Methods of Solving Ill-Posed Problems

TL;DR

This paper reviews various methods for solving ill-posed operator equations, emphasizing the dynamical systems method as a more efficient approach compared to traditional regularization techniques.

Contribution

It provides a comprehensive review of existing methods for ill-posed problems, highlighting recent developments and focusing on the dynamical systems method's advantages.

Findings

Dynamical systems method is more efficient than traditional regularization.

Review covers variational regularization, quasi-solution, and iterative methods.

Recent developments have improved solution stability and computational efficiency.

Abstract

Many physical problems can be formulated as operator equations of the form Au = f. If these operator equations are ill-posed, we then resort to finding the approximate solutions numerically. Ill-posed problems can be found in the fields of mathematical analysis, mathematical physics, geophysics, medicine, tomography, technology and ecology. The theory of ill-posed problems was developed in the 1960's by several mathematicians, mostly Soviet and American. In this report we review the methods of solving ill-posed problems and recent developments in this field. We review the variational regularization method, the method of quasi-solution, iterative regularization method and the dynamical systems method. We focus mainly on the dynamical systems method as it is found that the dynamical systems method is more efficient than the regularization procedure.