An adaptive finite element method for Fredholm integral equations of the first kind and its verification on experimental data

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1. [1] Asadzadeh M., Eriksson K., On adaptive finite element methods for Fredholm integral equations of the second kind, SIAM J. Numer. Anal., 1994, 31(3), 831–855 http://dx.doi.org/10.1137/0731045 Zbl0814.65134
2. [2] Atkinson K.E., The numerical solution of integral equations of the second kind, Cambridge Monogr. Appl. Comput. Math., 4, Cambridge University Press, Cambridge, 1997 
3. [3] Bakushinsky A.B., A posteriori error estimates for approximate solutions of irregular operator equations, Dokl. Math., 2011, 83(2), 192–193 http://dx.doi.org/10.1134/S1064562411020190 Zbl1252.65096
4. [4] Bakushinsky A.B., Kokurin M.Yu., Smirnova A., Iterative Methods for Ill-Posed Problems, Inverse Ill-Posed Probl. Ser., 54, Walter de Gruyter, Berlin, 2011 Zbl1215.47013
5. [5] Basistov Yu.A., Goncharsky A.V., Lekht E.E., Cherepashchuk A.M., Yagola A.G., Application of the regularization method for increasing of the radiotelescope resolution power, Astronomicheskii Zhurnal, 1979, 56(2), 443–449 (in Russian) 
6. [6] Beilina L., Klibanov M.V., A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem, Inverse Problems, 2010, 26(4), #045012 http://dx.doi.org/10.1088/0266-5611/26/4/045012 Zbl1193.65165
7. [7] Beilina L., Klibanov M.V., Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inverse Problems, 2010, 26(12), #125009 http://dx.doi.org/10.1088/0266-5611/26/12/125009 Zbl1209.78010
8. [8] Beilina L., Klibanov M.V., Kokurin M.Yu., Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem, J. Math. Sci. (N.Y.), 2010, 167(3), 279–325 http://dx.doi.org/10.1007/s10958-010-9921-1 Zbl1286.65147
9. [9] Beilina L., Klibanov M.V., Kuzhuget A., New a posteriori error estimates for adaptivity technique and global convergence for the hyperbolic coefficient inverse problem, J. Math. Sci. (N.Y.), 2011, 172(4), 449–476 http://dx.doi.org/10.1007/s10958-011-0203-3 Zbl1219.35350
10. [10] Bolotina A.V., Luk’yanov F.A., Rau E.I., Sennov R.A., Yagola A.G., Energy spectra of electrons backscattered from bulk solid targets, Moscow University Physics Bulletin, 2009, 64(5), 503–506 http://dx.doi.org/10.3103/S0027134909050075 
11. [11] Eriksson K., Estep D., Johnson C., Applied Mathematics: Body and Soul, 3, Springer, Berlin, 2004 http://dx.doi.org/10.1007/978-3-662-05796-4 Zbl1055.00005
12. [12] Goncharsky A.V., Cherepashchuk A.M., Yagola A.G., Ill-Posed Problems of Astrophysics, Moscow, Nauka, 1985 (in Russian) 
13. [13] Groetsch C.W., Inverse Problems in the Mathematical Sciences, Vieweg Math. Sci. Engrs., Friedr. Vieweg & Sohn, Braunschweig, 1993 Zbl0779.45001
14. [14] Johnson C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, Mineola, 2009 Zbl1191.65140
15. [15] Johnson C., Szepessy A., Adaptive finite element methods for conservation laws based on a posteriori error estimates, Comm. Pure Appl. Math., 1995, 48(3), 199–234 http://dx.doi.org/10.1002/cpa.3160480302 Zbl0826.65088
16. [16] Klibanov M.V., Bakushinsky A.B., Beilina L., Why a minimizer of the Tikhonov functional is closer to the exact solution than the first guess, J. Inverse Ill-Posed Probl., 2011, 19(1), 83–105 http://dx.doi.org/10.1515/jiip.2011.024 Zbl1279.35113
17. [17] Koshev N.A., Luk’yanov F.A., Rau E.I., Sennov R.A., Yagola A.G., Increasing spatial resolution in the backscattered electron mode of scanning electron microscopy, Bulletin of the Russian Academy of Sciences: Physics, 2011, 75(9), 1181–1184 http://dx.doi.org/10.3103/S1062873811090139 
18. [18] Koshev N.A., Orlikovsky N.A., Rau E.I., Yagola A.G., Solution of the inverse problem of restoring the signals from an electronic microscope in the backscattered electron mode on the class of bounded variation functions, Numerical Methods and Programming, 2011, 12, 362–367 (in Russian) 
19. [19] Kress R., Linear Integral Equations, Appl. Math. Sci., 82, Springer, Berlin, 1989 
20. [20] Tikhonov A.N., Goncharsky A.V., Stepanov V.V., Kochikov I.V., Ill-posed problems of image processing, Soviet Phys. Dokl., 1987, 32(6), 456–458 
21. [21] Tikhonov A.N., Goncharsky A.V., Stepanov V.V., Yagola A.G., Numerical Methods for the Solution of Ill-Posed Problems, Math. Appl., 328, Kluwer, Dordrecht, 1995 Zbl0831.65059
22. [22] Tikhonov A.N., Leonov A.S., Yagola A.G., Nonlinear Ill-Posed Problems, Appl. Math. Math. Comput., 14, Chapman & Hall, London, 1998 Zbl0920.65038
23. [23] Yagola A.G., Koshev N.A., Restoration of smeared and defocused color images, Numerical Methods and Programming, 2008, 9, 207–212 (in Russian)