(untitled) (CSIAM Transactions on Applied Mathematics, 2020)
The article is devoted to a review of the following new elements of the modern theory of solving inverse problems: (a) general theory of Tikhonov's regularization with practical examples is considered; (b) an overview of a-priori and a-posteriori error estimates for solutions of ill-posed problems is presented as well as a general scheme of a-posteriori error estimation; (c) a-posteriori error estimates for linear inverse problems and its finite-dimensional approximation are considered in detail together with practical a-posteriori error estimate algorithms; (d) optimality in order for the error estimator and extra-optimal regularizing algorithms are also discussed. In addition, the article contains applications of these theoretical results to solving two practical geophysical problems. First, for inverse problems of computer microtomography in microstructure analysis of shales, numerical experiments demonstrate that the use of functions with bounded VH-variation for a piecewise uniform regularization has a theoretical and practical advantage over methods using BV-variation. For these problems, a new algorithm of a-posteriori error estimation makes it possible to calculate the error of the solution in the form of a number. Second, in geophysical prospecting, Tikhonov's regularization is very effective in magnetic parameters inversion method with full tensor gradient data. In particular, the regularization algorithms allow to compare different models in this method and choose the best one, MGT-model.