This course provides an introduction to inverse problems with an emphasis on linear problems in finite dimensions. Special focus is placed on high-dimensional ill-posed inverse problems, such as those encountered in image processing, like inverse convolution and tomographic image reconstruction.
A central method employed is the use of regularization to handle ill-posedness and large condition numbers. The course covers mathematical and computational aspects of classical regularization methods, including truncated singular value decomposition, iterative methods, and variational models.
In addition to these classical approaches, the course also addresses the statistical perspective. Statistical models for the noise in data and the unknown signal are introduced. Techniques from Bayesian statistics can now be used to calculate the distribution of the signal based on the measured data. The course covers computational methods to simulate random outcomes from such a distribution, based on Markov-chain Monte Carlo methods and Bayesian inference. The course also explores alternative approaches to compute appropriate estimates related to the maximum likelihood method.