A Gradient Flow Approach to Quantization of Measures

Emanuele Caglioti (1) , François Golse (2) , Mikaela Iacobelli (1, 2)

1 Sapienza University of Rome - Dipartimento di Matematica "Guido Castelnuovo" [Roma I]
2 CMLS - Centre de Mathématiques Laurent Schwartz

In this paper we study a gradient flow approach to the problem of quantization of measures in one dimension. By embedding our problem in $L^2$ , we find a continuous version of it that corresponds to the limit as the number of particles tends to infinity. Under some suitable regularity assumptions on the density, we prove uniform stability and quantitative convergence result for the discrete and continuous dynamics.

Keywords: Gradient flow; Degenerate parabolic equation; Monge-Kantorovich distance; Quantization of measures

Domains: Mathematics [math]; Analysis of PDEs [math.AP]

Emanuele Caglioti, François Golse, Mikaela Iacobelli. A Gradient Flow Approach to Quantization of Measures. Mathematical Models and Methods in Applied Sciences, 2015, 25 (10), pp.1845-1885. ⟨10.1142/S0218202515500475⟩. ⟨hal-01109228⟩