Geodesic
The clustering effect for stationary points of discrepancy functionals associated with conditionally well-posed inverse problems
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 4, pp. 393-406.

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In the Hilbert space, we consider a class of conditionally well-posed inverse problems, for which the Hölder type estimate of conditional stability on a closed convex bounded subset holds. We investigate the Ivanov quasisolution method and its finite dimensional version associated with the minimizing a multi-extremal discrepancy functional over a conditional stability set and over the finite dimensional section of this set, respectively. For these optimization problems, we prove that each their stationary point that is located not too far from the desired solution of the original inverse problem, in reality belongs to a small neighborhood of the solution. Estimates for the diameter of this neighborhood in terms of error levels in input data are also given. 
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     author = {M. Yu. Kokurin},
     title = {The clustering effect for stationary points of discrepancy functionals associated with conditionally well-posed inverse problems},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
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     url = {http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a3/}
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M. Yu. Kokurin. The clustering effect for stationary points of discrepancy functionals associated with conditionally well-posed inverse problems. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 4, pp. 393-406. http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a3/

[1] Lavrentev M. M., Romanov V. G., Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR

[2] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sibirskoe nauchnoe izd-vo, Novosibirsk, 2008

[3] Romanov V. G., Ustoichivost v obratnykh zadachakh, Nauchnyi mir, M., 2004

[4] Isakov V., Inverse Problems for Partial Differential Equations, Springer, N.Y., 2006 | MR | Zbl

[5] Yakhno V. G., Obratnye zadachi dlya differentsialnykh uravnenii uprugosti, Nauka, Novosibirsk, 1990

[6] Kokurin M. Yu., “Conditionally well-posed and generalized well-posed problems”, Comput. Mathem. and Math. Phys., 53:6 (2013), 681–690 | DOI | DOI | MR | Zbl

[7] Kokurin M. Yu., “On a characteristic property of conditionally well-posed problems”, J. of Inverse and Ill-Posed Problems, 23:3 (2015), 245–262 | DOI | MR | Zbl

[8] Ivanov V. K., Vasin V. V., Tanana V. P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978

[9] Bakushinsky A. B., “A posteriori error estimates for approximate solutions of irregular operator equations”, Dokl. Math., 83:2 (2011), 192–193 | DOI | MR | Zbl

[10] Kokurin M. Yu., “On stable finite dimensional approximation of conditionally well-posed inverse problems”, Inverse Problems, 32:10 (2016), 105007 | DOI | MR | Zbl

[11] Kokurin M. Yu., “Stable gradient projection method for nonlinear conditionally well-posed inverse problems”, J. of Inverse and Ill-posed Problems, 24:3 (2016), 323–332 | DOI | MR | Zbl

[12] Bauschke H. H., Borwein J. M., “On the convergence of von Neumann's alternating projection algorithm for two sets”, Set-Valued Analysis, 1 (1993), 185–212 | DOI | MR | Zbl

[13] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961