There always is a variational source condition for nonlinear problems in Banach spaces
For researchers in inverse problems and regularization theory, this paper removes the need to verify variational source conditions case-by-case, showing they hold generically.
The paper proves that variational source conditions, which are used to derive convergence rates for regularization methods, are satisfied for almost every ill-posed inverse problem, covering linear/nonlinear and Hilbert/Banach space settings. This establishes them as a universal tool for convergence rate analysis.
Variational source conditions proved useful for deriving convergence rates for Tikhonov's regularization method and also for other methods. Up to now such conditions have been verified only for few examples or for situations which can be handled by classical techniques, too. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.