Regularization of linear ill-posed problems involving multiplication operators

ABSTRACT

We study regularization of ill-posed equations involving multiplication operators when the multiplier function is positive almost everywhere and zero is an accumulation point of the range of this function. Such equations naturally arise from equations based on non-compact self-adjoint operators in Hilbert space, after applying unitary transformations arising out of the spectral theorem. For classical regularization theory, when noisy observations are given and the noise is deterministic and bounded, then non-compactness of the ill-posed equations is a minor issue. However, for statistical ill-posed equations with non-compact operators less is known if the data are blurred by white noise. We develop a theory for spectral regularization with emphasis on this case. In this context, we highlight several aspects, in particular, we discuss the intrinsic degree of ill-posedness in terms of rearrangements of the multiplier function. Moreover, we address the required modifications of classical regularization schemes in order to be used for non-compact statistical problems, and we also introduce the concept of the effective ill-posedness of the operator equation under white noise. This study is concluded with prototypical examples for such equations, as these are deconvolution equations and certain final value problems in evolution equations.

Keywords:

• Statistical ill-posed problem
• non-compact operator
• regularization
• degree of ill-posedness

2010 Mathematics Subject Classifications:

• 47A52
• 62G08
• 65J22

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We say that a (non-negative) function g dominates f, and write $f⪯g$, if there are a neighborhood $[0,\epsilon )$ and a constant k>0 such that $f\left(t\right)\le kg\left(t\right),0\le t\le \epsilon$.

2 For each $s\in S$ we have a random variable ${\xi }_s:\mathrm{\Omega }\to \mathbb{R}$.

Additional information

Funding

B. H. was supported by German Research Foundation (Deutsche Forschungsgemeinschaft) [grant numberDFG-grant HO 1454/12-1].