Two new non-negativity preserving iterative regularization methods for ill-posed inverse problems

TL;DR

This paper introduces two innovative iterative regularization methods that preserve non-negativity and demonstrate strong convergence for solving ill-posed inverse problems, with applications to biosensor modeling and real data.

Contribution

The paper presents two novel non-negativity preserving iterative regularization methods with strong convergence and convergence rate results, improving upon existing approaches like the projected Landweber iteration.

Findings

Methods exhibit strong convergence even with noisy data.

Numerical examples show improved accuracy and speed.

Application to biosensor problem yields meaningful solutions.

Abstract

Many inverse problems are concerned with the estimation of non-negative parameter functions. In this paper, in order to obtain non-negative stable approximate solutions to ill-posed linear operator equations in a Hilbert space setting, we develop two novel non-negativity preserving iterative regularization methods. They are based on fixed point iterations in combination with preconditioning ideas. In contrast to the projected Landweber iteration, for which only weak convergence can be shown for the regularized solution when the noise level tends to zero, the introduced regularization methods exhibit strong convergence. There are presented convergence results, even for a combination of noisy right-hand side and imperfect forward operators, and for one of the approaches there are also convergence rates results. Specifically adapted discrepancy principles are used as a posteriori stopping rules of the established iterative regularization algorithms. For an application of the suggested new approaches, we consider a biosensor problem, which is modelled as a two dimensional linear Fredholm integral equation of the first kind. Several numerical examples, as well as a comparison with the projected Landweber method, are presented to show the accuracy and the acceleration effect of the novel methods. Case studies of a real data problem indicate that the developed methods can produce meaningful featured regularized solutions.