Multi-parameter Tikhonov Regularisation in Topological Spaces

TL;DR

This paper extends Tikhonov regularisation to topological spaces with multiple parameters, demonstrating well-posedness, convergence, and rates considering data and operator errors, thus broadening its theoretical foundation.

Contribution

It introduces a comprehensive analysis of multi-parameter Tikhonov regularisation in topological spaces, establishing well-posedness and convergence rates with operator errors included.

Findings

Regularisation is well-posed with continuous dependence on data.

Convergence to true solutions as noise decreases.

Derived convergence rates using variational inequalities.

Abstract

We study the behaviour of Tikhonov regularisation on topological spaces with multiple regularisation terms. The main result of the paper shows that multi-parameter regularisation is well-posed in the sense that the results depend continuously on the data and converge to a true solution of the equation to be solved as the noise level decreases to zero. Moreover, we derive convergence rates in terms of a generalised Bregman distance using the method of variational inequalities. All the results in the paper, including the convergence rates, consider not only noise in the data, but also errors in the operator.