Nonlinear impedance boundary conditions in acoustic scattering are used as a model for perfectly conducting objects coated with a thin layer of a nonlinear medium. We consider a scattering problem for the Helmholtz equation with a nonlinear impedance boundary condition of the form $$ \dfrac{\partial u}{\partial \nu} + ik\lambda u = g(\cdot,u) \quad \text{on} \ \partial D, $$ where $\nu$ denotes the unit normal vector, $\lambda \in L^{\infty}(\partial D)$ is a complex-valued impedance function, and $g: \partial D \times \mathbb{C} \to \mathbb{C}$ gives an additional nonlinear term with respect to the total field $u$. The contributed talk is devoted to the well-posedness of the direct problem, the determination of the domain derivative, and the inverse problem, which consists in reconstructing the shape of the scattering object from given far field data. Numerical results are presented relying on an all-at-once regularized Newton-type method based on the linearization of the forward problem and of the domain-to-far-field operator.

We consider scattering of time-harmonic acoustic waves by an ensemble of compactly supported penetrable scattering objects in 2D. These scattering objects are illuminated by an incident plane wave. The resulting total wave is the superposition of incident and scattered wave and solves a scattering problem for the Helmholtz equation. For guaranteeing uniqueness, the scattered wave must fulfill the Sommerfeld radiation condition at infinity.

In our consideration, measurements of the total wave are replaced by the corresponding far field operator. This operator contains all information about the scattered wave far away from the scattering objects for all possible illumination directions.

We are interested in two inverse problems. On the one hand, given a limited observation of this far field operator, we want to determine its missing part, which we refer to as operator completion problem. 'Limited observation' in this context means, that we do not have access to measurements for all illumination directions or that we cannot measure in all observation directions around the scattering objects. On the other hand, given the far field operator for the ensemble of scattering objects, we want to determine the far field operators of the individual scattering objects. This is what we refer to as operator splitting problem. Multiple reflection effects cause, in contrast to the first problem, the nonlinearity of this second problem.

We characterize spaces containing the individual, for the two problems relevant components of the far field operator. Operators in these spaces turn out to have a low rank and sparsity properties with respect to some known modulated Fourier frame. Furthermore, this rank and frame can be determined under knowledge of the locations and sizes of the scatterer's components.

In my talk I will suggest two reformulations of the inverse problems, a least squares norm formulation and a $l^1\times l^1$-norm minimization, and appropriate algorithms for solving these formulations numerically. Moreover, I will present stability results for these reconstructions and support them by numerical experiments.

Medical imaging and non-destructive testing holds the challenge of determining information of the interior of the body by taking measurements at its boundary. We consider an inverse coefficient problem that is motivated by impedance-based corrosion detection. The aim is to reconstruct an unknown transmission coefficient function in an elliptic PDE from finitely many measurements that correspond to voltage/current measurements on electrodes attached to the domain's outer boundary. We mathematically characterize how many electrodes are required to achieve a desired resolution, we derive stability and error estimates, and we discuss globally convergent numerical reconstruction methods by rewriting the problem as convex non-linear semidefinite optimization problem.

Iterative reconstruction methods for Electrical Impedance Tomography (EIT) often require solving the forward problem multiple times with different inner conductivity parameters using the Finite Element Method (FEM). The solution of the FEM problem requires solving a large sparse linear system of equations during each round of iteration. We present novel subspace methods for solving these linear systems using the same subspace for any conductivity parameter. We report that there seems to be structure in the problem that generally allows the size of these subspaces to stay small, enabling efficient computation.