Journal ArticleDOI
Block Thresholding and Sharp Adaptive Estimation in Severely Ill-Posed Inverse Problems
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In this paper, the authors 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.Abstract:
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.read more
Citations
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Journal ArticleDOI
Nonparametric statistical inverse problems
TL;DR: In this paper, the authors explain some basic theoretical issues regarding nonparametric statistics applied to inverse problems and present classical concepts such as the white noise model, risk estimation, minimax risk, model selection and optimal rates of convergence.
Journal ArticleDOI
Sharp Optimality in Density Deconvolution with Dominating Bias. II
TL;DR: In this paper, the authors consider estimation of the common probability density f of independent identically distributed random variables (X_i) that are observed with an additive independent noise and propose a kernel-type estimator, whose variance turns out to be asymptotically negligible with respect to its squared bias under both pointwise and √ √ L √ l 2 risk.
Journal ArticleDOI
Penalized contrast estimator for adaptive density deconvolution
TL;DR: In this article, the problem of estimating the density g of independent and identically disributed variables Xi, from a sample Z1,..., Z, such that Zi = Xi + aei for i = 1,.., n, and e is noise independent of X, with ae having a known distribution.
Journal ArticleDOI
Risk hull method and regularization by projections of ill-posed inverse problems
Laurent Cavalier,Yu. Golubev +1 more
TL;DR: In this paper, the authors study a standard method of regularization by projections of the linear inverse problem Y = Af + ∈, where ∈ is a white Gaussian noise, and A is a known compact operator with singular values converging to zero with polynomial decay.
Book ChapterDOI
Inverse Problems in Statistics
TL;DR: This work presents the framework of statistical inverse problems where the data are corrupted by some stochastic error, and explains some basic issues regarding nonparametric statistics applied to inverse problems.
References
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Journal ArticleDOI
Adaptive wavelet estimator for nonparametric density deconvolution
Marianna Pensky,Brani Vidakovic +1 more
TL;DR: The electronic version of this article is the complete one and can be found online at: http://projecteuclid.org/eaclid/1017939249.
Journal ArticleDOI
Error bounds for tikhonov regularization in hilbert scales
TL;DR: For an ill-posed problem Af = g, this article showed that the accuracy of Tikhonov regularization is asymptotically best possible, provided that ω is chosen optimally and p ≥ (q - a)/2.
Journal ArticleDOI
Statistical inverse estimation in Hilbert scales
B. A. Mair,Frits H. Ruymgaart +1 more
TL;DR: The recovery of signals from indirect measurements, blurred by random noise, is considered under the assumption that prior knowledge regarding the smoothness of the signal is avialable and the general problem is embedded in an abstract Hilbert scale.
Journal ArticleDOI
Oracle inequalities for inverse problems
TL;DR: In this article, the authors consider a sequence space model of statistical linear inverse problems where the objective is to mimic the estimator in the set of linear estimators that has the smallest risk on the true function.
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