Regularization of ill-posed problems with non-negative solutions

TL;DR

This survey reviews variational and iterative methods for reconstructing non-negative solutions to ill-posed problems, highlighting existing results and open challenges in infinite-dimensional spaces.

Contribution

It provides a comprehensive overview of entropy-based and projection methods for non-negative solution reconstruction, identifying key open problems.

Findings

Summarizes known theoretical results in the field.

Highlights open problems and research directions.

Categorizes methods into variational and iterative approaches.

Abstract

This survey reviews variational and iterative methods for reconstructing non-negative solutions of ill-posed problems in infinite-dimensional spaces. We focus on two classes of methods: variational methods based on entropy-minimization or constraints, and iterative methods involving projections or non-negativity-preserving multiplicative updates. We summarize known results and point out some open problems.