Geodesic
Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2020), pp. 29-38.

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We consider the problems of calculating global and local a-posteriori error estimates of approximate solutions to ill-posed inverse problems, introduced and investigated earlier by the author. For linear inverse problems in Hilbert spaces, they consist in maximizing a quadratic functional with two quadratic constraints. The article shows how under certain conditions these problems can be reduced to a problem of maximizing a special (written analytically) differentiable functional with one constraint. New algorithms for calculating global and local a-posteriori error estimates based on the solution of these problems are proposed. Their effectiveness is illustrated by numerical experiments on a-posteriori error estimation of solutions to the model two-dimensional inverse problem of potential continuation. Experiments show that the proposed algorithms give a-posteriori error estimates close to the true error values. Proposed algorithms for global a-posteriori error estimation turn out to be more rapid (3 to 5 times) than the previously known algorithms.
Keywords: linear ill-posed problems, regularizing algorithms, a-posteriori error estimates. 
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     title = {Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {29--38},
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     number = {2},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2020_2_a3/}
}
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A. S. Leonov. Effective algorithms for computing global and local posterior error estimates of solutions to linear ill-posed problems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2020), pp. 29-38. http://geodesic.mathdoc.fr/item/IVM_2020_2_a3/

[1] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979

[2] Leonov A. S., “Locally extra-optimal regularizing algorithms”, J. Inverse and III-posed Probl., 22:5 (2014), 713–737 | MR | Zbl

[3] Leonov A. S., “Lokalno ekstraoptimalnye regulyarizuyuschie algoritmy i aposteriornye otsenki tochnosti dlya nekorrektnykh zadach s razryvnymi resheniyami”, Zhurn. vychisl. matem. i matem. fiz., 56:1 (2016), 3–15 | DOI | Zbl

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

[5] Morozov V. A., Regulyarnye metody resheniya nekorrektno postavlennykh zadach, Nauka, M., 1987

[6] Bakushinskii A. B., Kokurin M. Yu., Iteratsionnye metody resheniya nekorrektnykh operatornykh uravnenii s gladkimi operatorami, Editorial URSS, M., 2002

[7] Engl H. W., Hanke M., Neubauer A., Regularization of inverse problems, Kluwer Academic Publ., Dordrecht, 1996 | MR | Zbl

[8] Dorofeev K. Yu., Titarenko V. N., Yagola A. G., “Algoritmy postroeniya aposteriornykh otsenok pogreshnostei dlya nekorrektnykh zadach”, Zhurn. vychisl. matem. i matem. fiz., 43:1 (2003), 12–25 | MR

[9] Yagola A. G., Nikolaeva N. N., Titarenko V. N., “Otsenka pogreshnosti resheniya uravneniya Abelya na mnozhestvakh monotonnykh i vypuklykh funktsii”, Sib. zhurn. vychisl. matem., 6:2 (2003), 171–180 | Zbl

[10] Bakushinskii A. B., “Aposteriornye otsenki tochnosti dlya priblizhennykh reshenii neregulyarnykh operatornykh uravnenii”, Dokl. RAN, 437:4 (2011), 439–440

[11] Leonov A. S., “Ob aposteriornykh otsenkakh tochnosti resheniya lineinykh nekorrektno postavlennykh zadach i ekstraoptimalnykh regulyarizuyuschikh algoritmakh”, Vychisl. metody i programmirovanie, 11 (2010), 14–24

[12] Leonov A. S., “Aposteriornye otsenki tochnosti resheniya nekorrektno postavlennykh obratnykh zadach i ekstraoptimalnye regulyarizuyuschie algoritmy ikh resheniya”, Sib. zhurn. vychisl. matem., 15:1 (2012), 85–102

[13] Leonov A. S., “Extra-optimal methods for solving ill-posed problems”, J. Inverse and III-posed Probl., 20:5–6 (2012), 637–665 | MR | Zbl

[14] Wang Yanfei, Stepanova I. E., Titarenko V. N., Yagola A. G., Inverse problems in geophysics and solution methods, Higher Education Press, Beijing, 2011

[15] Trenogin V. A., Funktsionalnyi analiz, Nauka, M., 1980

[16] Vasilev F. P., Metody optimizatsii, v. 1, 2, Izd-vo MTsNMO, M., 2011

[17] Fiakko A., Mak-Kormik G., Nelineinoe programmirovanie. Metody posledovatelnoi bezuslovnoi minimizatsii, Mir, M., 1972

[18] Khill E., Funktsionalnyi analiz i polugruppy, IL, M., 1951

[19] Tikhonov A. N., Goncharskii A. V., Stepanov V. V., Yagola A. G., Chislennye metody resheniya nekorrektnykh zadach, Nauka, M., 1990 | MR