Mathematics of Reconstruction in Dynamical and Active Models

In many situations, one is faced with the task of reconstructing relevant but not directly measurable parameters, for example in medical imaging using computed tomography or magnetic resonance imaging (MRI). A common approach is to compare suitable indirect measurement data with a mathematical model that describes the relationship between this measurement data and the parameters being searched for; this is referred to as an "inverse problem". Of particular interest here are models with dynamical and active components (i.e. those that are time-dependent and can be influenced externally), as these can be used to design the measurement in such a way that the reconstruction can be carried out as quickly, accurately, and robustly as possible. If the models are described by differential equations, this can be formulated as an "optimal control problem". For example in MRI -- whose mathematical model is the Bloch equations -- image acquisition is controlled by time-dependent magnetic fields, which can be optimized for the fastest possible acquisition with minimal noise or optimal resolution. Mathematical developments have already made significant contributions in parameter reconstruction and design of measurements, most recently particularly in the context of data-driven methods and machine learning. So far, however, these aspects have been considered largely in isolation from each other, and a holistic approach would open up far greater possibilities for more accurate and, above all, more robust parameter reconstruction. The aim of this special research area is therefore to develop a comprehensive theoretical framework and efficient numerical algorithms integrating data-driven methods for the entire measurement and reconstruction pipeline and their exemplary implementation for MRI. This requires the close collaboration of experts from optimization, inverse problems, calculus of variations, machine learning, and medical imaging, for which the Austrian research landscape and especially the location Graz is ideally positioned. This is expected to lead not only to a deeper mathematical understanding of the limits and possibilities of optimal reconstruction in dynamical models, but also to clinically relevant improvements in MRI by providing jointly optimized and practically feasible measurement and reconstruction protocols that overcome the current limitations due to motion, incomplete data, or noise sensitivity. In addition to scientific excellence, the community will be strengthened through sustained support for early career researchers who will be uniquely trained by working in an interdisciplinary environment with leading scientists.