4 research outputs found
Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints
The Tikhonov regularization of linear ill-posed problems with an
penalty is considered. We recall results for linear convergence rates and
results on exact recovery of the support. Moreover, we derive conditions for
exact support recovery which are especially applicable in the case of ill-posed
problems, where other conditions, e.g. based on the so-called coherence or the
restricted isometry property are usually not applicable. The obtained results
also show that the regularized solutions do not only converge in the
-norm but also in the vector space (when considered as the
strict inductive limit of the spaces as tends to infinity).
Additionally, the relations between different conditions for exact support
recovery and linear convergence rates are investigated.
With an imaging example from digital holography the applicability of the
obtained results is illustrated, i.e. that one may check a priori if the
experimental setup guarantees exact recovery with Tikhonov regularization with
sparsity constraints
Adaptive Frame Methods for Elliptic Operator Equations: The Steepest Descent Approach
Abstract: This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are particularly interested in discretization schemes based on wavelet frames. We show that by using three basic subroutines an implementable, convergent scheme can be derived, which, moreover, has optimal computational complexity. The scheme is based on adaptive steepest descent iterations. We illustrate our findings by numerical results for the computation of solutions of the Poisson equation with limited Sobolev smoothness on intervals in 1D and L-shaped domains in 2D