Regularization Methods for Ill-Posed Problems in Multiple Hilbert Scales

TL;DR

This paper introduces regularization methods within multiple Hilbert scales for ill-posed problems, establishing convergence results, orders of convergence, and relations between source conditions, applicable to single and multiple observations.

Contribution

It develops the concept of multiple Hilbert scales, defines regularization methods on these scales, and proves convergence and optimal convergence orders for both single and multiple observation cases.

Findings

Proved convergence of regularization methods in multiple Hilbert scales.

Derived orders of convergence and optimal orders.

Established relations between different source conditions.

Abstract

Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases convergence is proved and orders and optimal orders of convergence are shown.