Minimization of Tikhonov Functionals in Banach Spaces

By Thomas Bonesky, K. Kazimierski, Peter Maass, Frank Schöpfer, Tal Schuster. 2008

100 citations recorded by the source corpus.

Tikhonov functionals are known to be well suited for obtaining regularized solutions of linear operator equations. We analyze two iterative methods for finding the minimizer of norm-based Tikhonov functionals in Banach spaces. One is the steepest descent method, whereby the iterations are directly carried out in the underlying space, and the other one performs iterations in the dual space. We prove strong convergence of both methods.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1abf819d464e05d79896e5fb01e91f05f92ece58

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Citations in the corpus

On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems (2010, result): “For recent results we refer to the monograph [29] and in an exemplary manner to the papers [3, 4, 10, 12, 13, 14, 19, 21, 22, 26, 27, 28, 30, 31, 33] as well as to the thesis [24].”
Inverse problems in spaces of measures (2013, background): “While the basic regularization theory of inverse problems is well-established in Hilbert spaces [22] and for linear problems in Banach spaces [34], recent interest focuses on specific linear and non-linear problems usually set in Banach spaces [4,9,21,31,32,38,43,44,49].”
An analysis of electrical impedance tomography with applications to Tikhonov regularization (2012, methodology): “Actually this is true for any r ∈ (1, 2], once noting the following inequality for Bregman distance dξ(θ, θ†) [5], Lemma 2.”
An extension of the variational inequality approach for obtaining convergence rates in regularization of nonlinear ill-posed problems (2010, background): “MSC2000 subject classification: 47J06, 65J20, 47A52, 49N45 Keywords: Ill-posed problems, Tikhonov regularization, convergence rates, variational inequalities, source conditions, distance function, structure of nonlinearity, generalized Young inequality.”