Sourcewise representability conditions and power estimates of convergence rate in Tikhonov’s scheme for solving ill-posed extremal problems

Abstract

We study the rate of the convergence of approximations generated by the Tikhonov scheme for solving ill-posed optimization problems with smooth functionals given in a general form in a Hilbert space. We establish sourcewise representability conditions which are necessary and sufficient for the convergence of approximations at a power rate. Sufficient conditions are related to the estimate of the discrepancy with respect to the objective functional, while the necessary ones are formulated for the estimate with respect to the argument. We specify certain cases when sufficient and necessary conditions coincide in essence.