New a posteriori error estimates for approximate solutions to iregular operator equations

A brief overview of developed up to date a posteriori error estimates for approximate solutions to irregular operator equations is given. Among them are a posteriori estimates for some descriptive expanding compacts (A.G. Yagola, etc.), the evaluation using a posteriori residual values and regularizing functionals (A.S. Leonov), the estimates with more detailed a priori assumptions about solutions (A.B. Bakushinsky, etc.), estimating the accuracy of solutions to coefficient inverse problems for partial differential equations using the specifics of the Tikhonov regularization and the adaptive finite element method (L. Beilina, M. Klibanov, etc.). In this paper a new method for a posteriori estimates of the accuracy of approximate solutions calculated using the iterative procedures for irregular operator equations is proposed. The estimates are found using other a posteriori functionals of approximate solutions than in the overviewed papers. In this method, one can track the evolution of a posteriori estimates in solving the equation, which allows one to draw conclusions about iteration convergence and to introduce adequate improvements in the iterative procedures during their implementation.