ESAIM: Control, Optimisation and Calculus of Variations

Research Article

Approximation by finitely supported measures

Benoît Kloeckner

Institut Fourier, Université Joseph Fourier, BP 53, 38041 Grenoble, France. Benoit.Kloeckner@ujf-grenoble.fr

Abstract

We consider the problem of approximating a probability measure defined on a metric space by a measure supported on a finite number of points. More specifically we seek the asymptotic behavior of the minimal Wasserstein distance to an approximation when the number of points goes to infinity. The main result gives an equivalent when the space is a Riemannian manifold and the approximated measure is absolutely continuous and compactly supported.

(Received March 4 2010)

(Revised November 9 2010)

(Online publication April 13 2011)

Key Words:

  • Measures;
  • Wasserstein distance;
  • quantization;
  • location problem;
  • centroidal Voronoi tessellations

Mathematics Subject Classification:

  • 49Q20;
  • 90B85
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