Weak convergence rates for spectral regularization via sampling inequalities

TL;DR

This paper establishes weak convergence rates for spectral regularization in inverse problems using sampling inequalities, avoiding the need for source conditions, and applies kernel approximation techniques to derive these bounds.

Contribution

The paper generalizes sampling inequalities to spectral regularization and derives source-condition-independent weak convergence rates for inverse problems.

Findings

Weak convergence rates are established without source conditions.

Results apply to compact and trace class operators.

Sampling inequalities are extended to spectral regularization.

Abstract

Convergence rates in spectral regularization methods quantify the approximation error in inverse problems as a function of the noise level or the number of sampling points. Classical strong convergence rate results typically rely on source conditions, which are essential for estimating the truncation error. However, in the framework of kernel approximation, the truncation error in the case of Tikhonov regularization can be characterized entirely through sampling inequalities, without invoking source conditions. In this paper, we first generalize sampling inequalities to spectral regularization, and then, by exploiting the connection between inverse problems and kernel approximation, we derive weak convergence rate bounds for inverse problems, independently of source conditions. These weak convergence rates are established and analyzed when the forward operator is compact and uniformly bounded, or the kernel operator is of trace class.