@Article{CSIAM-AM-1-1,
author = {Yanfei, Wang and S., Leonov, Alexander and V., Lukyanenko, Dmitry and Yagola, G., Anatoly},
title = {General Tikhonov Regularization with Applications in Geoscience},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2020},
volume = {1},
number = {1},
pages = {53--85},
abstract = {<p style="text-align: justify;">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&#39;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&#39;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.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.2020-0004},
url = {https://global-sci.com/article/82370/general-tikhonov-regularization-with-applications-in-geoscience}
}