Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces

Abstract

Even if the recent literature on enhanced convergence rates for Tikhonov regularization of ill-posed problems in Banach spaces shows substantial progress, not all factors influencing the best possible convergence rates under sufficiently strong smoothness assumptions were clearly determined. In particular, it was an open problem whether the structure of the residual term can limit the rates. For residual norms of power type in the functional to be minimized for obtaining regularized solutions, the latest rates results for nonlinear problems by Neubauer [On enhanced convergence rates for Tikhonov regularization of non-linear ill-posed problems in Banach spaces, Inverse Prob. 25 (2009), p. 065009] indicate an apparent qualification of the method caused by the residual norm exponent p. The new message of the present article is that optimal rates are shown to be independent of that exponent in the range 1 ≤ p < ∞. However, on the one hand, the smoothness of the image space influences the rates, and on the other hand best possible rates require specific choices of the regularization parameters α > 0. While for all p > 1 the regularization parameters have to decay to zero with some prescribed speed depending on p when the noise level tends to zero in order to obtain the best rates, the limiting case p = 1 shows some curious behaviour. For that case, the α-values must get asymptotically frozen at a fixed positive value characterized by the properties of the solution as the noise level decreases.

Keywords:

• Tikhonov regularization
• Banach spaces
• convergence rates

AMS Subject Classifications::

• 47J06
• 47A52

Acknowledgements

The work has been conducted during the Mini Special Semester on Inverse Problems, 18 May–15 July 2009, organized by RICAM (Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences) Linz, Austria. B. Hofmann and U. Tautenhahn as long-term guests of that scientific event thank RICAM for kind financial support. Research of B. Hofmann was partly supported by Deutsche Forschungsgemeinschaft (DFG) under Grant HO1454/7-2.