Conditional stability versus ill-posedness for operator equations with monotone operators in Hilbert space

TL;DR

This paper investigates the effects of conditional stability and ill-posedness on Lavrentiev regularization for operator equations with monotone operators in Hilbert spaces, providing new convergence rate results including for non-selfadjoint cases.

Contribution

It introduces a detailed analysis of stability phenomena in regularization, extending convergence rate results to non-selfadjoint linear monotone operators and nonlinear problems.

Findings

Derived general convergence rates including logarithmic rates.

Extended results to non-selfadjoint linear monotone operators.

Demonstrated theoretical findings with examples like fractional integral and Cesàro operators.

Abstract

In the literature on singular perturbation (Lavrentiev regularization) for the stable approximate solution of operator equations with monotone operators in the Hilbert space the phenomena of conditional stability and local well-posedness and ill-posedness are rarely investigated. Our goal is to present some studies which try to bridge this gap. So we discuss the impact of conditional stability on error estimates and convergence rates for the Lavrentiev regularization and distinguish for linear problems well-posedness and ill-posedness in a specific manner motivated by a saturation result. The role of the regularization error in the noise-free case, called bias, is a crucial point in the paper for nonlinear and linear problems. In particular, for linear operator equations general convergence rates, including logarithmic rates, are derived by means of the method of approximate source conditions. This allows us to extend well-known convergence rates results for the Lavrentiev regularization that were based on general source conditions to the case of non-selfadjoint linear monotone forward operators for which general source conditions fail. Examples presenting the self-adjoint multiplication operator as well as the non-selfadjoint fractional integral operator and Ces\`aro operator illustrate the theoretical results. Extensions to the nonlinear case under specific conditions on the nonlinearity structure complete the paper.