Regularisation theory in the data driven setting

逆问题处理从间接测量的数据重建一些感兴趣的量。一个典型的例子是医学成像，其中没有直接访问感兴趣的量（病人的身体内部）和成像技术，如X射线成像和磁共振成像（MRI），使用。反问题的经典方法使用描述测量物理的模型。例如，在X射线成像中，该模型将描述X射线如何穿过身体。然而，在大数据时代，越来越流行的做法是不对物理模型进行建模，而是使用大量数据，将已知图像与相应的测量结果联系起来。然而，这种数据驱动方法的理论还没有得到很好的发展。还没有很好地理解，在训练数据的哪些条件下，这些方法相对于测量中的小变化是稳定的，以及它们如何很好地适应不同于训练图像的图像。理解这一点很重要，因为否则重建算法可能会错过图像的重要特征，如果它们不存在于训练集中，例如以前看不见的位置的肿瘤。在这个项目中，我将扩展最先进的基于模型的理论到这个数据驱动的设置。我将研究在什么条件下数据驱动的方法可以实现正则化，即什么时候它们可以稳定地解决一个不稳定的问题。这将使分析数据驱动重建方法的稳定性变得更加容易，并有助于开发具有数学保证的新颖、稳定的数据驱动反演方法。我还将与剑桥的国家物理实验室和化学工程与生物技术系合作，将我的方法应用于成像，以减少获取图像所需的时间，并使重建更加可靠。

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.