We consider the Saint-Venant model in 2 dimensions for nonlinear elasticity. Under the hypothesis the fluid is incompressible, we recover the displaced field and the Lame parameter $\mu$ from power density measurements. A stability estimate is shown to hold for small displacement fields, under some natural hypotheses on the direction of the displacement. The techniques introduced show the difficulties of using hybrid imaging techniques for non-linear inverse problems.

We consider the inverse wave problem of reconstructing the properties of a viscoelastic medium. The data acquisition corresponds to probing waves that are sent and measured outside of the sample of interest, in the configuration of non-intrusive inversion. In media with attenuation, waves lose energy when propagating through the domain. Attenuation is a frequency-dependent phenomenon with several models existing [1], each leading to different models of wave equations. Therefore, in addition to adding unknown coefficients to the inverse problem, the attenuation law characterizing a medium is typically unknown prior to the reconstruction, hence further increasing the ill-posedness. In this work, we consider time-harmonic waves which are convenient to unify the different models of attenuation using complex-valued parameters. We illustrate the difference in wave propagation depending on the attenuation law and carry out the reconstruction with attenuation model uncertainty [2]. That is, we perform the reconstruction procedure with different attenuation models used for the (synthetic) data generation and for the reconstruction. In this way, we show the robustness of the reconstruction method. Furthermore, we investigate a configuration with reflecting boundary surrounding the sample. To handle the resulting multiple reflections, we introduce a strategy of reconstruction with a progression of complex frequencies. We illustrate with experiments of ultrasound imaging.

[1] J. M. Carcione. Wave Fields in Real Media: Wave Propagation in Anisotropic, Anelastic, Porous and Electromagnetic Media, third ed., Elsevier, 2015.

[2] F. Faucher, O. Scherzer. Quantitative inverse problem in visco-acoustic media under attenuation model uncertainty, Journal of Computational Physics 472: 111685, 2023. https://doi.org/10.1016/j.jcp.2022.111685

We consider optical coherence elastography, which is an emerging research field but still lacking precision and reproducibility. Elastography as an imaging modality aims at mapping of the biomechanical properties of a given sample. This problem is widely used in Medicine, in particular for the non-invasive identification of malignant formations inside the human skin or tissue biopsies during surgeries. In term of diagnostics accuracy, one is interested in quantitative values mapped on top of the visualisation of the sample rather then only qualitative images.

In this work, we discuss a general intensity-based approach to the inverse problem of quasi-static elastography, under any deformation model. From a pair of tomographic scans obtained by an imaging modality of choice, e.g. as X-ray, ultrasound, magnetic resonance, optical imaging or other, we aim to recover one or a set of unknown material parameters describing the sample. This approach has been briefly introduced in [1], under the name of intensity-based inversion method, and applied for recovery of the Young's modulus of a set of samples imaged with Optical Coherence Tomography. Here, we mainly focus on investigating the intensity-based inversion approach in the Inverse Problems framework. Furthermore, we illustrate the performance of the inversion method on twelve silicone elastomer phantoms with inclusions of varying size and stiffness.

[1] L. Krainz, E. Sherina, S. Hubmer, M. Liu, W. Drexler, O. Scherzer. Quantitative Optical Coherence Elastography: A Novel Intensity-Based Inversion Method Versus Strain-Based Reconstructions. IEEE J. Sel. Topics Quantum Electron. 29(4): 1-16, 2023. DOI: 10.1109/JSTQE.2022.3225108.

I analyze the geometric inverse problem of recovering cavities and inclusions embedded in a linear elastic isotropic medium from boundary displacement measurements. Starting from a constrained minimization problem involving a boundary quadratic misfit functional with a regularization term penalizing the perimeter of the cavity or inclusion we consider a family of relaxed functionals using a phase-field approach and derive a robust algorithm for the reconstruction of elastic inclusions and cavities modeled as inclusions with a very small elasticity tensor.