On complex-valued deautoconvolution of compactly supported functions with sparse Fourier representation. (5th August 2016)

• Record Type: Journal Article
• Authors: Bürger, Steven; Flemming, Jens; Hofmann, Bernd
• Is Part Of: Inverse problems. Volume 32:Number 10(2016:Oct.)
• Publication Date: 2016-08-05
• Subjects: ill-posed inverse problems -- deautoconvolution -- nonlinearity conditions -- sparsity constraints -- tikhonov regularization -- convergence rates -- variational source conditions; 47J06 -- 65J15 -- 47A52 -- 45Q05; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/0266-5611/32/10/104006
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: Convergence rates results for the Tikhonov regularization of nonlinear ill-posed operator equations are missing, even for a Hilbert space setting, if a range type source condition fails and if moreover nonlinearity conditions of tangential cone type cannot be shown. This situation applies for a deautoconvolution problem in complex-valued L 2 -spaces over finite real intervals, occurring in a slightly generalized version in laser optics. For this problem we show that the lack of applicable convergence rates results can be overcome under the assumption that the solution of the operator equation has a sparse Fourier representation. Precisely, we derive a variational source condition for that case, which implies a convergence rate immediately. The surprising observation is that a sparsity assumption imposed on the solution leads to success, although the used norm square is not known to be a sparsity promoting penalty in the Tikhonov functional.