Direct and converse theorems for iterative methods of solving irregular operator equations and finite difference methods for solving ill-posed Cauchy problems

Results obtained in recent years concerning necessary and sufficient conditions for the convergence (at a given rate) of approximation methods for solutions of irregular operator equations are overviewed. The exposition is given in the context of classical direct and converse theorems of approximation theory. Due to the proximity of the resulting necessary and sufficient conditions to each other, the solutions on which a certain convergence rate of the methods is reached can be characterized nearly completely. The problems under consideration include irregular linear and nonlinear operator equations and ill-posed Cauchy problems for first- and second-order differential operator equations. Procedures for stable approximation of solutions of general irregular linear equations, classes of finite-difference regularization methods and the quasi-reversibility method for ill-posed Cauchy problems, and the class of iteratively regularized Gauss–Newton type methods for irregular nonlinear operator equations are examined.