Inverse problems in spaces of measures
Institute of Mathematics and Scientific Computing, University of
2 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria
Revised: 17 October 2011
The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate 𝒪(n-1) in terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.
Mathematics Subject Classification: 65J20 / 46E27 / 49M05
Key words: Inverse problems / vector-valued finite Radon measures / Tikhonov regularization / delta-peak solutions / generalized conditional gradient method / iterative soft-thresholding / sparse deconvolution
© EDP Sciences, SMAI, 2012