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Choosing this minimizer as the next iterate yields, again by using the proximal mapping of [PHI] and (2.
k] is shown to approximate the minimizer of the functional.
This characterizes a minimizer u for [THETA] = F + [PHI] by
The minimizer in the first step of the algorithm can now be obtained by considering the first order optimality condition
For nonlinear operators, the minimizer cannot be computed explicitly in general, and the authors of [38] suggest to use a fixed point iteration based on the first order optimality condition of the surrogate functional.
As mentioned above, this procedure reconstructs a global minimizer of [J.
3 show that the TIGRA algorithm can in principle be applied to the minimization Of the transformed Tikhonov functional for the case q = 2 and reconstructs a global minimizer.
Additionally, the fact that the reconstructed solutions are always close to the true solution suggests that the algorithm reconstructs the global minimizer fitting the constructed data and thus provides a reconstruction of good quality.
The minimizer (or at least a critical point of the functional) will be computed by the method of surrogate functionals, that is we consider
k]) is certainly a critical point of the surrogate functional, but it remains to show that it is also a global minimizer of [J.
k]) has a unique fixed point, and the fixed point iteration converges towards the minimizer.
As a consequence, the functional might have several, even local, minimizer.