Inverse problems (IP) are ubiquitous in science and engineering, and appear when a quantity has to be reconstructed from indirect measurements. Whenever physics plays a crucial role in the description of an inverse problem, the mathematical model is based on a partial differential equation (PDE). Many imaging modalities belong to this category, including ultrasonography, electrical impedance tomography and photoacoustic tomography. Many different PDE appear, depending on the physical domain. The theoretical analysis of many IP in PDE is very well developed. Uniqueness results, stability estimates and reconstruction algorithms abound. However these results are often too abstract to have a strong impact in the applications, where many safe and effective modalities have had very limited use due to low reconstruction quality. This is mainly due to the following two fundamental limitations:
1. Most theoretical studies are carried out in idealistic settings, using approximated PDE models. While this makes the analysis simpler, it does not allow applying these results to most real-world scenarios, which are described by more complicated models. The analysis of such models, especially regarding the related inverse problems, is at its infancy.
2. Even in the cases when these simplified models can be used, there is always a gap between the abstract theoretical results (typically on uniqueness, stability and infinite-dimensional reconstruction algorithms) and numerical implementations.
The main objective of this project is to overcome these limitations within a multidisciplinary approach. On the one hand, we will use advanced PDE theory methods such as unique continuation to provide a rigorous mathematical analysis of several inverse problems associated with complicated PDE models. On the other hand, we will combine PDE theory techniques with methods of numerical analysis (including regularisation theory and optimisation), applied harmonic analysis (including wavelet theory and compressed sensing) and machine learning (including deep learning) in order to bridge the gap discussed above, by developing reconstruction algorithms that are supported by rigorous theoretical understanding and effective from the numerical point of view. While the main objective of this project is to develop the mathematics of IP in PDE, a long lasting outcome of this project is the implementation of the findings of the project in real-world scenarios.
The research of the PI, focused on PDE theory, on abstract and applied harmonic analysis and on abstract and numerical inverse problems, will guarantee an active coordination of all the tasks of this project. Furthermore, the several expertises brought by the three units on all the methodologies mentioned above will allow for a fruitful interaction of all the researchers and, eventually, ensure the success of this project.