We investigates the inverse problem of reconstructing the electrical conductivity of an object in hybrid imaging methods. These techniques have been developed in recent years to produce clearer images than those produced by electrical impedance tomography. We focus on the inverse problem arising in the quantitative step of many hybrid imaging techniques. The problem is formulated as recovering the isotropic conductivity of an object given internal current densities generated by applying different boundary conditions to the electrostatic equation. We will present two specific examples of these techniques, current density impedance imaging and magneto-acousto-electric tomography, to illustrate the different boundary conditions that can be used. We provide a local Lipschitz stability for the general inverse problem in both full and partial data cases.

We study the classical inverse problem to determine the shape of a three-dimensional scattering obstacle from measurements of scattered waves or their far-field patterns. Previous research on this subject has mostly assumed the object to be star-shaped and imposed a Sobolev penalty on the radial function or has defined the penalty term in some other ad-hoc manner which is not invariant under coordinate transformations.

For the case of curves in $\mathbb{R}^2$, reference [1] suggests to use the bending energy as regularisation functional and proposes Tikhonov regularization and regularized Newton methods on a shape manifold. The case of surfaces in $\mathbb{R}^3$ is considerably more demanding. First, a suitable space (manifold) of shapes is not obvious. The second problem is to find a stabilizing functional for generalised Tikhonov regularisation which on the one hand should be bending-sensitive and on the other hand prevent the surface from self-intersections during the reconstruction.

The tangent-point energy is a parametrization-invariant and repulsive surface energy that is constructed as the double integral over a power of the tangent point radius with respect to two points on the surface, i.e. the smallest radius of a sphere being tangent to the first point and intersecting the other. The finiteness of this energy also provides $C^{1,\alpha}$ Hölder regularity of the surfaces.[2] Using this energy as the stabilising functional, we choose general surfaces of Sobolev-Slobodeckij reguality, which are naturally connected to this energy.

The proposed approach works for surfaces of arbitrary (known) topology. In numerical examples we demonstrate that the flexibility of our approach in handling rather general shapes.

[1] J Eckhardt, R Hiptmair, T Hohage, H Schumacher, M Wardetzky. Elastic energy regularization for inverse obstacle scattering problems. 2019

[2] P. Strzlecki, H. von der Mosel. Tangent-point repulsive potentials for a class of smooth $m$-dimensional sets in $\mathbb{R}^n$. Part 1: Smoothing and self-avoidance effects. 2011

We consider the problem of finding a compactly supported potential in the multidimensional Schrodinger equation from its differential scattering cross section (squared modulus of the scattering amplitude) at fixed energy. In the Born approximation this problem simplifies to the phase retrieval problem of reconstructing the potential from the absolute value of its Fourier transform on a ball. To compensate for the missing phase information we use the method of a priori known background scatterers. In particular, we propose an iterative scheme for finding the potential from measurements of a single differential scattering cross section corresponding to the sum of the unknown potential and a known background potential, which is sufficiently disjoint. If this condition is relaxed, then we give similar results for finding the potential from additional monochromatic measurements of the differential scattering cross section of the unknown potential without the background potential. The performance of the proposed algorithms is demonstrated in numerical examples. This talk is based on the work

Hohage, Novikov, Sivkin, preprint 2022, hal-03806616

We discuss several iterative optimization algorithms for the minimization of a cost function consisting of a linear combination of up to three convex terms with at least one differentiable and a second one prox-simple. Such optimization problems frequently occur in the numerical solution of inverse problems (data misfit term plus penalty or constraint term).

We present several new results on the convergence of proximal-gradient-like algorithms in the context of a optimization-by-continuation strategy. The algorithms special feature lies in their ability to approximate, in a single iteration run, the minimizers of the cost function for many different values of the parameters determining the relative weight of the three terms in the cost function (penalty parameters). As a special case, one recovers a generalization of the primal-dual algorithm of Chambolle and Pock.