Projection methods for ill-posed problems revisited

TL;DR

This paper revisits projection methods for solving linear ill-posed problems, providing new convergence conditions and characterizations, including practical criteria and examples of non-convergence.

Contribution

It offers new equivalent conditions for convergence, characterizes global convergence via oblique projections, and extends results on local convergence for discretized least-squares solutions.

Findings

Discretized solutions can be characterized as oblique projections.

Global convergence is linked to subspaces with bounded angles.

An example demonstrates non-convergence of discretized solutions.

Abstract

The discretization of least-squares problems for linear ill-posed operator equations in Hilbert spaces is considered. The main subject of this article concerns conditions for convergence of the associated discretized minimum-norm least-squares solution to the exact solution using exact attainable data. The two cases of global convergence (convergence for all exact solution) or local convergence (convergence for a specific exact solution) are investigated. We review the existing results and prove new equivalent condition when the discretized solution always converges to the exact solution. An important tool is to recognize the discrete solution operator as oblique projection. Hence, global convergence can be characterized by certain subspaces having uniformly bounded angles. We furthermore derive practically useful conditions when this holds and put them into the context of known results. For local convergence we generalize results on the characterization of weak or strong convergence and state some new sufficient conditions. We furthermore provide an example of a bounded sequence of discretized solutions which does not converge at all, not even weakly.