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16
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 ..."
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Cited by 53 (14 self)
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. 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 ..."
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Cited by 16 (2 self)
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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.
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 ..."
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Cited by 7 (2 self)
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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.
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 ..."
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Cited by 5 (2 self)
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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.
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 ..."
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Cited by 5 (3 self)
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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.
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 ..."
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Cited by 4 (0 self)
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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
Convergence rate estimates for the gradient differential inclusion
 Optim. Methods Softw
, 2005
"... Let f: H → R ∪ {∞} be a proper, lower semi–continuous, convex function in a Hilbert space H. The gradient differential inclusion is x ′ (t) ∈ −∂f(x(t)), x(0) = x, where x ∈ dom(f). If f is differentiable, the inclusion can be considered as the continuous version of the steepest descent method for ..."
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Cited by 4 (0 self)
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Let f: H → R ∪ {∞} be a proper, lower semi–continuous, convex function in a Hilbert space H. The gradient differential inclusion is x ′ (t) ∈ −∂f(x(t)), x(0) = x, where x ∈ dom(f). If f is differentiable, the inclusion can be considered as the continuous version of the steepest descent method for minimizing f on H. Even if f is not differentiable, the inclusion has a unique solution {x(t) : t> 0} which converges weakly to a minimizer of f if such a minimizer exists. In general, the inclusion can be interpreted as the the continuous version of the proximal point method for minimization f on H. There is a remarkable similarity between the behavior of the inclusion and its discrete counterparts as well in the methods used in both cases. As a simple consequence of our previous results on the proximal point method, we prove the convergence rate estimate f(x(t))−f(u) ≤ (1/2t)u−x  2 −(1/2t)u−x(t)  2 −(t/2)∂f 0 (x(t))  2, where ∂f 0 (x(t)) is the least norm element of ∂f(x(t)). If f has a minimizer x ∗ , this implies f(x(t))−f(x ∗ ) = O(1/t), a result due to Brézis. If x(t) converges strongly to x ∗ , we give a better estimate f(x(t)) − f(x ∗ ) ≤ 1 / ( ∫ t 0 x(s) − x ∗   −2 ds) = o(1/t). Key words. Global convergence rate, subdifferential, nonlinear semigroups, strong convergence. Abbreviated title: Convergence rate estimates for the gradient inclusion.
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 ..."
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Cited by 3 (0 self)
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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.
The Heavy Ball With Friction Dynamical System for Convex Constrained Minimization Problems
"... . The \heavy ball with friction" equation u+ _ u+r(u) = 0 is a nonlinear oscillator with damping ( > 0). In [2], Alvarez proved that when H is a real Hilbert space (possibly innite dimensional) and : H ! IR is a smooth convex function whose minimal value is achieved, then each trajectory t ! u(t ..."
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Cited by 2 (2 self)
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. The \heavy ball with friction" equation u+ _ u+r(u) = 0 is a nonlinear oscillator with damping ( > 0). In [2], Alvarez proved that when H is a real Hilbert space (possibly innite dimensional) and : H ! IR is a smooth convex function whose minimal value is achieved, then each trajectory t ! u(t) weakly converges to a minimizer of . We prove a similar result in the convex constrained case by considering the gradientprojection dynamical system u + _ u + u proj C (u r(u)) = 0; where C is a closed convex subset of H. This result extends, by using dierent technics, previous results by Antipin [1]. AMS classication: 34C35, 34D05, 49M10, 49M30, 49J40, 90C25. Key words: Heavy ball with friction equation, dissipative dynamical systems, asymptotical behavior, steepest descent, constrained convex minimization, projection method, xed point of a contraction. ACSIOMCNRS EP 2066, Departement de Mathematiques, Universite Montpellier II, France; email: attouch@math.univm...