Geodesic
Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems
Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 534-556

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We consider the problem of solving linear operator equations from noisy data under the assumptions that the singular values of the operator decrease exponentially fast and that the underlying solution is also exponentially smooth in the Fourier domain. We suggest an estimator of the solution based on a running version of block thresholding in the space of Fourier coefficients. This estimator is shown to be sharp adaptive to the unknown smoothness of the solution.
Keywords: linear operator equation, white Gaussian noise, adaptive estimation, running block thresholding. 
@article{TVP_2003_48_3_a5,
     author = {L. Cavalier and Yu. F. Golubev and O. V. Lepskiǐ and A. Tsybakov},
     title = {Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {534--556},
     publisher = {mathdoc},
     volume = {48},
     number = {3},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a5/}
}
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L. Cavalier; Yu. F. Golubev; O. V. Lepskiǐ; A. Tsybakov. Block thresholding and sharp adaptive estimation in severely ill-posed inverse problems. Teoriâ veroâtnostej i ee primeneniâ, Tome 48 (2003) no. 3, pp. 534-556. http://geodesic.mathdoc.fr/item/TVP_2003_48_3_a5/