Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
The aim of this paper is to investigate the asymptotic behavior of an inertial alternating algorithm based on the composition of resolvents of monotone operators. The proposed algorithm is a generalization of those proposed in Attouch et al. (2007) and Bauschke et al. (2005). As a special case, we also recover the classical alternating minimization algorithm (Acker, 1980), which itself is a natural extension of the alternating projection algorithm of von Neumann (1950) [4]. An application to equilibrium problems is also proposed.
ISSN: 0893-9659

hal-00776658
https://hal.univ-antilles.fr/hal-00776658
DOI : 10.1016/j.aml.2010.01.023

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
In the setting of a real Hilbert space H, we investigate the asymptotic behavior, as time t goes to infinity, of trajectories of second-order damped evolution equations, which are governed by the sum of the gradient operator of a convex differentiable potential function , and a maximal monotone operator which is assumed to be cocoercive. Under a sharp condition involving the viscous and the cocoercive parameters, it is proved that each trajectory weakly converges to an equilibrium. Passing from weak to strong convergence is obtained by introducing an asymptotically vanishing Tikhonov-like regularizing term. As special cases, we recover the asymptotic analysis of the heavy ball with friction dynamic attached to a convex potential, the second-order gradient-projection dynamic, and the second-order dynamic governed by the Yosida approximation of a general maximal monotone operator. The breadth and flexibility of the proposed framework is illustrated through applications in the areas of constrained optimization, dynamical approach to Nash equilibria for noncooperative games, and asymptotic stabilization in the case of a continuum of equilibria.
ISSN: 1292-8119

hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774
https://hal.archives-ouvertes.fr/hal-00803774/document
https://hal.archives-ouvertes.fr/hal-00803774/file/ATTOUCH-MAINGE-09-04-2010-COCV.pdf
DOI : 10.1051/cocv/2010024

Éditeur(s) : HAL CCSD
Résumé : The geological storage of CO 2 is considered as one of the solutions to the problem of global warming. The most possible scenarios of geological behavior of soil in the long term conditions has to be precisely defined to avoid catastrophic damages in the future. Chemical properties of saline water in the aquifer after the CO 2 injection becomes more acidic with time. Therefore, chemical degradation of rock occurs, which leads to modifications of its mechanical properties (chemo-mechanical coupling) [1]. In this paper an incremental model of chemo-mechanical coupling using homogenization approach (asymptotic expansion method [2,3]) is proposed. The paper is focused on the numerical computations for the X-ray tomography image of carbonate rock microstructure [4]. The chemical degradation of the microstructure is obtained as a homogeneous dissolution in each direction (Fig.1. – Fig.2.) For every porosity state numerical computations to obtain full, orthotropic elasticity tensor are performed. Therefore, the degradations functions for all components of the rigidity tensor were defined (Fig.3.). Moreover, results of numerical calculations of diffusion, tortuosity and permeability tensors are presented [5,6]. The link between geometrical properties and mechanical and transport parameters are discussed. The numerical analysis of the carbonate rock sample in long term conditions shows the increase of porosity and associated mechanical weakening of the rock. Moreover, the increase of diffusion, tortuosity and permeability tensors elements with time were observed. Finally, the solution of the one-dimensional macroscopic problem of soil settlement due to chemical dissolution of rock at constant load is presented. The method is universal, in the sense that the non-dimensional time scale is used.
ALERT Workshop 2013
Aussois, France

hal-01103175
https://hal.archives-ouvertes.fr/hal-01103175