Optimal discretization and Degrees of ill-posedness for inverse estimation in Hilbert scales in the presence of random noise
- Mathé, Peter
- Pereverzev, Sergei V.
2010 Mathematics Subject Classification
- 62G05 65J10
- Ill-posed problems, inverse estimation, operator equations, Gaussian noise, optimal difficulty, regularized inverse estimator, histogram estimator
The problem of minimizing the difficulty of the inverse estimation of some unknown element x0 from noisy observations yδ = Ax0 + δξ in dependence of the nature of the random noise ξ is considered. It is shown that a combination of a Tikhonov regularization estimator with a certain projection scheme is order optimal in the sense of difficulty for a wide class of operators A acting along Hilbert scales.