Abstract:
We investigate rate of convergence estimates for approximations generated by Tikhonov's scheme for solving ill-posed optimization problems with general smooth functionals in a Hilbert space. Sourcewise representability conditions necessary and sufficient for convergence of approximations at the power rate are established. Sufficient conditions are related to estimates of a discrepancy by the objective functional while necessary ones are formulated for estimates by the argument. The cases are specified where sufficient and necessary conditions coincide in the main.
\Bibitem{Kok14}
\by M.~Yu.~Kokurin
\paper Conditions of sourcewise representability and power estimates of convergence rate in Tikhonov's scheme for solving ill-posed extremal problems
\jour Izv. Vyssh. Uchebn. Zaved. Mat.
\yr 2014
\issue 7
\pages 72--82
\mathnet{http://mi.mathnet.ru/ivm8913}
\transl
\jour Russian Math. (Iz. VUZ)
\yr 2014
\vol 58
\issue 7
\pages 61--70
\crossref{https://doi.org/10.3103/S1066369X1407007X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84903719450}
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https://www.mathnet.ru/eng/ivm8913
https://www.mathnet.ru/eng/ivm/y2014/i7/p72
This publication is cited in the following 4 articles:
Mikhail M. Kokurin, “A posteriori choice of time-discretization step in finite difference methods for solving ill-posed Cauchy problems in Hilbert space”, Journal of Inverse and Ill-posed Problems, 29:6 (2021), 935
M. Y. Kokurin, “Source conditions and accuracy estimates in Tikhonov's scheme of solving ill-posed nonconvex optimization problems”, J. Inverse Ill-Posed Probl., 26:4 (2018), 463–475
M. Yu. Kokurin, “Necessary and sufficient conditions for power convergence rate of approximations in Tikhonov's scheme for solving ill-posed optimization problems”, Russian Math. (Iz. VUZ), 61:6 (2017), 51–59
M. Yu. Kokurin, “Convergence rate estimates for Tikhonov's scheme as applied to ill-posed nonconvex optimization problems”, Comput. Math. Math. Phys., 57:7 (2017), 1101–1110
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