Abstract
We study Tikhonov regularization for possibly nonlinear inverse problems with weighted ell ^1-penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an L^2-space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted ell ^2-norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces B^r_{1,1}. In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.
Highlights
In this paper we analyze numerical solutions of ill-posed operator equationsF(x) = g with a forward operator F mapping sequences x = (x j ) j∈Λ indexed by a countable set Λ to a Banach space Y
In this subsection, which may be skipped in first reading, we provide more details on the motivating example (2): Suppose the operator F is the composition of a forward operator G mapping functions on a domain Ω to elements of the Hilbert space Y and a wavelet synthesis operator S
We prove that the condition x† ∈ kt is necessary and sufficient for the Hölder type approximation rate O(α1−t ): Theorem 21 (Converse result for exact data) Suppose Assumption 3 and 4 hold true
Summary
In contrast to [29] and related works, we do not require that (r j ) j∈Λ is uniformly bounded away from zero This allows us to consider Besov B10,1-norm penalties given by wavelet coefficients. The class of operators satisfying this condition includes in particular the Radon transform and nonlinear parameter identification problems for partial differential equations with distributed measurements In this setting Besov B1r,1-norms can be written in the form of the penalty term in (1). It becomes much more likely that the penalty term fails to be finite at the exact solution, and it is desirable to derive error bounds for this situation This case has only rarely been considered in variational regularization theory. None of these assumptions is to be understood as a standing assumption, but each assumption is referenced whenever it is needed
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
