# Research

Our research is focused on theory and applications of *inverse problems*. An inverse problem means the computation of a quantity from a parameter space, which is not directly oservable, from measured data that contain information about the searched quantity. A mathematical model, which usually is represented by an operator equation or recently also by a neural network, describes the connection between the parameter and the data space. The reconstruction process then consists by a stable inversion procedure that maps the given data to the searched quantity. Usually inverse problems are *ill-posed* what means that even small data errors deteriorate the inversion process leading to a useless solution. This is why stable solution schemes have to be developed, so called *regularization methods*. In our research we follow all the way from mathematical modeling, analysis of the underlying mathematical operator, development of problem adapted regularization methods down to the implementation and validation of these methods using innovative numerical solvers. Our fundamental research is currently focused at regularization methods in Banach and Bochner spaces, where the latter play a prominent role in the emerging field of *dynamic inverse problems*, where we consider applications with time-dependent data to compute time-dependent parameters. Applications are

- Vector and tensor tomography in inhomogeneous media (funded by DFG, in cooperation with the Sobolev institute Novosibirsk, Prof. Dr. Bernadette Hahn-Rigaud and Prof. Dr. Alexander Katsevich)
- Terahertz tomography (in cooperation with the Plastics Center Würzburg, SKZ)
- Structural Health Monitoring in composite materials (parameter identification for elastic wave equations)
- Magnetic Particle Imaging (funded by the Federal Minstry of Education and Research, BMBF)
- Parameter identification for nonlinear heat conduction problems (in cooperation with Dillinger Hütte AG)
- Sequential subspace optimization methods in Hilbert and Banach spaces
- Inverse problems and machine learning
- Development of an imaging material testing system based on the photothermal measurement method (funded by the European Fund for Regional Development from the Operational Program
**EFRE Saarland 2014-2020**with the objective "Investments in Growth and Employment" (“Investitionen in Wachstum und Beschäftigung”)) - Deep Learning in Tomography (funded by the Federal Ministry of Education and Research, BMBF)

## Vector and tensor tomography

In contrast to traditional computer tomography, where scalar quantities such as absorption or refraction coefficients, are computed, vector (tensor) tomography aims at the reconstruction and visualization of vector-valued (resp. tensor-valued) quantities such as velocity fields of moving fluids, magnetic fields, strain tensors or permeability tensors. Applications range from medical technology, industrial applications to oceanographics and plasma physics.

## Terahertz tomography

**Grant program: AiF-IGF**

Terahertz tomography is an imaging method allowing non-destructive testing of materials, especially plastics or ceramics. The THz range includes frequencies between 0.1 and 10 THz, i.e. it is right between infrared light and microwave radiation. Due to the large wave length the lateral spatial resolution is quite limited. This can be overcome by time-resolved depth information (analoguous to ultrasound techniques) yielding a high resolution in the propagation direction as well. In the project we develop new mathematical models as well as numerical solution methods for parameter identification problems.

## Structural Health Monitoring in Composite Materials

**Parameter identification for elastic wave equations**

In many industrial productions fibre composites are being used more and more often and also in ever more diverse setups. For example in air plane construction carbon reinforced fibre composites are of great importance. These high-performane materials come at a high price. Hence, especially from an economic point of view, it is important to detect damages as soon as possible before greater parts of a composite have to be replaced.

## Inverse problems and regularization methods

Inverse problems ask about how to get from an indirect measurement to its cause where the relation between the two, the measurement process, is known. This relation is described by a mathematical model, given by a linear or even non-linear operator between topological spaces.

## Hyperspectral imaging: Mathematical methods for innovation in medicine and industry

In hyperspectral imaging not just three (red, green, and blue) but thousands of so-called spectral channels are measured at each pixel of a spatially-resolved image. This allows using spectral fingerprints to recognize objects in the image, ranging from molecules in spatially-resolved Raman spectroscopy or telling streets apart from trees in satellite imagery. However, this additional information comes at the price of increased data volume, easily going into tens of gigabytes per image.

In the joint project HYPERMATH the challenge of this "big data" problem is answered from a mathematical point of view. The problem has great importance as multitudinous applications in medical technology, material testing or even movie camera development are eager to take advantage of this technique. Next-generation products require next-generation algorithms and methods.

## Model-based Parameter-Identification in Magnetic Particle Imaging (MPI)²

Grant program: Bundesministerium für Bildung und Forschung (BMBF)

Magnetic particle imaging is a novel medical imaging technique. Magnetic iron-oxide particles are injected into the bloodstream of a patient. A reconstruction of their position inside the body yields information about the inner structure, particularly of the cardiovascular system, without using ionizing radiation. MPI is highly sensitive and offers a high temporal resolution. The aim of this project is to take into account nonlinear modeling and relaxation effects in order to increase the resolution of the reconstructions.

The official website of this project can be found here.

Project partners:

- Zentrum für Technomathematik (ZeTeM) in Bremen
- Institut für Biomedizinische Bildgebung, Universitätsklinikum Hamburg-Eppendorf
- Arbeitsgruppe für Numerische Mathematik, Universität des Saarlandes
- Arbeitsgruppe Stark, Hochschule Aschaffenburg

## Development of an imaging material testing system based on the photothermal measurement method

Photothermal measurement is an established method for non-contact measurement of the coating thickness of paints or powder coatings on metal or plastic substrates. It is based on the fact that thermal waves are excited by an exciting light source inside a test specimen. On contact with interfaces, these show the same behavior as, for example, acoustic or optical waves, in particular they are reflected. The phase angle difference between the excitation signal and the reflected signal can be used to determine the position of the interface. Such an interface does not necessarily have to be between the coating and the substrate, it can also be between two coating layers in a multi-layered coating system or a material and a defect, for example an air inclusion.

## Deep Learning in Tomography

Collaborative project funded by Federal Ministry of Education and Research (2020-2023)