[Vers la régularisation de Tikhonov pour des problèmes non linéaires mal posés : une approche en optimisation dc]
La méthode de régularisation de Tikhonov pour les problèmes non linéaires mal posés requiert une solution optimale globale des problèmes d'optimisation non convexe qui ont été très peu étudiés dans la communauté des problèmes inverses. Dans ce papier nous suggérons une méthode qui est applicable à une large classe des problèmes non linéaires mal posés. C'est une classe de problèmes dans lesquels la fonctionnelle de Tikhonov peut être représentée comme différences de fonctionnelles convexes (dc). Notre méthode pour ces problèmes est une combinaison de l'algorithme DCA, récemment développé en optimisation dc, et les techniques de séparation et évaluation.
The Tikhonov regularization method for non-linear ill-posed problems requires us to globally solve non-convex optimization problem which have been very little studied in the inverse problems community. In this paper we suggest a method which is applicable to the Tikhonov method for a wide class of non-linear ill-posed problems. This is a class of problems when the Tikhonov functional for them can be represented by the difference of two convex functionals. Our method for these problems is a combination of the recently developed algorithm DCA in dc programming with the branch-and-bound techniques.
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@article{CRMATH_2002__335_12_1073_0,
author = {Le Thi Hoai An and Pham Dinh Tao and Dinh Nho H\`ao},
title = {Towards {Tikhonov} regularization of non-linear ill-posed problems: a dc programming approach},
journal = {Comptes Rendus. Math\'ematique},
pages = {1073--1078},
publisher = {Elsevier},
volume = {335},
number = {12},
year = {2002},
doi = {10.1016/S1631-073X(02)02611-0},
language = {en},
url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02611-0/}
}
TY - JOUR AU - Le Thi Hoai An AU - Pham Dinh Tao AU - Dinh Nho Hào TI - Towards Tikhonov regularization of non-linear ill-posed problems: a dc programming approach JO - Comptes Rendus. Mathématique PY - 2002 SP - 1073 EP - 1078 VL - 335 IS - 12 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02611-0/ DO - 10.1016/S1631-073X(02)02611-0 LA - en ID - CRMATH_2002__335_12_1073_0 ER -
%0 Journal Article %A Le Thi Hoai An %A Pham Dinh Tao %A Dinh Nho Hào %T Towards Tikhonov regularization of non-linear ill-posed problems: a dc programming approach %J Comptes Rendus. Mathématique %D 2002 %P 1073-1078 %V 335 %N 12 %I Elsevier %U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02611-0/ %R 10.1016/S1631-073X(02)02611-0 %G en %F CRMATH_2002__335_12_1073_0
Le Thi Hoai An; Pham Dinh Tao; Dinh Nho Hào. Towards Tikhonov regularization of non-linear ill-posed problems: a dc programming approach. Comptes Rendus. Mathématique, Tome 335 (2002) no. 12, pp. 1073-1078. doi : 10.1016/S1631-073X(02)02611-0. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02611-0/
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