Nonlinear parameter tomography is a technique for enhancing ultrasound imaging and amounts to identifying the spatially varying coefficient $\eta=\eta(x)$ in the Westervelt equation $ p_{tt}-c^2\triangle p - b\triangle p_t = \eta(p^2)_{tt} + h$ in a domain $(0,T)\times\Omega$. Here $p$ is the acoustic pressure, $c$ the speed of sound, $b$ the diffusivity of sound, and $h$ the excitation. Observations consist of pressure measurements on some manifold $\Sigma$ immersed in the acoustic domain $\Omega$.

Our imaging goal is to show unique recovery when $\eta(x)$ is a finite set $\{a_i\chi(D_i)\}_i$ and where each $D_i$ is starlike with respect to its centroid.

Assuming periodic excitations of the form $h(x,t) = A e^{i\omega t}$ for some fixed frequency $\omega$ one can convert this to an infinite system of coupled linear Helmholtz equations. We will give both uniqueness and reconstructions results and note that this work was inspired by a previous paper of one author and Rainer Kress.

We combine data-driven reduced order models with the Lippmann- Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann-Schwinger equation, in a direct or iterative framework. The approach also allows us to process more general transfer functions, i.e., to remove the main limitation of the earlier versions of the ROM based inversion algorithms. We describe how the generation of internal solutions simplifies in the time domain, and show how Lanczos orthogonalization in the spectral domain relates to time stepping. We give examples of its use given mono static data, targeting synthetic aperture radar.

The concept of topological derivative has proved effective as a qualitative inversion tool for wave-based identification of finite-sized objects. This approach is often based on a heuristic interpretation of the topological derivative. Its mathematical justification has however also been studied, in particular in cases where the true obstacle is small enough for asymptotic approximations of wave scattering to be applicable, and also for finite-sized objects in the scalar wave framework. This work extends our previous efforts in the latter direction to the identification of elastic inhomogeneities embedded in elastic media interrogated by elastic waves. The data used for identification, assumed to be of near-field nature (i.e. no far-field approximation is introduced), is introduced through a misfit functional $J$. The imaging functional that reveals embedded inhomogeneities then consists of the topological derivative $\mathcal{T}_J$ of $J$ (in particular, the actual minimization of $J$ is not performed, making the procedure significantly faster than standard inversion based on PDE-constrained minimization). Our main contribution consists in an analysis of $\mathcal{T}_J$ using a suitable factorization of the near fields, achievable thanks to a convenient reformulation of the volume integral equation formulation of the forward elastodynamic scattering problem established earlier. Our results include justification of both the sign heuristics for $\mathbf{z}\mapsto\mathcal{T}_J(\mathbf{z})$ (which is expected to be most negative at points $\mathbf{z}$ inside, or close to, the support of the sought flaw) and the spatial decay of $\mathcal{T}_J(\mathbf{z})$ as $\mathbf{z}$ moves away from the flaw support. This result, subject to a limitation on the strength of the inhomogeneity to be identified, provides a theoretical conditional validation of the usual heuristic interpretation of $\mathcal{T}_J$ as an imaging functional. Our findings are demonstrated on 3D computational experiments.