Regularisation theory in the data driven setting

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Inverse problems deal with the reconstruction of some quantity of interest from indirectly measured data. A typical example is medical imaging, where there is no direct access to the quantity of interest (the inside of the patient's body) and imaging techniques, such X-Ray imaging and magnetic resonance imaging (MRI), are used. The classical approach to inverse problems uses models that describe the physics of the measurement. For example, in X-Ray imaging this model would describe how X-Rays pass through the body. In the era of big data, however, it becomes increasingly popular not to model the physics but to use vast amounts of data instead that relate known images with corresponding measurements. The theory of such data driven methods, however, is not well developed yet. It is not well understood, under which conditions on the training data such methods are stable with respect to small changes in the measurement and how well they adapt to images that are different from the training images. It is important to understand this, since otherwise the reconstruction algorithm can miss important features of the image if they weren't present in the training set, such as tumours at previously unseen locations.In this project I will extend the state-of-the-art model based theory to this data driven setting. I will study under which conditions can data driven methods achieve regularisation, i.e. when can they stably solve an otherwise unstable problem. This will make it easier to analyse stability of data driven reconstruction methods and help developing novel, stable data driven inversion methods with mathematical guarantees. I will also collaborate with the National Physical Laboratory and the Department of Chemical Engineering and Biotechnology in Cambridge on applications of my methods in imaging to reduce the time needed to acquire an image and make the reconstructions more reliable.