# Numerical inverse wave problems for quantitative viscoelastic reconstructions

Florian Faucher, Otmar Scherzer, Ha Pham, Hélène Barucq

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Florian Faucher, Otmar Scherzer, Ha Pham, Hélène Barucq. Numerical inverse wave problems for quantitative viscoelastic reconstructions. IPMS 2024 - 11th International Conference "Inverse Problems: Modeling and Simulation", The Eurasian Association on Inverse Problems, May 2024, Il-Mellieħa, Malta. hal-04854272

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https://inria.hal.science/hal-04854272v1

Submitted on 23 Dec 2024

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Distributed under a Creative Commons Attribution 4.0 International License
Numerical inverse wave problems for quantitative
viscoelastic reconstructions

Florian Faucher<fnref n="1" />, Otmar Scherzer<fnref n="2" />, Ha Pham<fnref n="1" />, Hélène Barucq<fnref n="1" />

We consider the inverse problem for the quantitative reconstruction of the material properties that characterize a viscoelastic medium from wave measurements. Waves in such media display loss of energy (attenuation), and their propagation is mathematically modeled with linear viscoelastic equations, [1]. The time-harmonic formulation of the wave problem is considered to facilitate the investigation of the different attenuation models which are frequency-dependent. For the numerical discretization of the viscoelastic wave equations composed of the equation of motion and constitutive law, the hybridizable discontinuous Galerkin (HDG, [2]) method is employed and allows to reduce the computational cost. For the stabilization of HDG numerical traces, we further construct a hybridized Godunov-upwind flux for anisotropic elasticity, [3]. Then, the nonlinear inversion follows a minimization algorithm to iteratively update the elastic moduli. We carry out reconstructions with attenuation model uncertainty ([4]), that is, using different models of attenuation between the modeling and inversion. We illustrate with ultrasound imaging experiment of synthetic breast sample reconstruction.

References

[1] J. M. Carcione, Wave fields in real media: Wave propagation in anisotropic, anelastic, porous and electromagnetic media, *Elsevier* (2007).

[2] B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, *SIAM Journal on Numerical Analysis* 47 (2009).

[3] H. Pham, F. Faucher and H. Barucq, Numerical investigation of stabilization in the Hybridizable Discontinuous Galerkin method for linear anisotropic elastic equation, *arXiv preprint arXiv:2403.02862* (2024).

[4] F. Faucher and O. Scherzer, Quantitative inverse problem in visco-acoustic media under attenuation model uncertainty, *Journal of Computational Physics* 472 (2023).

<footnote n="1">1) Team Makutu, Inria, Université de Pau et des Pays de l'Adour, TotalEnergies, CNRS, France.
florian.faucher@inria.fr</footnote>

<footnote n="2">²) Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz, Austria.</footnote>