Convergence in gradient-like systems which are asymptotically autonomous and analytic

Huang, Sen-Zhong; Takáč, Peter

doi:10.1016/s0362-546x(00)00145-0

Cited by 69 publications

(60 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The reader is referred to [44,45,38] for the smooth cases, and to [15,17] for nonsmooth inequalities. Kurdyka-Lojasiewicz inequalities have been successfully used to analyze various types of asymptotic behavior: gradient-like systems [15,34,35,39], PDE [55,22], gradient methods [1,48], proximal methods [3], projection methods or alternating methods [5,14].…”

Section: Introductionmentioning

confidence: 99%

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

View full text Add to dashboard Cite

To cite this version:Hedy Attouch, Jérôme Bolte, Benar Svaiter. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods. Math. Program., Ser. A, 2011, 137 (1), pp.91-124. <10.1007/s10107-011-0484-9>. Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward-backward splitting, and regularized Gauss-Seidel methods , 15, 2010; revised: July, 21, 2011 Abstract In view of the minimization of a nonsmooth nonconvex function f , we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka-Lojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward-backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss-Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.2010 Mathematics Subject Classification: 34G25, 47J25, 47J30, 47J35, 49M15, 49M37, 65K15, 90C25, 90C53.

show abstract

Section: Introductionmentioning

confidence: 99%

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

2011

View full text Add to dashboard Cite

show abstract

“…5.1] of the Lojasiewicz-Simon inequality (see, for instance, [9,11,25,29,30, 31]), we are able to show that the ω-limit reduces to a singleton. Then, following the methods developed in [10,24] and refined in [23] for the long time analysis of asymptotically autonomous PDEs, we also prove that, if the L 2 -norm of f (t) decays to 0 as t goes to ∞ with a prescribed rate, then also the convergence rate of the L 2 -norm of χ (t) − χ ∞ can be quantitatively estimated. In order to motivate the convexity assumption on W , we remark that, due to the lack of spatial regularization effects in (1.2), a strong convergence of χ (t) to χ ∞ is not immediate to prove.…”

Section: A Nonlocal Phase-field System With Inertial Term 453mentioning

confidence: 78%

A nonlocal phase-field system with inertial term

2007

View full text Add to dashboard Cite

Abstract. We study a phase-field system where the energy balance equation has the standard (parabolic) form, while the kinetic equation ruling the evolution of the order parameter χ is a nonlocal and nonlinear second-order ODE. The main features of the latter equation are a space convolution term which models long-range interactions of particles and a singular configuration potential that forces χ to take values in (−1, 1). We first prove the global existence and uniqueness of a regular solution to a suitable initial and boundary value problem associated with the system. Then, we investigate its long time behavior from the point of view of ω-limits. In particular, using a nonsmooth version of the Lojasiewicz-Simon inequality, we show that the ω-limit of any trajectory contains one and only one stationary solution, provided that the configuration potential in the kinetic equation is convex and analytic.

show abstract

“…A direct generalization to infinite dimensions does not hold (take T ∈ L(H, K) to be compact and self-adjoint, H, K infinite dimensional Hilbert spaces and define φ(x) = T x 2 K /2, x ∈ H). In [16], this inequality was extended to some infinite dimensional cases to study asymptotic limits of solutions to time-dependent PDE (see also [2], [5]). Theorem 8 gives a sufficient condition for a gradient inequality to hold.…”

Section: Theorem 3 Suppose φ Is Bounded From Below and ∇φ Is Locallymentioning

confidence: 99%

“…Numerical considerations for directly computing critical points of energy functionals such as (15) are quite similar to those for (5). Sometimes for an energy functional it is helpful to define a second function J:…”

Section: ∇G(u) ∇Umentioning

confidence: 99%

Prospects for a central theory of partial differential equations

Neuberger

2005

The Mathematical Intelligencer

View full text Add to dashboard Cite

Convergence in gradient-like systems which are asymptotically autonomous and analytic

Cited by 69 publications

References 18 publications

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

A nonlocal phase-field system with inertial term

Prospects for a central theory of partial differential equations

Contact Info

Product

Resources

About