Methods of Solving Ill-Posed Problems

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.