Advertisement

A Gradient Flow Perspective on the Quantization Problem

  • Mikaela Iacobelli
Chapter
Part of the Springer INdAM Series book series (SINDAMS, volume 28)

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.

Keywords

Quantization of measures Gradient flows Riemannian manifolds 

Notes

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.

References

  1. 1.
    Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (2008)Google Scholar
  2. 2.
    Bouchitté, G., Jimenez, C., Rajesh, M.: Asymptotique d’un problème de positionnement optimal. C. R. Math. Acad. Sci. Paris 335(10), 853–858 (2002)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bouchitté, G., Jimenez, C., Rajesh, M.: Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95(4), 382–419 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bourne, D.P., Peletier, M. A., Theil, F.: Optimality of the triangular lattice for a particle system with Wasserstein interaction. Commun. Math. Phys. 329, 117–140 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brancolini, A., Buttazzo, G., Santambrogio, F., Stepanov, E.: Long-term planning versus short-term planning in the asymptotical location problem. ESAIM Control Optim. Calc. Var. 15(3), 509–524 (2009)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bucklew, J., Wise, G.: Multidimensional asymptotic quantization theory with r-th power distortion measures. IEEE Inf. Theory 28(2), 239–247 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Buttazzo, G., Santambrogio, F.: A mass transportation model for the optimal planning of an urban region. SIAM Rev. 51, 593–610 (2009).MathSciNetCrossRefGoogle Scholar
  8. 8.
    Caglioti, E., Golse, F., Iacobelli, M.: A gradient flow approach to quantization of measures. Math. Models Methods Appl. Sci. 25, 1845–1885 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Caglioti, E., Golse, F., Iacobelli, M.: Quantization of measures and gradient flows: a perturbative approach in the 2-dimensional case. Ann. Inst. H. Poincaré Anal. Non Linéaire 35(6), 1531–1555Google Scholar
  10. 10.
    Fejes Tóth, L.: Lagerungen in der Ebene, auf der Kugel und im Raum. Springer, Berlin (1953). 2nd ed. 1972Google Scholar
  11. 11.
    Fejes Tóth, G.: Astability criterion to the moment theorem. Stud. Sci. Math. Hungar. 38, 209–224 (2001)zbMATHGoogle Scholar
  12. 12.
    Gersho, A., Gray, R.M.: Vector Quantization and Signal Processing. The Springer International Series in Engineering and Computer Science, vol. 1. Springer, New York (1992)Google Scholar
  13. 13.
    Gruber, P.M.: A short analytic proof of Fejes Tóth’s theorem on sums of moments. Aequationes Math. 58, 291–295 (1999)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gruber, P.M.: Optimal configurations of finite sets in Riemannian 2-manifolds. Geom. Dedicata 84(1–3), 271–320 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Gruber, P.M.: Convex and Discrete Geometry. Springer, New York (2007)zbMATHGoogle Scholar
  16. 16.
    Graf, S., Luschgy, H.: Foundations of Quantization for Probability Distributions. Lecture Notes in Mathematics, vol. 1730. Springer, Berlin (2000)Google Scholar
  17. 17.
    Iacobelli, M.: Asymptotic quantization for probability measures on Riemannian manifolds. ESAIM Control Optim. Calc. Var. 22(3), 770–785 (2016)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Iacobelli, M.: Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete Contin. Dyn. Syst. A (to appear)Google Scholar
  19. 19.
    Kloeckner, B.: Approximation by finitely supported measures. ESAIM Control Optim. Calc. Var. 18(2), 343–359 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Morgan, F., Bolton, R.: Hexagonal economic regions solve the location problem. Am. Math. Mon. 109(2), 165–172 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mosconi, S., Tilli, P.: Γ-convergence for the irrigation problem. J. Conv. Anal. 12(1), 145–158 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Pagès, G., Pham, H., Printems, J.: Optimal quantization methods and applications to numerical problems in finance. In: Handbook on Numerical Methods in Finance, pp. 253–298. Birkhäuser, Boston (2004)CrossRefGoogle Scholar
  23. 23.
    Santambrogio, F.: Optimal Transport for Applied Mathematicians. Calculus of Variations, PDEs, and Modeling. Progress in Nonlinear Differential Equations and their Applications, vol. 87, xxvii+353 pp. Birkhäuser/Springer, Cham (2015)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Durham UniversityDepartment of Mathematical SciencesDurhamUK

Personalised recommendations