Abstract:
It is proved that a priori global accuracy estimate for approximate solutions to linear inverse problems with perturbed data can be of the same order as approximate data errors for well-posed in the sense of Tikhonov problems only. A method for assessing the quality of selected sets of correctness is proposed. The use of the generalized residual method on a set of correctness allows us to solve the inverse problem and to obtain a posteriori accuracy estimate for approximate solutions, which is comparable with the accuracy of the problem data. The approach proposed is illustrated by a numerical example.
Key words:
linear inverse problems, correctness in the sense of Tikhonov, a priori and a posteriori accuracy estimate.
Citation:
A. S. Leonov, “Which of inverse problems can have a priori approximate solution accuracy estimates comparable in order with the data accuracy”, Sib. Zh. Vychisl. Mat., 17:4 (2014), 339–348; Num. Anal. Appl., 7:4 (2014), 284–292
\Bibitem{Leo14}
\by A.~S.~Leonov
\paper Which of inverse problems can have a~priori approximate solution accuracy estimates comparable in order with the data accuracy
\jour Sib. Zh. Vychisl. Mat.
\yr 2014
\vol 17
\issue 4
\pages 339--348
\mathnet{http://mi.mathnet.ru/sjvm554}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3409492}
\transl
\jour Num. Anal. Appl.
\yr 2014
\vol 7
\issue 4
\pages 284--292
\crossref{https://doi.org/10.1134/S199542391404003X}
Linking options:
https://www.mathnet.ru/eng/sjvm554
https://www.mathnet.ru/eng/sjvm/v17/i4/p339
This publication is cited in the following 6 articles:
P. I. Balk, A. S. Dolgal, “Double-Sided Estimates of Inclusion Type for Localizing and Detailing the Location of the Gravitational Field Sources”, Izv., Phys. Solid Earth, 58:3 (2022), 394
Zakharova S.A., Davydova M.A., Lukyanenko D.V., “Use of Asymptotic Analysis For Solving the Inverse Problem of Source Parameters Determination of Nitrogen Oxide Emission in the Atmosphere”, Inverse Probl. Sci. Eng., 29:3 (2021), 365–377
T. M. Bannikova, V. M. Nemtsov, N. A. Baranova, G. N. Konygin, O. M. Nemtsova, “Metod otsenki statisticheskoi pogreshnosti resheniya v obratnoi zadache spektroskopii”, Izv. IMI UdGU, 58 (2021), 3–17
Argun R. Gorbachev A. Lukyanenko D. Shishlenin M., “On Some Features of the Numerical Solving of Coefficient Inverse Problems For An Equation of the Reaction-Diffusion-Advection-Type With Data on the Position of a Reaction Front”, Mathematics, 9:22 (2021), 2894
Argun R. Gorbachev A. Levashova N. Lukyanenko D., “Inverse Problem For An Equation of the Reaction-Diffusion-Advection Type With Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement”, Mathematics, 9:18 (2021), 2342
Levashova N. Gorbachev A. Argun R. Lukyanenko D., “The Problem of the Non-Uniqueness of the Solution to the Inverse Problem of Recovering the Symmetric States of a Bistable Medium With Data on the Position of An Autowave Front”, Symmetry-Basel, 13:5 (2021), 860