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A Gradient Flow Perspective on the Quantization Problem

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PDE Models for Multi-Agent Phenomena

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Abstract

In this paper we review recent results by the author on the problem of quantization of measures. More precisely, we propose a dynamical approach, and we investigate it in dimensions 1 and 2. Moreover, we discuss a recent general result on the static problem on arbitrary Riemannian manifolds.

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Notes

  1. 1.

    Equivalently known as Monge-Kantorovich distances; we shall use both terms interchangeably.

  2. 2.

    The vertices of the triangular lattice are the centres of a hexagonal tiling.

  3. 3.

    Note that this corresponds to the quantization of ρ ≡ 1 with d = r = 2 for N ≈ n 2 →∞.

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Acknowledgements

The author would like to thank Megan Griffin-Pickering for her useful comments on a preliminary version of this paper and the L’Oréal Foundation for partially supporting this project by awarding the L’Oréal-UNESCO For Women in Science France fellowship.

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Authors and Affiliations

  1. Durham University, Department of Mathematical Sciences, Durham, UK

    Mikaela Iacobelli

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  1. Mikaela Iacobelli
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Correspondence to Mikaela Iacobelli .

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Editors and Affiliations

  1. Ceremade, Université Paris-Dauphine, Paris, France

    Pierre Cardaliaguet

  2. Dept. of Mathematics, Universitá di Roma “Tor Vergata”, Roma, Italy

    Alessio Porretta

  3. Dept. of Mathematics “F. Casorati”, University of Pavia, Pavia, Italy

    Francesco Salvarani

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Iacobelli, M. (2018). A Gradient Flow Perspective on the Quantization Problem. In: Cardaliaguet, P., Porretta, A., Salvarani, F. (eds) PDE Models for Multi-Agent Phenomena. Springer INdAM Series, vol 28. Springer, Cham. https://doi.org/10.1007/978-3-030-01947-1_7

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