TY  - JOUR
A1  - Rastogi, Abhishake
T1  - Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems
JF  - Communications on Pure and Applied Analysis
N2  - In this paper, we consider the nonlinear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered to reconstruct the estimator from the random noisy data. In this statistical learning setting, we derive the rates of convergence for the regularized solution under certain assumptions on the nonlinear forward operator and the prior assumptions. We discuss estimates of the reconstruction error using the approach of reproducing kernel Hilbert spaces.
KW  - Statistical inverse problem
KW  - Tikhonov regularization
KW  - Hilbert Scales
KW  - reproducing kernel Hilbert space
KW  - minimax convergence rates
Y1  - 2020
U6  - https://doi.org/10.3934/cpaa.2020183
SN  - 1534-0392
SN  - 1553-5258
VL  - 19
IS  - 8
SP  - 4111
EP  - 4126
PB  - American Institute of Mathematical Sciences
CY  - Springfield
ER  - 
TY  - GEN
A1  - Rastogi, Abhishake
T1  - Tikhonov regularization with oversmoothing penalty for linear statistical inverse learning problems
T2  - AIP Conference Proceedings : third international Conference of mathematical sciences (ICMS 2019)
N2  - In this paper, we consider the linear ill-posed inverse problem with noisy data in the statistical learning setting. The Tikhonov regularization scheme in Hilbert scales is considered in the reproducing kernel Hilbert space framework to reconstruct the estimator from the random noisy data. We discuss the rates of convergence for the regularized solution under the prior assumptions and link condition. For regression functions with smoothness given in terms of source conditions the error bound can explicitly be established.
KW  - Statistical inverse problem
KW  - Tikhonov regularization
KW  - Hilbert Scales
KW  - Reproducing kernel Hilbert space
KW  - Minimax convergence rates
Y1  - 2019
SN  - 978-0-7354-1930-8
U6  - https://doi.org/10.1063/1.5136221
SN  - 0094-243X
VL  - 2183
PB  - American Institute of Physics
CY  - Melville
ER  - 
TY  - INPR
A1  - Dicken, Volker
T1  - Simultaneous activity and attenuation reconstruction in emission tomography
N2  - In single photon emission computed tomography (SPECT) one is interested in reconstructing the activity distribution f of some radiopharmaceutical. The data gathered suffer from attenuation due to the tissue density µ. Each imaged slice incorporates noisy sample values of the nonlinear attenuated Radon transform (formular at this place in the original abstract) Traditional theory for SPECT reconstruction treats µ as a known parameter. In practical applications, however, µ is not known, but either crudely estimated, determined in costly additional measurements or plainly neglected. We demonstrate that an approximation of both f and µ from SPECT data alone is feasible, leading to quantitatively more accurate SPECT images. The result is based on nonlinear Tikhonov regularization techniques for parameter estimation problems in differential equations combined with Gauss-Newton-CG minimization.
T3  - NLD Preprints - 50 
KW  - SPECT
KW  - tomogrphy
KW  - attenuated Radon transform
KW  - nonlinear invers problem
KW  - Tikhonov regularization
KW  - nonlinear optimization
Y1  - 1998
U6  - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus-14747
ER  -