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Effective rates for continuous-time quasi-Fejér monotone dynamical systems

Anton Freund, Nicholas Pischke

Abstract

We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces.

Effective rates for continuous-time quasi-Fejér monotone dynamical systems

Abstract

We provide quantitative convergence results for continuous-time dynamical systems in metric spaces that satisfy a continuous-time analog of quasi-Fejér monotonicity. More precisely, we provide a (strong) convergence result for such dynamical systems over compact metric spaces which is quantitatively outfitted with a continuous-time rate of metastability, which moreover can be explicitly and effectively constructed in a very uniform way, only depending on a few moduli representing quantitative witnesses to key properties of the dynamical system and a measure for the compactness of the space. We further show how this convergence result can be extended to non-compact spaces under a regularity assumption of the associated problem, where moreover rates of convergence can then be explicitly constructed which are similarly uniform. In both cases, already the associated ``infinitary'' convergence result is qualitatively novel in its present generality. Beyond this abstract quantitative theory for such dynamical systems, we motivate how the presently studied continuous-time variant of quasi-Fejér monotonicity naturally occurs as a unifying property of many dynamical systems and differential equations and inclusions, and in that way can be used to provide a comprehensive quantitative theory for many such dynamical systems. We illustrate this with three case studies for both classical first- and second-order dynamical systems in Hilbert spaces as well as (generalized) gradient flows and associated semigroups in nonlinear Hadamard spaces.
Paper Structure (13 sections, 49 theorems, 308 equations)

This paper contains 13 sections, 49 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. Continuous-time Fejér and quasi-Fejér monotonicity
  3. Definition and motivation
  4. Quantitative results for differential inequalities
  5. A quantitative convergence theory for quasi-Fejér monotone dynamical systems
  6. Convergence under relative compactness assumptions
  7. Convergence under metric regularity assumptions
  8. Applications to dynamical systems
  9. First-order systems over nonexpansive operators
  10. Second-order dynamical systems for cocoercive operators
  11. (Generalized) gradient flows in Hadamard spaces
  12. The gradient flow of a convex function
  13. A class of nonlinear semigroups for a nonexpansive mapping

Key Result

Lemma 2.2

Assume $\mathcal{E}:[0,\infty)\to\mathbb R$ is locally absolutely continuous and bounded below such that almost everywhere for some $\overline{e}\in L^1([0,\infty))$. Then $\lim_{t\to\infty}\mathcal{E}(t)$ exists. If further $\mathcal{E}$ is nonnegative and $\mathcal{E}\in L^p([0,\infty))$ for $p\in [1,+\infty)$, then $\lim_{t\to\infty}\mathcal{E}(t)=0$. In that case, we may also have $\overline{

Theorems & Definitions (109)

  • Definition 2.1: Quasi-Fejér monotone dynamical system
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Definition 3.1
  • Remark 3.2
  • Definition 3.3
  • ...and 99 more