General Tikhonov Regularization with Applications in Geoscience (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 regularizationwith practical examples is considered; (b) an overview of a-priori and a-posteriori errorestimates for solutions of ill-posed problems is presented as well as a general schemeof a-posteriori error estimation; (c) a-posteriori error estimates for linear inverse problems and its finite-dimensional approximation are considered in detail together withpractical a-posteriori error estimate algorithms; (d) optimality in order for the errorestimator and extra-optimal regularizing algorithms are also discussed. In addition,the article contains applications of these theoretical results to solving two practicalgeophysical problems. First, for inverse problems of computer microtomography inmicrostructure analysis of shales, numerical experiments demonstrate that the use offunctions with bounded $VH$-variation for a piecewise uniform regularization has atheoretical and practical advantage over methods using $BV$-variation. For these problems, a new algorithm of a-posteriori error estimation makes it possible to calculatethe error of the solution in the form of a number. Second, in geophysical prospecting,Tikhonov's regularization is very effective in magnetic parameters inversion methodwith full tensor gradient data. In particular, the regularization algorithms allow tocompare different models in this method and choose the best one, MGT-model.
Keywords: Regularization; a-posteriori error estimates; extra-optimal methods; microtomography; magnetic parameters inversion.