14 Pages
English
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Models and applications of optimal transport in economics traffic and urban planning

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14 Pages
English

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Niveau: Supérieur, Doctorat, Bac+8
Models and applications of Optimal Transport Theory Filippo Santambrogio? Grenoble, June 2009 These lecture notes will present the main issues and ideas of some variational problems that use or touch the theory of Optimal Transportation. Just ideas, almost no proofs. 1 The urban planning of residents and services A very simplified model that has been proposed for studying the distribution of residents and services in a given urban region ? passes through the minimization of a total quantity F(µ, ?) concerning two unknown densities µ and ?. the two measures µ and ? will be searched among probabilities on ?. This means that the total amounts of population and production are fixed as problem data. The definition of the total cost functional to optimize takes into account some criteria we want the two densities µ and ? to satisfy: (i) there is a transportation cost T for moving from the residential areas to the services areas; (ii) people do not want to live in areas where the density of population is too high; (iii) services need to be concentrated as much as possible in order to increase efficiency and decrease management costs. Fact (i) is described, in its easiest version, through a p-Wasserstein distance (p ≥ 1). We will look at T (µ, ?) = W pp (µ, ?).

  • problem

  • functionals favoring

  • optimal transportation

  • transport plan

  • service

  • densities µ

  • single citizen

  • searched among


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ModelsandapplicationsofOptimalTransportTheoryFilippoSantambrogioGrenoble,June2009TheselecturenoteswillpresentthemainissuesandideasofsomevariationalproblemsthatuseortouchthetheoryofOptimalTransportation.Justideas,almostnoproofs.1TheurbanplanningofresidentsandservicesAverysimplifiedmodelthathasbeenproposedforstudyingthedistributionofresidentsandservicesinagivenurbanregionΩpassesthroughtheminimizationofatotalquantityF(µ,ν)concerningtwounknowndensitiesµandν.thetwomeasuresµandνwillbesearchedamongprobabilitiesonΩ.Thismeansthatthetotalamountsofpopulationandproductionarefixedasproblemdata.Thedefinitionofthetotalcostfunctionaltooptimizetakesintoaccountsomecriteriawewantthetwodensitiesµandνtosatisfy:(i)thereisatransportationcostTformovingfromtheresidentialareastotheservicesareas;(ii)peopledonotwanttoliveinareaswherethedensityofpopulationistoohigh;(iii)servicesneedtobeconcentratedasmuchaspossibleinordertoincreaseefficiencyanddecreasemanagementcosts.Fact(i)isdescribed,initseasiestversion,throughap-Wassersteindistance(p1).WewilllookatT(µ,ν)=Wpp(µ,ν).Fact(ii)willbedescribedbyapenalizationfunctional,akindoftotalunhappinessofcitizensduetohighdensityofpopulation,obtainedbyintegratingwithrespecttothecitizens’densitytheirpersonalunhappiness.Fact(iii)ismodeledbyathirdtermrepresentingcostsformanagingservicesoncetheyarelocatedaccordingtothedistributionν,takingintoaccountthatefficiencydependsstronglyonhowmuchνisconcentrated.ThecostfunctionaltobeconsideredisthenF(µ,ν)=T(µ,ν)+F(µ)+G(ν),(1.1)CEREMADE,UMRCNRS7534,Universite´Paris-Dauphine,Pl.deLattredeTassigny,75775ParisCedex16,FRANCEfilippo@ceremade.dauphine.fr,http://www.ceremade.dauphine.fr/filippo.1