Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Based on the very recent work by Censor-Gibali-Reich (http://arxiv.org/abs/1009.3780), we propose an extension of their new variational problem (Split Variational Inequality Problem) to monotone variational inclusions. Relying on the Krasnosel'skii-Mann Theorem for averaged operators, we analyze an algorithm for solving new split monotone inclusions under weaker conditions. Our weak convergence results improve and develop previously discussed Split Variational Inequality Problems, feasibility problems and related problems and algorithms.
ISSN: 0022-3239

hal-00776651
https://hal.univ-antilles.fr/hal-00776651
DOI : 10.1007/s10957-011-9814-6

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Let H1,H2,H3 be real Hilbert spaces, let C⊂H1, Q⊂H2 be two nonempty closed convex level sets, let A:H1→H3, B:H2→H3 be two bounded linear operators. Our interest is in solving the following new convex feasibility problem Find x∈C,y∈Q such that Ax=By, which allows asymmetric and partial relations between the variables x and y. In this paper, we present and study the convergence of a relaxed alternating CQ-algorithm (RACQA) and show that the sequences generated by such an algorithm weakly converge to a solution of (1.1). The interest of RACQA is that we just need projections onto half-spaces, thus making the relaxed CQ-algorithm implementable. Note that, by taking B=I, in (1.1), we recover the split convex feasibility problem originally introduced in Censor and Elfving (1994) [13] and used later in intensity-modulated radiation therapy (Censor et al. (2006) [11]). We also recover the relaxed CQ-algorithm introduced by Yang (2004) [8] by particularizing both B and a given parameter.
ISSN: 0362-546X

hal-00776638
https://hal.univ-antilles.fr/hal-00776638
DOI : 10.1016/j.na.2012.11.013

Résumé : Cette thèse est consacrée à la résolution d'un problème fondamental de l'analyse variationnelle qu'est la recherchede zéros d'opérateurs maximaux monotones dans un espace de Hilbert. Nous nous sommes tout d'abord intéressés au cas de l'opérateur somme étendue de deux opérateurs maximaux monotones; la recherche d'un zéro de cet opérateur est un problème dont la bibliographie est peu fournie: nous proposons une version modifiée de l'algorithme d'éclatement forward-backward utilisant à chaque itération, l'epsilon-élargissement d'un opérateur maximal monotone,afin de construire une solution. Nous avons ensuite étudié la convergence d'un nouvel algorithme de faisceaux pour construire ID zéro d'un opérateur maximal monotone quelconque en dimension finie. Cet algorithme fait intervenir une double approximation polyédrale de l'epsilon-élargissement de l'opérateur considéré
This thesis is devoted to solving a basic problem of variational analysis which is the search of zeros of maximal monotone operators in a Hilbert space. First of aIl, we concentrate on the case of the extended som of two maximal monotone operators; the search of a zero of this operator is a problem for which the bibliography is not abondant: we purpose a modified version of the forward-backward splitting algorithm using at each iteration, the epsilon-enlargement of a maximal monotone operator, in order to construet a solution. Secondly, we study the convergence of a new bondie algorithm to construet a zero of an arbitrary maximal monotone operator in a finite dimensional space. In this algorithm, intervenes a double polyhedral approximation of the epsilon-enlargement of the considered operator

http://www.theses.fr/2012AGUY0548/document

Éditeur(s) : HAL CCSD Institute for Scientific Computing and Information
Résumé : International audience
In this paper, we solve a problem describing the mechanical deformations of a porous medium in the presence of a monophasic linear flow or a two phase nonlinear flow with the purpose of modelizing subsidence of hydrocarbon reservoirs. An essential characteristics of this problem is that the mechanical deformation and the flow have different time scales. In petroleum industry, one uses different very efficient simulators for the flow problem and the mechanical deformations, which enables to handle complex models. Therefore it is necessary to be able to combine as efficiently as possible the exploitation of these simulators. We propose two alternative splitting approaches. The first one is the staggered algorithm used by engineers, which amounts to a Gauss-Seidel method in the one phase linear case. The second approach is based upon the preconditioned conjugate gradient method. We use a numerical multi-scale method in both of these algorithms. We compare these two approaches and we show that the preconditioned conjugate gradient algorithm is faster and more robust than the staggered algorithm. Applying the preconditioned conjugate gradient algorithm therefore seems to compensate for the fact that the inf-sup condition for the mixed discretization method is not satisfied when combining the simulators for the mechanical deformations and for the flow computations.
ISSN: 1705-5105

Droits : http://creativecommons.org/licenses/by/
hal-01449039
https://hal.archives-ouvertes.fr/hal-01449039
https://hal.archives-ouvertes.fr/hal-01449039/document
https://hal.archives-ouvertes.fr/hal-01449039/file/DHLE.pdf

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
Algorithms issued from the NonLinear Galerkin method have been used in many situations and with different discretizations for the resolution of evolutionary nonlinear equations. The main idea of these methods is to use a splitting of the solution in order to model the equation. According to the splitting of the solution, a splitting of the equation is obtained. The modeling principle is to freeze terms which have a small time variation. In this work we use wavelet discretizations of the 2-D Burgers equations and compare the results with the hierarchical finite elements method. The numerical tests indicate that wavelets give better results than finite elements
ISSN: 0885-7474

hal-00383423
https://hal.archives-ouvertes.fr/hal-00383423
DOI : 10.1007/s10915-004-4806-4

Éditeur(s) : HAL CCSD IOP Publishing
Résumé : International audience
Based on the very recent work by Censor and Segal (2009 J. Convex Anal. 16 587-600) and inspired by Xu (2006 Inverse Problems 22 2021-34) and Yang (2004 Inverse Problems 20 1261-6), we investigate an algorithm for solving the split common fixed-point problem for the class of demicontractive operators in a Hilbert space. Our results improve and/or develop previously discussed feasibility problems and related algorithms. It is worth mentioning that the convex feasibility formalism is at the core of the modeling of many inverse problems and has been used to model significant real-world problems, for instance, in sensor networks, in radiation therapy treatment planning, in computerized tomography and data compression, see Censor et al (2006 Phys. Med. Biol. 51 2353-65) and Combettes (1996 Adv. Imaging Electron. Phys. 95 155-270) and references therein.
ISSN: 0266-5611

hal-00778174
https://hal.univ-antilles.fr/hal-00778174
DOI : 10.1088/0266-5611/26/5/055007

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
We propose an extended forward-backward algorithm for approximating a zero of a maximal monotone operator which can be split as the extended sum of two maximal monotone operators. We establish the weak convergence in average of the sequence generated by the algorithm under assumptions similar to those used in classical forward-backward algorithms. This provides as a special case an algorithm for solving convex constrained minimization problems without qualification condition.
Journal of Mathematical Analysis and applications

hal-00761657
https://hal.archives-ouvertes.fr/hal-00761657
ARXIV : 1212.0523
DOI : 10.1016/j.jmaa.2013.02.022

Éditeur(s) : HAL CCSD Lavoisier
Résumé : National audience
Dans cette étude, nous présentons, une méthode permettant de découvrir, par le biais de variables agrégées et d’un algorithme génétique, des informations non connues a priori et utiles pour la prédiction de crues. Cette méthode se prête particulièrement bien aux données dans lesquelles des informations à propos d’un même élément sont dispersées à travers plusieurs lignes du jeu de données initial.
In this paper, we present an heuristic based approach for feature selection in the context of flood prediction. Features are complex variables that represent aggregate values. We apply a preprocessing method on data in order to elicit relevant information that could not be easily accessible initially because it is split through several lines of a dataset. A genetic algorithm is used in order to search for the features that may prove the best performances for flood prediction.
ISSN: 1633-1311

hal-01184510
https://hal.archives-ouvertes.fr/hal-01184510

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Two splitting procedures for solving equilibrium problems involving the sum of two bifunctions are proposed and their convergence is established under mild assumptions.
Journal of Mathematical Analysis and applications

hal-00779982
https://hal.univ-antilles.fr/hal-00779982
DOI : 10.1016/j.jmaa.2009.06.005

Éditeur(s) : HAL CCSD
Résumé : International audience
To solve general variational inequalities considered in Robinson [9], we propose a generalized version of an algorithm introduced by Bermudez and Moreno [3]. Our results extend, improve and develop some known results in this field.
Advanced Modeling and Optimization

hal-00783892
https://hal.univ-antilles.fr/hal-00783892

Éditeur(s) : HAL CCSD
Résumé : The convex feasibility problem (CFP) is to find a member of the intersection of finitely many closed convex sets in Euclidean space. When the intersection is empty, one can minimize a proximity function to obtain an approximate solution to the problem. The split feasibility problem (SFP) and the split equality problem (SEP) are generalizations of the CFP. The approximate SFP (ASFP) and approximate SEP (ASEP) involve finding only approximate solutions to the SFP and SEP, respectively. We present here the SSEA, a simultaneous iterative algorithm for solving the ASEP. When this algorithm is applied to the ASFP it resembles closely, but is not equivalent to, the CQ algorithm. The SSEA involves orthogonal projection onto the given closed convex sets. The relaxed SSEA (RSSEA) is an easily implementable variant of the SSEA that uses orthogonal projection onto half-spaces at each step to solve the SEP. The perturbed version of the SSEA (PSSEA) is similar to the RSSEA, but uses orthogonal projection onto a sequence of epi-convergent closed convex sets.
https://hal.univ-antilles.fr/hal-00776640

Droits : info:eu-repo/semantics/OpenAccess
hal-00776640
https://hal.univ-antilles.fr/hal-00776640
https://hal.univ-antilles.fr/hal-00776640/document
https://hal.univ-antilles.fr/hal-00776640/file/CBAM10.pdf

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Relying on the Yosida approximate, we investigate the problem of finding zeroes for a difference of two maximal monotone operators in Hilbert spaces. These zeroes are compared to those of the corresponding regularized and dual problems. The behavior of the regularized operator is also studied and a splitting algorithm involving the resolvents of both operators is suggested via a fixed-point formulation of the regularized problem. A particular attention is given to the DC programming case.
ISSN: 0096-3003

hal-00779986
https://hal.univ-antilles.fr/hal-00779986
DOI : 10.1016/j.amc.2008.01.024

Éditeur(s) : HAL CCSD Lavoisier
Résumé : International audience
In this paper, we present an heuristic based approach for feature selection in the context of flood prediction. Features are complex variables that represent aggregate values. We apply a preprocessing method on data in order to elicit relevant information that could not be easily accessible initially because it is split through several lines of a dataset. A genetic algorithm is used in order to search for the features that may prove the best performances for flood prediction.
ISSN: 1633-1311

hal-00840738
https://hal.archives-ouvertes.fr/hal-00840738
DOI : 10.3166/isi.16.3.113-139

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
Based on the very recent work by Censor and Segal (2009), and inspired by Xu (2006), Zhao and Yang (2005) , and Bauschke and Combettes (2001), we introduce and analyze an algorithm for solving the split common fixed-point problem for the wide class of quasi-nonexpansive operators in Hilbert spaces. Our results improve and develop previously discussed feasibility problems and related algorithms.
ISSN: 0362-546X

hal-00776645
https://hal.univ-antilles.fr/hal-00776645
DOI : 10.1016/j.na.2011.03.041

Éditeur(s) : HAL CCSD
Résumé : In this work, we develop a variant of a bundle method in order to find a zero of a maximal monotone operator. This algorithm relies on two polyhedral approximations of the epsilon-enlargement of the considered operator, via a systematic use of the transportation formula. Moreover, the use of a double polyhedral approximation in our algorithm could inspire other bundle methods for the case where the given operator can be split as the sum of two other maximal monotone operators.
https://hal.archives-ouvertes.fr/hal-00828930

hal-00828930
https://hal.archives-ouvertes.fr/hal-00828930
ARXIV : 1305.5810

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
A forward-backward inertial procedure for solving the problem of finding a zero of the sum of two maximal monotone operators is proposed and its convergence is established under a cocoercivity condition with respect to the solution set.
ISSN: 0377-0427

hal-00780778
https://hal.univ-antilles.fr/hal-00780778
https://hal.univ-antilles.fr/hal-00780778/document
https://hal.univ-antilles.fr/hal-00780778/file/10.1.1.12.4189.pdf
DOI : 10.1016/S0377-0427(02)00906-8