In this talk, we will discuss the problem of finding the optimal shape of a system of acoustic lenses in a dissipative medium. The problem is tackled by introducing a phase-field formulation through diffuse interfaces between the lenses and the surrounding fluid. The resulting formulation is shown to be well-posed and we rigorously derive first-order optimality conditions for this problem. Additionally, a relation between the diffuse interface problem and a perimeter-regularized sharp interface shape optimization problem can be established via the $\Gamma$-limit of the reduced objective.

Helioseismology aims at recovering the properties of the solar interior from the observations of line-of-sight Doppler velocities at the surface. Interpreting these observations requires first to solve a forward problem describing the propagation of waves in a highly-stratified medium representing the interior of the Sun. Considering only acoustic waves, the forward problem can be written as $$\mathcal{L}\psi := -\frac{1}{\rho c^2} (\omega^2 + 2 i \omega \gamma + 2 i \omega \mathbf{u} \cdot \nabla) \psi - \nabla \left( \frac{1}{\rho} \nabla \psi \right) = s,$$ where $\rho$ is the density, $c$ the sound speed, $\mathbf{u}$ the flow and $\psi$ the Lagrangian pressure perturbation. The source term $s$ is stochastic and caused by the excitation of waves by convection. As the signal is incoherent, we cannot study directly the wavefield $\psi$ but only its cross-covariance $C(r_1,r_2,\omega) = \psi(r_1,\omega)^\ast \psi(r_2,\omega)$. Under the hypothesis of energy equipartition, the expectation value of the cross-covariance is proportional to the imaginary part of the Green's function associated to $\mathcal{L}$. The inverse problem is then to reconstruct the parameters $q \in \{\rho, c, \mathbf{u} \}$ from the observations of Im[$G(r_1,r_2,\omega)$] for any two points $r_1$, $r_2$ at the solar surface. To increase the signal-to-noise ratio and reduce the size of the input data, wave travel times are usually extracted from the cross-covariances and serve as input data for the inversions. We will present inversions of large-scale flows (differential rotation and meridional circulation) from travel-time measurements using synthetic and observed data.

Magnetic particle imaging (MPI) is a tracer based imaging modality which exploits the magnetization behavior of magnetic nanoparticles (MNPs) to obtain spatially distributed concentration images from voltage measurements. Proper modeling, which is still an unsolved problem in MPI, relies on magnetization dynamics of individual nanoparticles which typically include Neel and Brownian magnetic moment rotation dynamics. In the context of MPI large ensembles of MNPs and their magnetization behavior need to be considered. Taking into account Neel/Brownian rotation, the ensembles magnetization behavior can be described by a Fokker-Planck equation, i.e., a linear parabolic PDE which models the temporal evolution of the probability that the magnetic moment of a nanoparticle has a certain orientation. The resulting behavior is strongly influenced by time-dependent parameters in the PDE. In this talk we discuss the physical modeling as well as time-dependent parameter identification problems related to the magnetization dynamics based on a Fokker-Planck equation.

In this talk, we consider the problem of determining a polyhedral conductivity inclusion embedded in a homogeneous isotropic medium from boundary measurements. Specifically, we prove global Lipschitz stability for the polyhedral inclusion from the local Dirichlet-to-Neumann map.