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Density of convex intersections and applications.

M Hintermüller1,2, C N Rautenberg1, S Rösel1

  • 1Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099 Berlin, Germany.

Proceedings. Mathematical, Physical, and Engineering Sciences
|October 10, 2017
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Summary

This study explores density properties of convex set intersections in function spaces using Gamma-convergence. It reveals how these properties arise in optimization and variational inequalities, with applications in elasto-plasticity and image restoration.

Keywords:
convex constraintsdensityelasto-plasticityfinite elementsimage restorationvariational inequalities

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Area of Science:

  • Mathematical Analysis
  • Optimization Theory
  • Numerical Analysis

Background:

  • Density properties of convex sets are crucial in various mathematical fields.
  • Understanding these properties is essential for solving optimization and variational inequality problems.
  • Existing methods often lack a general framework to analyze these density issues.

Purpose of the Study:

  • To investigate density properties of intersections of convex sets in function spaces.
  • To establish a general framework for understanding density issues using Gamma-convergence.
  • To present density results and counterexamples for pointwise constraints in Sobolev spaces.

Main Methods:

  • Utilizing the concept of Gamma-convergence.
  • Analyzing regularization, discretization, and dualization of constrained optimization problems.
  • Examining perturbed variational inequalities.

Main Results:

  • A general framework is established demonstrating how density issues arise naturally.
  • Various density results and counterexamples for pointwise constraints in Sobolev spaces are presented.
  • Regularity requirements on the upper bound for density properties are identified.

Conclusions:

  • The findings provide a deeper understanding of density properties in function spaces.
  • The results have implications for finite-element discretizations of convex constraints.
  • Applications in elasto-plasticity and image restoration highlight the practical relevance of the study.