Implicit iteration methods in Hilbert scales under general smoothness conditions

TL;DR

This paper develops implicit iteration regularization methods in Hilbert scales for solving linear ill-posed problems with noisy data, providing order optimal error bounds and efficient algorithms for parameter choice.

Contribution

It introduces a new framework for implicit iteration methods under general smoothness conditions using operator monotonicity and interpolation in Hilbert scales.

Findings

Order optimal error bounds achieved with a priori and a posteriori parameter choices.

A fast Newton-based algorithm for implementing the discrepancy principle.

Effective regularization in ill-posed problems with general smoothness assumptions.

Abstract

For solving linear ill-posed problems regularization methods are required when the right hand side is with some noise. In the present paper regularized solutions are obtained by implicit iteration methods in Hilbert scales. % By exploiting operator monotonicity of certain functions and interpolation techniques in variable Hilbert scales, we study these methods under general smoothness conditions. Order optimal error bounds are given in case the regularization parameter is chosen either {\it a priori} or {\it a posteriori} by the discrepancy principle. For realizing the discrepancy principle, some fast algorithm is proposed which is based on Newton's method applied to some properly transformed equations.