Geodesic
An iterative regularization method for a class of inverse boundary value problems of elliptic type
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 1, pp. 66-85 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper deals with the problem of determining an unknown source and an unknown boundary condition $u(0)$ in a boundary value problem of elliptic type from extra measurements at internal points. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. For solving the considered problem an iterative method is proposed. Using this method a regularized solution is constructed and an a priori error estimate between the exact solution and its regularization approximation is obtained. Moreover, the numerical results are presented to illustrate the accuracy and efficiency of this method. 
@article{JMAG_2020_16_1_a4,
     author = {Fairouz Zouyed and Souheyla Debbouche},
     title = {An iterative regularization method for a class of inverse boundary value problems of elliptic type},
     journal = {\v{Z}urnal matemati\v{c}eskoj fiziki, analiza, geometrii},
     pages = {66--85},
     year = {2020},
     volume = {16},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JMAG_2020_16_1_a4/}
}
TY  - JOUR
AU  - Fairouz Zouyed
AU  - Souheyla Debbouche
TI  - An iterative regularization method for a class of inverse boundary value problems of elliptic type
JO  - Žurnal matematičeskoj fiziki, analiza, geometrii
PY  - 2020
SP  - 66
EP  - 85
VL  - 16
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/JMAG_2020_16_1_a4/
LA  - en
ID  - JMAG_2020_16_1_a4
ER  - 
%0 Journal Article
%A Fairouz Zouyed
%A Souheyla Debbouche
%T An iterative regularization method for a class of inverse boundary value problems of elliptic type
%J Žurnal matematičeskoj fiziki, analiza, geometrii
%D 2020
%P 66-85
%V 16
%N 1
%U http://geodesic.mathdoc.fr/item/JMAG_2020_16_1_a4/
%G en
%F JMAG_2020_16_1_a4
Fairouz Zouyed; Souheyla Debbouche. An iterative regularization method for a class of inverse boundary value problems of elliptic type. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 16 (2020) no. 1, pp. 66-85. http://geodesic.mathdoc.fr/item/JMAG_2020_16_1_a4/

[1] A.B. Bakushinsky, M.Y. Kokurin, Iterative Methods for Approximate Solution of Inverse Problems, Mathematics and Its Applications, 577, Springer, Berlin, Germany, 2004 | MR | Zbl

[2] G. Bastay, Iterative Methods for Ill-posed Boundary Value Problems, Linköping Studies in Science and Technology, Dissertations, 392, Linköping Univ., Linköping, 1995 | MR

[3] J. Baumeister, A. Leitao, “On iterative methods for solving ill-posed problems modeled by partial differential equations”, J. Inverse Ill-Posed Probl., 9:1 (2001), 13–29 | DOI | MR | Zbl

[4] F. Berntsson, V.A. Kozlov, L. Mpinganzima, B.O. Turesson, “An alternating iterative procedure for the Cauchy problem for the Helmholtz equation”, Inverse Probl. Sci. Eng., 22:1 (2014), 45–62 | DOI | MR | Zbl

[5] A. Bouzitouna, N. Boussetila, F. Rebbani, “Two regularization methods for a class of inverse boundary value problems of elliptic type”, Bound. Value Probl., 2013 (2013), 178 | DOI | MR | Zbl

[6] P. Deuflhard, H.W. Engl, O. Scherzer, “A convergence analysis of iterative methods for the solution of nonlinear ill-posed problems under a nely invariant conditions”, Inverse Problems, 14 (1998), 1081–1106 | DOI | MR | Zbl

[7] R. Gorenflo, “Funktionentheoretische Bestimmung des Aussenfeldes zu einer zwei- dimensionalen magnetohydrostatischen Konfiguration”, Z. Angew. Math. Phys., 16:2 (1965), 279–290 | DOI | MR | Zbl

[8] J.A. Goldstein, Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985 | MR | Zbl

[9] C.R. Johnson, “Computational and numerical methods for bioelectric field problems”, Critical Reviews in Biomedical Engineering, 25 (1997), 181 | DOI

[10] V.A. Kozlov, V.G. Maz'ya, “Iterative procedures for solving ill-posed boundary value problems that preserve the differential equations”, Leningrad Math. J., 1 (1990), 1207–1228 | MR | Zbl

[11] V.A. Kozlov, V.G. Maz'ya, A.V. Fomin, “An iterative method for solving the Cauchy problem for elliptic equations”, U.S.S.R. Comput. Math. Math. Phys., 31 (1991), 45–52 | MR | Zbl

[12] M.A. Krasnosel'skii, Ya.B. Rutitskii, V. Ya. Stetsenko, G.M. Vainikko, P.P. Zabreiko, Approximate Solutions of Operator Equations, Wolters-Noordhoff Publishing, Groningen, 1972 | MR

[13] M.M. Lavrentev, V.G. Romanov, S.P. Shishatskii, Ill-Posed Problems of Mathematical Physics and Analysis, Translations of Mathematical Monographs, 64, Amer. Math. Soc., Providence, RI, USA, 1986 | DOI | MR | Zbl

[14] A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, 1983 | MR

[15] A.P.S. Selvadurai, Partial differential equation in Mechanics, v. 2, The biharmonic equation, Poisson's equation, Springer-Verlag, Heidelberg, 2000 | MR

[16] H.W. Zhang, T. Wei, “Two iterative methods for a Cauchy problem of the elliptic equation with variable coeffcients in a strip region”, Numer. Algor., 65 (2014), 875–892 | DOI | MR | Zbl

[17] F. Zouyed, S. Djemoui, “An iterative regularization method for identifying the source term in a second order differential equation”, Math. Probl. Eng., 2015 (2015), 713403 | DOI | MR | Zbl