×

Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces. (English) Zbl 1214.65031

Ill-posed problems \(F(x)= y\) in Banach spaces are considered. For the noisy data \(y^\delta\) of \(y\in Y\) with \(\| y-y^\delta\|\leq\delta\) regularized solutions are obtained by minimizing the Tikhonov-type functional
\[ T^\delta_\alpha(x)=\tfrac1p \| F(x)- y^\delta\|^P+\alpha R(x). \]
The functional depends on the positive parameters \(\alpha\), \(\delta\), \(p\) and on the function \(R\). In Hilbert spaces we can choose \(p= 2\) and \(R(x)=\| x-x^*\|^2\) and we can prove the convergence of minimizers \(x^\delta_\alpha\) of \(T^\delta_\alpha\) to a solution \(x^*\) of \(F(x)= y\).
The main message of the present paper is that optimal rates are shown to be independent of the exponent \(p\) in the range \(1\leq p<\infty\). Best possible rates require suitable choices of the regularization parameter \(\alpha\). The case \(p= 1\) shows some curious behavior. In that case the \(\alpha\)-values must get asymptotically frozen at a fixed value. Essential questions are unsolved in the case \(0< p< 1\). A discussion of different parameter choices is added. At the end, a generalized version of a Tikhonov functional with monotone functions \(f\) is studied:
\[ f(\| F(x)- y^\delta\|)+\alpha R(x). \]
Assume \(f\) to be convex, under some assumptions convergence can be proved. There are no concrete examples.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
65J20 Numerical solutions of ill-posed problems in abstract spaces; regularization
47J06 Nonlinear ill-posed problems

Citations:

Zbl 1176.65071
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1088/0266-5611/5/4/007 · Zbl 0695.65037 · doi:10.1088/0266-5611/5/4/007
[2] Engl HW, Regularization of Inverse Problems (1996)
[3] DOI: 10.1088/0266-5611/23/3/009 · Zbl 1131.65046 · doi:10.1088/0266-5611/23/3/009
[4] Scherzer O, Variational Methods in Imaging, Volume 167, Applied Mathematical Sciences (2008)
[5] DOI: 10.1088/0266-5611/5/4/008 · Zbl 0695.65038 · doi:10.1088/0266-5611/5/4/008
[6] DOI: 10.1515/jiip.1995.3.1.47 · Zbl 0833.65049 · doi:10.1515/jiip.1995.3.1.47
[7] Tautenhahn U, Parameter Identification and Inverse Problems in Hydrology, Geology and Ecology pp 261– (1996)
[8] DOI: 10.1088/0266-5611/19/1/301 · Zbl 1030.65061 · doi:10.1088/0266-5611/19/1/301
[9] DOI: 10.1088/0266-5611/21/4/007 · Zbl 1082.65055 · doi:10.1088/0266-5611/21/4/007
[10] DOI: 10.1080/00036810802555474 · Zbl 1200.65040 · doi:10.1080/00036810802555474
[11] DOI: 10.1088/0266-5611/25/6/065009 · Zbl 1176.65071 · doi:10.1088/0266-5611/25/6/065009
[12] DOI: 10.1088/0266-5611/22/3/004 · Zbl 1103.65062 · doi:10.1088/0266-5611/22/3/004
[13] Hein T, Inverse Probl. 25 (2009)
[14] Cioranescu I, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems 62 (1990) · doi:10.1007/978-94-009-2121-4
[15] Hein T, Inverse Probl. 25 (2009)
[16] DOI: 10.1016/0022-247X(91)90144-O · Zbl 0757.46034 · doi:10.1016/0022-247X(91)90144-O
[17] DOI: 10.1137/S0036142993253928 · Zbl 0878.65038 · doi:10.1137/S0036142993253928
[18] DOI: 10.1088/0266-5611/20/5/005 · Zbl 1068.65085 · doi:10.1088/0266-5611/20/5/005
[19] DOI: 10.1016/0041-5553(84)90253-2 · Zbl 0595.65064 · doi:10.1016/0041-5553(84)90253-2
[20] Tikhonov AN, Nonlinear Ill-posed Problems, Applied Mathematics and Mathematical Computation, Vol. 14 (Translated from the Russian) (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.