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2013 | 11 | 8 | 1489-1509
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We propose an adaptive finite element method for the solution of a linear Fredholm integral equation of the first kind. We derive a posteriori error estimates in the functional to be minimized and in the regularized solution to this functional, and formulate corresponding adaptive algorithms. To do this we specify nonlinear results obtained earlier for the case of a linear bounded operator. Numerical experiments justify the efficiency of our a posteriori estimates applied both to the computationally simulated and experimental backscattered data measured in microtomography.
Wydawca
Czasopismo
Rocznik
Tom
11
Numer
8
Strony
1489-1509
Opis fizyczny
Daty
wydano
2013-08-01
online
2013-05-22
Twórcy
Bibliografia
  • [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
  • [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] 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
  • [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
  • [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] 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
  • [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
  • [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
  • [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
  • [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] 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
  • [12] Goncharsky A.V., Cherepashchuk A.M., Yagola A.G., Ill-Posed Problems of Astrophysics, Moscow, Nauka, 1985 (in Russian)
  • [13] Groetsch C.W., Inverse Problems in the Mathematical Sciences, Vieweg Math. Sci. Engrs., Friedr. Vieweg & Sohn, Braunschweig, 1993
  • [14] Johnson C., Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover, Mineola, 2009
  • [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
  • [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
  • [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] 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] Kress R., Linear Integral Equations, Appl. Math. Sci., 82, Springer, Berlin, 1989
  • [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] 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
  • [22] Tikhonov A.N., Leonov A.S., Yagola A.G., Nonlinear Ill-Posed Problems, Appl. Math. Math. Comput., 14, Chapman & Hall, London, 1998
  • [23] Yagola A.G., Koshev N.A., Restoration of smeared and defocused color images, Numerical Methods and Programming, 2008, 9, 207–212 (in Russian)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-013-0247-3
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