this work, we numerically study the inverse problem of locating small circular obstacles in a homogeneous medium using noisy backscattered data collected at several frequencies. The main novelty of our work is the implementation of a single-layer potential based fast solver (called FSSL) in a Full-Waveform inversion procedure, to give high quality reconstruction with low-time cost. The efficiency of FSSL was studied in our previous works. We show reconstruction results with up to 12 obstacles in structured or random configurations with several initial guesses, all allowed to be far and different in nature from the target. This last assumption is not expected in results using nonlinear optimization schemes in general. For results with 6 obstacles, we also investigate several optimization methods, comparing between nonlinear gradient descent and quasi-Newton, as well as their convergence with different line search algorithms.

The resulting research report is [45].