Asymptotics for transportation cost in high dimensions

Dobric, Vladimir; Yukich, J. E.

doi:10.1007/bf02213456

Cited by 82 publications

(89 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Interestingly, very similar asymptotic formulas appear in the bipartite matching problem [DY95], the traveling salesman problem [BHH59] (both for p = 1) and in rather general combinatorial problems [BB11].…”

Section: Remarkmentioning

confidence: 72%

See 1 more Smart Citation

Constructive quantization: Approximation by empirical measures

Dereich¹,

Scheutzow²,

Schottstedt³

2013

Ann. Inst. H. Poincaré Probab. Statist.

125

138

View full text Add to dashboard Cite

In this article, we study the approximation of a probability measure µ on R d by its empirical measureμN interpreted as a random quantization. As error criterion we consider an averaged p-th moment Wasserstein metric. In the case where 2p < d, we establish fine upper and lower bounds for the error, a high-resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions.

show abstract

Section: Remarkmentioning

confidence: 72%

“…An upper bound for more general distributions can be found in [HK94]. Closely related problems are the bipartite matching problem [DY95] and the traveling salesman problem [BHH59]. Interestingly, there has been progress on this class of problems in several aspects [BB11,BG11] parallel to our research.…”

Section: Introductionmentioning

confidence: 71%

Constructive quantization: Approximation by empirical measures

Dereich¹,

Scheutzow²,

Schottstedt³

2013

Ann. Inst. H. Poincaré Probab. Statist.

125

138

View full text Add to dashboard Cite

show abstract

“…(3) As was kindly pointed out by M. Hauray, estimate (4.12) should also be compared with some estimates in [22] where the related quantity…”

Section: 2mentioning

confidence: 95%

Kac’s program in kinetic theory

Mischler

Mouhot²

2012

Invent. math.

158

292

View full text Add to dashboard Cite

Abstract. This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean [42,53] giving rise to the Boltzmann equation. We solve the conjecture raised by Kac [42], motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.Motivated by the inspirative paper by Grünbaum [35], we develop an abstract method that reduces the question of propagation of chaos to that of proving a purely functional estimate on generator operators (consistency estimates), along with differentiability estimates on the flow of the nonlinear limit equation (stability estimates). This allows us to exploit dissipativity at the level of the mean-field limit equation rather than the level of the particle system (as proposed by Kac).Using this method we show: (1) Quantitative estimates, that are uniform in time, on the chaoticity of a family of states. (2) Propagation of entropic chaoticity, as defined in [10]. (3) Estimates on the time of relaxation to equilibrium, that are independent of the number of particles in the system. Our results cover the two main Boltzmann physical collision processes with unbounded collision rates: hard spheres and true Maxwell molecules interactions. The proof of the stability estimates for these models requires significant analytic efforts and new estimates.

show abstract

“…Let Q = (X , Y). In [6] it is shown that there exists a constant β M,d such that the optimal bipartite matching cost…”

Section: B the Euclidean Bipartite Matching Problemmentioning

confidence: 99%

“…A significant component of the paper will be to generalize the results in [6] to address the case that the distribution of X points is different from that of Y points. The following notion of transportation complexity will prove useful.…”

Section: Euclidean Wasserstein Distancementioning

confidence: 99%

Cost bounds for Pickup and Delivery Problems with application to large-scale transportation systems

Treleaven

Pavone

2012

2012 American Control Conference (ACC)

View full text Add to dashboard Cite

Abstract-Demand-responsive transport (DRT) systems, where users generate requests for transportation from a pickup point to a delivery point, are expected to increase in usage dramatically as the inconvenience of privately-owned cars in metropolitan areas becomes excessive. However, despite the increasing role of DRT systems, there are very few rigorous results characterizing achievable performance (in terms, e.g., of stability conditions). In this paper, our aim is to bridge this gap for a rather general model of DRT systems, which takes the form of a generalized Dynamic Pickup and Delivery Problem. The key strategy is to develop analytical bounds for the optimal cost of the Euclidean Stacker Crane Problem (ESCP), which represents a general static model for DRT systems. By leveraging such bounds, we characterize a necessary and sufficient condition for the stability of DRT systems; the condition depends only on the workspace geometry, the stochastic distributions of pickup and delivery points, customers' arrival rate, and the number of vehicles. Our results exhibit some surprising features that are absent in traditional spatiallydistributed queueing systems.

show abstract

Asymptotics for transportation cost in high dimensions

Cited by 82 publications

References 12 publications

Constructive quantization: Approximation by empirical measures

Constructive quantization: Approximation by empirical measures

Kac’s program in kinetic theory

Cost bounds for Pickup and Delivery Problems with application to large-scale transportation systems

Contact Info

Product

Resources

About