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

Dahmani, Abdelnasser1 ; Bouhmila, Fatah1
1 Department of Mathematics, Laboratory of Applied Mathematics, University of Bejaia, 06000 Bejaia, Algeria
Comptes Rendus. Mathématique, Tome 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
Dahmani, Abdelnasser 1 ; Bouhmila, Fatah 1

1 Department of Mathematics, Laboratory of Applied Mathematics, University of Bejaia, 06000 Bejaia, Algeria 
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Dahmani, Abdelnasser; Bouhmila, Fatah. Consistency of Landweber algorithm in an ill-posed problem with random data. Comptes Rendus. Mathématique, Tome 343 (2006) no. 7, pp. 487-491. doi : 10.1016/j.crma.2006.09.010. http://www.numdam.org/articles/10.1016/j.crma.2006.09.010/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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