Working Group Dynamic Inverse Problems

In a large variety of industrial, medical, or economical problems one is interested in a quantity which is not directly accessible and therefore has to be reconstructed from indirect measurements. A common example is Computed Tomography, in which a patient’s three-dimensional anatomy has to be computationally reconstructed based on two-dimensional X-ray images taken from all directions. Such problems are called inverse problems (IPs); their solution tremendously contributes to technological and scientific progress, from medical imaging-based personalized medicine over option pricing or determining physical constants all the way to monitoring of smart structures, to name but a few examples.

IPs are commonly formulated in a static setting where a wealth of theoretical results and numerical tools are available. With the current increase of sensitivity and temporal resolution in measurement technology we witness a shift of emphasis from static to (far more complicated) dynamic inverse problems (DIPs) in which the quantity of interest can change dynamically over time during the measurement process. This requires to additionally determine the dynamics from the spatiotemporal measurement data: Sometimes these are merely needed to explain and remove artefacts of a reconstruction (as in motion compensation in medical imaging), sometimes the dynamics are the actual quantity of interest (as in the imaging of leukocyte trafficking or fracture growth). The vast scientific and technological opportunities opened by observing temporal processes will drastically boost the demand for DIP research in the next few years.

Incorporating time-dependency leads to substantially increased problem sizes, an increased number of unknown model parameters, or a higher degree of model uncertainties compared to static settings. Even worse, indiscriminate inclusion of time as another dimension ignores causality and thus results in information loss and changed IP characteristics. Hence, DIPs require the development of a new comprehensive framework unfolding from modelling and learning through reduction methods to stable and efficient numerical solvers to tackle application problems currently out of reach. The mathematical tools, e.g. in IPs, optimization, machine learning, or model reduction, have now sufficiently matured to pursue this development and tackle DIPs in an integrated way. From today’s viewpoint, our approach provides the potential to improve a wide range of crucial applications such as motion compensation in tomography, dynamic magnetic resonance, or dynamic load monitoring in carbon fibre composites.

The overarching goal of the working group is the development of a holistic approach for dynamic image reconstruction and information extraction which enables efficient treatment of high-dimensional DIPs. This is achieved through the development of both model-based and data-driven approaches, followed by techniques to derive a computationally feasible approximation. The working group’s subjects are both at the forefront of research in inverse problems and crucial for a wide range of future technological developments currently out of reach.