Introduction to Regularization and Learning Methods for Inverse Problems
This provides an educational overview of regularization and learning methods for inverse problems, targeting students and researchers in applied mathematics and machine learning, but it is incremental as it reviews existing concepts without new results.
The lecture notes introduce mathematical concepts for inverse problems, covering classical regularization methods like Tikhonov and sparsity promoting approaches, and modern deep-learning techniques such as learned regularization and plug-n-play methods.
These lecture notes evolve around mathematical concepts arising in inverse problems. We start by introducing inverse problems through examples such as differentiation, deconvolution, computed tomography and phase retrieval. This then leads us to the framework of well-posedness and first considerations regarding reconstruction and inversion approaches. The second chapter then first deals with classical regularization theory of inverse problems in Hilbert spaces. After introducing the pseudo-inverse, we review the concept of convergent regularization. Within this chapter we then proceed to ask the question of how to realize practical reconstruction algorithms. Here, we mainly focus on Tikhonov and sparsity promoting regularization in finite dimensional spaces. In the third chapter, we dive into modern deep-learning methods, which allow solving inverse problems in a data-dependent approach. The intersection between inverse problems and machine learning is a rapidly growing field and our exposition here restricts itself to a very limited selection of topics. Among them are learned regularization, fully-learned Bayesian estimation, post-processing strategies and plug-n-play methods.