{"title":"在有域约束条件的过渡紧凑支承上实现最佳传输的变式面进化方法","authors":"Anthony Yezzi","doi":"10.3390/fluids9050118","DOIUrl":null,"url":null,"abstract":"We examine the optimal mass transport problem in Rn between densities with transitioning compact support by considering the geometry of a continuous interpolating support boundary Γ in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space-time embedding in terms of points, vectors, and sets in Rn+1=R×Rn and blend the mass density and velocity as well into a space-time solenoidal vector field W|Ω→Rn+1 over a compact set Ω⊂Rn+1. We then formulate a joint optimization for W and its support using the shaped gradient of the space-time surface Γ outlining the support boundary ∂Ω. This easily accommodates spatiotemporal constraints, including obstacles or mandatory regions to visit.","PeriodicalId":12397,"journal":{"name":"Fluids","volume":null,"pages":null},"PeriodicalIF":1.8000,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints\",\"authors\":\"Anthony Yezzi\",\"doi\":\"10.3390/fluids9050118\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We examine the optimal mass transport problem in Rn between densities with transitioning compact support by considering the geometry of a continuous interpolating support boundary Γ in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space-time embedding in terms of points, vectors, and sets in Rn+1=R×Rn and blend the mass density and velocity as well into a space-time solenoidal vector field W|Ω→Rn+1 over a compact set Ω⊂Rn+1. We then formulate a joint optimization for W and its support using the shaped gradient of the space-time surface Γ outlining the support boundary ∂Ω. This easily accommodates spatiotemporal constraints, including obstacles or mandatory regions to visit.\",\"PeriodicalId\":12397,\"journal\":{\"name\":\"Fluids\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.8000,\"publicationDate\":\"2024-05-16\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fluids\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3390/fluids9050118\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MECHANICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fluids","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3390/fluids9050118","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MECHANICS","Score":null,"Total":0}
A Variational Surface-Evolution Approach to Optimal Transport over Transitioning Compact Supports with Domain Constraints
We examine the optimal mass transport problem in Rn between densities with transitioning compact support by considering the geometry of a continuous interpolating support boundary Γ in space-time within which the mass density evolves according to the fluid dynamical framework of Benamou and Brenier. We treat the geometry of this space-time embedding in terms of points, vectors, and sets in Rn+1=R×Rn and blend the mass density and velocity as well into a space-time solenoidal vector field W|Ω→Rn+1 over a compact set Ω⊂Rn+1. We then formulate a joint optimization for W and its support using the shaped gradient of the space-time surface Γ outlining the support boundary ∂Ω. This easily accommodates spatiotemporal constraints, including obstacles or mandatory regions to visit.