# Regularization of linear ill-posed problems involving multiplication operators

## TL;DR

This paper develops a regularization framework for ill-posed equations involving multiplication operators, especially addressing the challenges posed by white noise in statistical settings and introducing concepts like effective ill-posedness.

## Contribution

It introduces a new regularization theory for non-compact operators with white noise, including the notion of effective ill-posedness and modifications to classical schemes.

## Key findings

- Analysis of intrinsic ill-posedness via multiplier function rearrangements
- Development of regularization methods for statistical ill-posed problems
- Application to deconvolution and evolution equations

## 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 regularization theory 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.