Maximal spaces for approximation rates in $$\ell ^1$$-regularization

Abstract We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically $$L^2$$ L 2 -space, is assumed satisfy two-sided Lipschitz condition respect ^2$$ -norm and the norm of image space. show that in this setting approximation rates arbitrarily high Hölder-type order parameter can be achieved, we characterize maximal subspaces sequences on which these are attained. On method also converges optimal terms noise level discrepancy principle as choice rule. Our analysis includes case penalty term not finite at exact solution (’oversmoothing’). As standard example discuss wavelet Besov spaces $$B^r_{1,1}$$ B , r . In demonstrate numerical simulations identification problem differential equation our theoretical results correctly predict improved convergence piecewise smooth unknown coefficients.