Asymptotic analysis of a class of optimal location problems

作者：Bouchitte Guy^*; Jimenez Chloe; Mahadevan Rajesh

来源：Journal de Mathematiques Pures et Appliquees, 2011, 95(4): 382-419.

DOI：10.1016/j.matpur.2010.10.009

• Monge-Kantorovich transportation
• Wasserstein distance
• Entropy
• Optimal partitions
• Rate of approximation
• Asymptotics
• Gamma-convergence
• Functionals on measures
• Integral representation

Given a density function f on a compact subset of R(d) we look at the problem of finding the best approximation of f by discrete measures v = Sigma c(i)delta(xi) in the sense of the p-Wasserstein distance, subject to size constraints of the form Sigma h(c(i)) <= alpha where h is a given weight function. This is an important problem with applications in economic planning of locations and in information theory. The efficiency of the approximation can be measured by studying the rate at which the minimal distance tends to zero as alpha tends to infinity. In this paper, we introduce a rescaled distance which depends on a small parameter and establish a representation formula for its limit as a function of the local statistics for the distribution of the c(i)'s. This allows to treat the asymptotic problem as alpha -> infinity for a quite large class of weight functions h.

• 出版日期2011-4