Results 1  10
of
42
NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two differ ..."
Abstract

Cited by 93 (22 self)
 Add to MetaCart
. We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a noncompact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...
On The Minimizing Property Of A Second Order Dissipative System In Hilbert Spaces
 SIAM J. CONTROL AND OPTIMIZATION
, 1998
"... We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below then the solution trajectories are minimizing ..."
Abstract

Cited by 39 (4 self)
 Add to MetaCart
(Show Context)
We study the asymptotic behavior at infinity of solutions of a second order evolution equation with linear damping and convex potential. The differential system is defined in a real Hilbert space. It is proved that if the potential is bounded from below then the solution trajectories are minimizing for it, and converge weakly towards a minimizer of Phi if there exists one; this convergence is strong when Phi is even or when the optimal set has a nonempty interior. We introduce a second order proximallike iterative algorithm for the minimization of a convex function. It is defined by an implicit discretization of the continuous evolution problem and is valid for any closed proper convex function. We find conditions on some parameters of the algorithm in order to have a convergence result similar to the continuous case.
Characterizations of Lojasiewicz inequalities: subgradient flows, talweg, convexity.
"... Abstract The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to ominimal structures (Kurdyka) have a considerable impact on the analysis of gradientlike methods and related problems: minimization methods, complexity theory, asymptotic anal ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
Abstract The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to ominimal structures (Kurdyka) have a considerable impact on the analysis of gradientlike methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In the framework of metric spaces, we show that a generalized form of the Lojasiewicz inequality (hereby called the KurdykaLojasiewicz inequality) is related to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by −∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the KurdykaLojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines —a concept linked to the location of the less steepest points at the level sets of f— and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish a kind of asymptotic equivalence for discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the KurdykaLojasiewicz inequality. Key words Lojasiewicz inequality, gradient inequalities, metric regularity, subgradient curve, talweg, gradient method, convex functions, global convergence, proximal method.
Asymptotic behaviour of selfcontracted planar curves and gradient orbits of convex functions
 J. Math. Pures Appl
"... Abstract. We hereby introduce and study the notion of selfcontracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded selfcontracted planar curves have a finite length. We also give an example of a convex function defined ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
Abstract. We hereby introduce and study the notion of selfcontracted curves, which encompasses orbits of gradient systems of convex and quasiconvex functions. Our main result shows that bounded selfcontracted planar curves have a finite length. We also give an example of a convex function defined in the plane whose gradient orbits spiral infinitely many times around the unique minimum of the function.
A Parabolic Quasilinear Problem for Linear Growth Functionals
"... We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
(Show Context)
We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f(x; ) = p 1 + kk 2 , which corresponds with the timedependent minimal surface equation. We also study the asymptotic behavoiur of the solutions.
HESSIAN RIEMANNIAN GRADIENT FLOWS IN CONVEX PROGRAMMING ∗
"... Abstract. In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have ..."
Abstract

Cited by 12 (0 self)
 Add to MetaCart
(Show Context)
Abstract. In view of solving theoretically constrained minimization problems, we investigate the properties of the gradient flows with respect to Hessian Riemannian metrics induced by Legendre functions. The first result characterizes Hessian Riemannian structures on convex sets as metrics that have a specific integration property with respect to variational inequalities, giving a new motivation for the introduction of Bregmantype distances. Then, the general evolution problem is introduced, and global convergence is established under quasiconvexity conditions, with interesting refinements in the case of convex minimization. Some explicit examples of these gradient flows are discussed. Dual trajectories are identified, and sufficient conditions for dual convergence are examined for a convex program with positivity and equality constraints. Some convergence rate results are established. In the case of a linear objective function, several optimality characterizations of the orbits are given: optimal path of viscosity methods, continuoustime model of Bregmantype proximal algorithms, geodesics for some adequate metrics, and projections of ˙qtrajectories of some Lagrange equations and completely integrable Hamiltonian systems.
A secondorder gradientlike dissipative dynamical system with hessiandriven damping. application to optimization and mechanics
 J. Math. Pures Appl
"... Given H a real Hilbert space and Φ: H → R a smooth C 2 function, we study the dynamical system (DIN) ¨x(t) + α ˙x(t) + β ∇ 2 Φ(x(t)) ˙x(t) + ∇Φ(x(t)) = 0 where α and β are positive parameters. The inertial term ¨x(t) acts as a singular perturbation and, in fact, regularization of the possibly degen ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
(Show Context)
Given H a real Hilbert space and Φ: H → R a smooth C 2 function, we study the dynamical system (DIN) ¨x(t) + α ˙x(t) + β ∇ 2 Φ(x(t)) ˙x(t) + ∇Φ(x(t)) = 0 where α and β are positive parameters. The inertial term ¨x(t) acts as a singular perturbation and, in fact, regularization of the possibly degenerate classical Newton continuous dynamical system ∇ 2 Φ(x(t)) ˙x(t)+∇Φ(x(t)) = 0. We show that (DIN) is a wellposed dynamical system. Due to their dissipative aspect, trajectories of (DIN) enjoy remarkable optimization properties. For example, when Φ is convex and argmin Φ = ∅, then each trajectory of (DIN) weakly converges to a minimizer of Φ. If Φ is real analytic, then each trajectory converges to a critical point of Φ. A remarkable feature of (DIN) is that one can produce an equivalent system which is firstorder in time and with no occurrence of the Hessian, namely ˙x(t) + c∇Φ(x(t)) + ax(t) + by(t) = 0 ˙y(t) + ax(t) + by(t) = 0 where a, b, c are parameters which can be explicitly expressed in terms of α and β. This allows to consider (DIN) when Φ is C 1 only, or more generally, nonsmooth or subject to constraints. This is first illustrated by a gradient projection dynamical system exhibiting both viable trajectories, inertial aspects, optimization properties, and secondly by a mechanical system with impact.
ASYMPTOTIC BEHAVIOR OF COUPLED DYNAMICAL SYSTEMS WITH MULTISCALE ASPECTS
, 904
"... Abstract. We study the asymptotic behaviour, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, Φ: H → R ∪ {+∞} and Ψ: H → R ∪ {+∞} two closed convex functions, and β a function of t which tends to + ..."
Abstract

Cited by 10 (4 self)
 Add to MetaCart
(Show Context)
Abstract. We study the asymptotic behaviour, as time variable t goes to +∞, of nonautonomous dynamical systems involving multiscale features. As a benchmark case, given H a general Hilbert space, Φ: H → R ∪ {+∞} and Ψ: H → R ∪ {+∞} two closed convex functions, and β a function of t which tends to + ∞ as t goes to +∞, we consider the differential inclusion ˙x(t) + ∂Φ(x(t)) + β(t)∂Ψ(x(t)) ∋ 0. This system models the emergence of various collective behaviors in game theory, as well as the asymptotic control of coupled systems. We show several results ranging from weak ergodic to strong convergence of the trajectories. As a key ingredient we assume that, for every p belonging to the range of NC β(t)
A CONTINUOUS DYNAMICAL NEWTONLIKE APPROACH TO SOLVING MONOTONE INCLUSIONS
, 2010
"... We introduce nonautonomous continuous dynamical systems which are linked to Newton and LevenbergMarquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we sh ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
We introduce nonautonomous continuous dynamical systems which are linked to Newton and LevenbergMarquardt methods. They aim at solving inclusions governed by maximal monotone operators in Hilbert spaces. Relying on Minty representation of maximal monotone operators as lipschitzian manifolds, we show that these dynamics can be formulated as firstorder in time differential systems, which are relevant to CauchyLipschitz theorem. By using Lyapunov asymptotical analysis, we prove that their trajectories converge weakly to equilibria. Time discretization of these dynamics gives algorithms providing new insight on Newton’s method for solving monotone inclusions.
An Asymptotical Variational Principle Associated with the Steepest Descent Method for a Convex Function
, 1996
"... Introduction Let X be a real Hilbert space endowed with inner product h:; :i and associated norm k:k, and let f be a proper closed convex function on X. The paper considers the problem of minimizing f , that is, of finding inf X f and some element in the optimal set S := Argminf , this set assume ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
Introduction Let X be a real Hilbert space endowed with inner product h:; :i and associated norm k:k, and let f be a proper closed convex function on X. The paper considers the problem of minimizing f , that is, of finding inf X f and some element in the optimal set S := Argminf , this set assumed being non empty. Letting @f denote the subdifferential operator associated with f , we focus on the continuous steepest descent method associated with f , i.e., the differential inclusion \Gamma du dt 2 @f(u); t ? 0 with initial condition u(0) = u 0 : This method is known to yield convergence under broad conditions summarized in the followin