Abstract
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.
Highlights
This article is concerned with the stable solution of operator equations of the first kind in Banach spaces
Ψ is Gateaux differentiable and the steepest descent method applied to 1.2 coincides with the well-known Landweber method xn 1 xn − μn∇Ψ xn xn − μnA∗ Axn − y
Only weak convergence of the proposed scheme is shown. Another interesting approach to obtain convergence results of descent methods in general Banach spaces can be found in the recent papers by Reich and Zaslavski 12, 13
Summary
This article is concerned with the stable solution of operator equations of the first kind in Banach spaces. If X and Y are Hilbert spaces, many results exist for problem 1.2 concerning solution methods, convergence, and stability of them and parameter choice rules for α can be found in the literature. Ψ is Gateaux differentiable and the steepest descent method applied to 1.2 coincides with the well-known Landweber method xn 1 xn − μn∇Ψ xn xn − μnA∗ Axn − y This iterative method converges to the unique minimizer of problem 1.2 , if the stepsize μn is chosen properly. Only weak convergence of the proposed scheme is shown Another interesting approach to obtain convergence results of descent methods in general Banach spaces can be found in the recent papers by Reich and Zaslavski 12, 13.
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