On ill-posedness concepts, stable solvability and saturation

TL;DR

This paper explores various concepts of well-posedness and ill-posedness for linear and nonlinear operator equations in Hilbert spaces, analyzing their properties, examples, and implications for regularization techniques.

Contribution

It clarifies the relationships between different well-posedness concepts, investigates local stability notions for nonlinear problems, and examines how non-injectivity affects regularization saturation.

Findings

Linear well-posedness is a global property influenced by the nullspace.

Non-injectivity impacts the saturation behavior of Tikhonov and Lavrentiev regularization.

Examples include autoconvolution and quadratic equations in Hilbert spaces.

Abstract

We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.