  We provide an overview of recent progress in statistical inverse problems
with random experimental design, covering both linear and nonlinear inverse
problems. Different regularization schemes have been studied to produce robust
and stable solutions. We discuss recent results in spectral regularization
methods and regularization by projection, exploring both approaches within the
context of Hilbert scales and presenting new insights particularly in
regularization by projection. Additionally, we overview recent advancements in
regularization using convex penalties. Convergence rates are analyzed in terms
of the sample size in a probabilistic sense, yielding minimax rates in both
expectation and probability. To achieve these results, the structure of
reproducing kernel Hilbert spaces is leveraged to establish minimax rates in
the statistical learning setting. We detail the assumptions underpinning these
key elements of our proofs. Finally, we demonstrate the application of these
concepts to nonlinear inverse problems in pharmacokinetic/pharmacodynamic
(PK/PD) models, where the task is to predict changes in drug concentrations in
patients.
