Solving Inverse Problems with Frames Decompositions

Whether in medical imaging using computed tomography (CT) or in geophysics when exploring the Earth`s interior researchers often need to infer hidden structures from indirect measurements. These tasks are known as inverse problems. They are particularly challenging because even small errors in the measurement data can lead to large deviations in the results. This makes it difficult to draw reliable conclusions even though accuracy is especially crucial in fields like medical diagnostics or environmental analysis. Therefore, specialized mathematical methods are necessary to obtain reliable and stable information from incomplete and noisy data. At the core lies a mathematical model that must be inverted. However, such inverse models are often inherently unstable small errors in the data may be amplified or cannot be directly reconstructed at all. To still obtain stable information from imperfect data, so-called regularization methods are used. These methods provide a stable approximation of the desired inverse model. This project focuses specifically on regularization methods based on so-called frame decompositions. Here, the underlying model is deliberately broken down into smaller components to better identify and filter out both noise and relevant information. Compared to classical approaches, these methods offer greater flexibility and can be more precisely tailored to different inverse problems. In addition, they hold great potential for further development and broader application. The goal of this project is to deepen the understanding of frame-based methods in order to enhance and extend their applicability in complex scenarios. In addition to the development and analysis of such regularization techniques, the project also investigates the existence and mathematical foundations of frame decompositions for various underlying models. Furthermore, it explores how these methods can be combined with modern learning techniques to achieve even more accurate and reliable results. The research integrates several mathematical disciplines such as optimization, numerical analysis, and harmonic analysis. The project will initially be carried out during a two-year international research stay in Mannheim, and subsequently continued for another year in Innsbruck.