ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2013 - Volume 19, Issue 01  

Research Article

Inverse problems in spaces of measures

Kristian Brediesa1 and Hanna Katriina Pikkarainena2

a1 Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstraße 36, 8010 Graz, Austria. kristian.bredies@uni-graz.at

a2 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria; hanna.pikkarainen@ricam.oeaw.ac.at

Abstract

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.

(Received November 16 2010)

(Revised October 17 2011)

(Online publication March 27 2012)

Key Words:

  • Inverse problems;
  • vector-valued finite Radon measures;
  • Tikhonov regularization;
  • delta-peak solutions;
  • generalized conditional gradient method;
  • iterative soft-thresholding;
  • sparse deconvolution

Mathematics Subject Classification:

  • 65J20;
  • 46E27;
  • 49M05