Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We present iterative methods for finding the critical points and/or the minima of extended real valued functions of the form $phi = psi+ g-h$, where $psi$ is a differentiable function and g and h are convex, proper, and lower semicontinuous. The underlying idea relies upon the discretization of a first order dissipative dynamical system which allows us to preserve the local feature and to obtain some convergence results. The main theorems not only recover known convergence results in this field but also provide a theoretical basis for the development of new iterative methods.
ISSN: 1052-6234

hal-00779987
https://hal.univ-antilles.fr/hal-00779987
DOI : 10.1137/060655183

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
ISSN: 0363-0129

hal-01147931
https://hal.archives-ouvertes.fr/hal-01147931

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We discuss the control of distributed systems with incomplete data following the notion of no-regret control (or, equivalently, Pareto control) used by Lions in [C. R. Acad. Sci. Paris Ser. I Math., 302 (1986), pp. 223-227] and [C. R. Acad. Sci. Paris Ser. I Math., 302 (1992), pp. 1253-1257]. We associate with the no-regret control a sequence of low-regret controls defined by a quadratic perturbation previously used by Nakoulima, Omrane, and Velin in [C. R. Acad. Sci. Paris Ser. I Math., 330 (2000), pp. 801-806]. In the first part, we prove that the perturbed system corresponds to a sequence of standard control problems and converges to the no-regret (or Pareto) control for which we obtain a singular optimality system. We give also some applications. In the second part, we show how the method can be extended to the evolution case. Equations of parabolic type, Petrowsky type, or hyperbolic type are considered.
ISSN: 0363-0129

hal-00764259
https://hal.univ-antilles.fr/hal-00764259
https://hal.univ-antilles.fr/hal-00764259/document
https://hal.univ-antilles.fr/hal-00764259/file/38018.pdf
DOI : 10.1137/S0363012900380188

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
We establish a general Fermat rule for the problem of minimizing a lower semicontinuous function on a convex subset of a Banach space. Our basic tool is a constrained variational principle derived from the "smooth" variational principle of Borwein and Preiss. Specializing the Fermat rule to the case when the convex set is a "drop," we obtain a multidirectional Rolle-type inequality from which, in turn, we deduce a multidirectional mean value inequality, in the line of Clarke and Ledyaev. We follow the abstract approach of our previous paper [Trans. Amer. Math. Soc., 347 (1995), pp. 4147-4161], thus covering all standard situations met in applications, while stressing the links between the results and the few key properties that are needed.
ISSN: 1052-6234

hal-00699204
https://hal.archives-ouvertes.fr/hal-00699204

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
ISSN: 0363-0129

hal-00577233
https://hal.archives-ouvertes.fr/hal-00577233

Éditeur(s) : HAL CCSD Society for Industrial and Applied Mathematics
Résumé : International audience
In this paper, we developed a theoretical study for nonconservative sytems in one dimension in order to construct numerical schemes for solving the Riemann problem. The nonconservative form of our model system required the use of a well-adapted theory in order to give us a sense of our problem. We chose a framework of generalized functions for solving a scalar hyperbolic equation with a discontinuous coefficient $sigma_t +usigma_x approx 0$, where u is the velocity solution of a Burgers's equation. After an explicit solution of the Riemann problem, we derived Godunov split schemes for computing an approximate solution of the Cauchy problem. We applied our study to a system modeling elasticity and a system modeling gas dynamics. Some stability properties of a scheme and its convergence to a generalized solution are proved for the first model. Numerical experiments confirmed this convergence result. For the second model, calculations of flows containing weak-to-moderate shocks showed that conservation errors are reduced when the mesh is refined but were not entirely eliminated.
ISSN: 0036-1429

hal-00770266
https://hal.univ-antilles.fr/hal-00770266
DOI : 10.1137/S0036142900378637