Comptes Rendus
no. 7
Probability Theory
Consistency of Landweber algorithm in an ill-posed problem with random data
[Consistance de l'algorithme de Landweber pour un problème mal posé avec des erreurs aléatoires]

Présenté par : Paul Deheuvels

Abdelnasser Dahmani1 ; Fatah Bouhmila1
1 Department of Mathematics, Laboratory of Applied Mathematics, University of Bejaia, 06000 Bejaia, Algeria
Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 487-491.

Dans cette Note, nous considérons un problème mal posé linéaire décrit par une équation à opérateur où le second membre est mesuré avec des erreurs aléatoires. Nous montrons l'existence et l'unicité de la pseudo-solution du problème puis nous l'estimons en utilisant l'algorithme de Landweber. Par ailleurs, nous montrons la convergence presque complète (p.co) de celui-ci tout en précisant la vitesse de convergence et nous construisons un domaine de confiance pour ladite pseudo-solution.

This Note deals with the linear ill-posed problem, described by operator equations in which the second member is measured with random errors. We first show the existence and the unicity of the pseudo-solution for such a problem and later estimate it using Landweber algorithm. We also show the ‘almost complete convergence’ (a.co) of this algorithm specifying its convergence rate. We finally build a confidence domain for the so mentioned pseudo-solution.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2006.09.010
Abdelnasser Dahmani 1 ; Fatah Bouhmila 1

1 Department of Mathematics, Laboratory of Applied Mathematics, University of Bejaia, 06000 Bejaia, Algeria 
@article{CRMATH_2006__343_7_487_0,
     author = {Abdelnasser Dahmani and Fatah Bouhmila},
     title = {Consistency of {Landweber} algorithm in an ill-posed problem with random data},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {487--491},
     publisher = {Elsevier},
     volume = {343},
     number = {7},
     year = {2006},
     doi = {10.1016/j.crma.2006.09.010},
     language = {en},
}
TY  - JOUR
AU  - Abdelnasser Dahmani
AU  - Fatah Bouhmila
TI  - Consistency of Landweber algorithm in an ill-posed problem with random data
JO  - Comptes Rendus. Mathématique
PY  - 2006
SP  - 487
EP  - 491
VL  - 343
IS  - 7
PB  - Elsevier
DO  - 10.1016/j.crma.2006.09.010
LA  - en
ID  - CRMATH_2006__343_7_487_0
ER  - 
%0 Journal Article
%A Abdelnasser Dahmani
%A Fatah Bouhmila
%T Consistency of Landweber algorithm in an ill-posed problem with random data
%J Comptes Rendus. Mathématique
%D 2006
%P 487-491
%V 343
%N 7
%I Elsevier
%R 10.1016/j.crma.2006.09.010
%G en
%F CRMATH_2006__343_7_487_0
Abdelnasser Dahmani; Fatah Bouhmila. Consistency of Landweber algorithm in an ill-posed problem with random data. Comptes Rendus. Mathématique, Volume 343 (2006) no. 7, pp. 487-491. doi : 10.1016/j.crma.2006.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.09.010/

[1] R. Arcangeli Pseudo-solution de l'équation Ax=y, C. R. Acad. Sci. Paris, Ser. A, Volume 263 (1966), pp. 282-285

[2] N. Bissantz; T. Hohage; A. Munk Consistency and rates of convergence of non linear Tikhonov regularization with random noise, Inverse Problems, Volume 20 (2004), pp. 1773-1789

[3] H. Cardot Spatially adaptive splines for statistical linear inverse problems, J. Multivariate Anal., Volume 81 (2002), pp. 100-119

[4] H.W. Engl; M. Hanke; A. Neubauer Regularization of Inverse Problems, Kluwer, Dordrecht, 1996

[5] M. Hanke; A. Neubauer; O. Scherzer A convergence analysis of the Landweber iteration for nonlinear ill-posed problems, Numer. Math., Volume 72 (1995), pp. 21-73

[6] V.K. Ivanov Sur les problèmes mal posés linéaires, Rapports de l'Académie des Sciences de l'URSS, Volume 145 (1962) no. 2, pp. 270-272

[7] J. Kaipio; E. Somersalo Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005

[8] L. Landweber An iteration formula for Fredholm integral equations of the first kind, Amer. J. Math., Volume 73 (1951), pp. 615-624

[9] V.A. Morozov Sur les pseudo-solutions, JVM et MPH, Volume 9 (1969) no. 6, pp. 1381-1392

[10] A.G. Ramm Inverse Problems Mathematical and Analytical Techniques with Applications to Engineering, Springer, 2005

[11] C. Takiya; O. Helene; E.Do. Nascimento; V.R. Vanin Minimum variance regularization in linear inverse problems, Nucl. Instrum. Methods Phys. Res., Sect. A, Volume 523 (2004), pp. 186-192

[12] A. Tarantola Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, 2005

[13] U. Tautenhahn On the method of Lavrentiev regularization for nonlinear ill-posed problems, Inverse Problems, Volume 18 (2002) no. 1, pp. 191-207

[14] A.N. Tikhonov; V.Y. Arsenin Solution for Ill-Posed Problems, Wiley, New York, 1977

[15] V.V. Yurinskii Exponential inequalities for sums of random vectors, J. Multivariate Anal., Volume 6 (1976), pp. 473-499

Cité par Sources :

Commentaires - Politique