  These lecture notes for a graduate class present the regularization theory
for linear and nonlinear ill-posed operator equations in Hilbert spaces.
Covered are the general framework of regularization methods and their analysis
via spectral filters as well as the concrete examples of Tikhonov
regularization, Landweber iteration, regularization by discretization for
linear inverse problems. In the nonlinear setting, Tikhonov regularization and
iterative regularization (Landweber, Levenberg-Marquardt, and iteratively
regularized Gau{\ss}-Newton methods) are discussed. The necessary background
from functional analysis is also briefly summarized. The notes end with a brief
outlook to statistical inverse problems from both a frequentist and a Bayesian
point of view.
