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  -