Bailey Miller, Rohan Sawhney, Keenan Crane, Ioannis Gkioulekas
{"title":"漫步罗宾:与罗宾一起漫步星空边界条件","authors":"Bailey Miller, Rohan Sawhney, Keenan Crane, Ioannis Gkioulekas","doi":"10.1145/3658153","DOIUrl":null,"url":null,"abstract":"\n Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the\n walk on stars (WoSt)\n algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces\n bounded-variance\n estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the\n walk on boundary\n algorithm. We also develop\n bidirectional\n and\n boundary value caching\n strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.\n","PeriodicalId":7,"journal":{"name":"ACS Applied Polymer Materials","volume":" March","pages":""},"PeriodicalIF":4.4000,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Walkin’ Robin: Walk on Stars with Robin Boundary Conditions\",\"authors\":\"Bailey Miller, Rohan Sawhney, Keenan Crane, Ioannis Gkioulekas\",\"doi\":\"10.1145/3658153\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"\\n Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the\\n walk on stars (WoSt)\\n algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces\\n bounded-variance\\n estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the\\n walk on boundary\\n algorithm. We also develop\\n bidirectional\\n and\\n boundary value caching\\n strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.\\n\",\"PeriodicalId\":7,\"journal\":{\"name\":\"ACS Applied Polymer Materials\",\"volume\":\" March\",\"pages\":\"\"},\"PeriodicalIF\":4.4000,\"publicationDate\":\"2024-07-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Polymer Materials\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1145/3658153\",\"RegionNum\":2,\"RegionCategory\":\"化学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATERIALS SCIENCE, MULTIDISCIPLINARY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Polymer Materials","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1145/3658153","RegionNum":2,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, MULTIDISCIPLINARY","Score":null,"Total":0}
Walkin’ Robin: Walk on Stars with Robin Boundary Conditions
Numerous scientific and engineering applications require solutions to boundary value problems (BVPs) involving elliptic partial differential equations, such as the Laplace or Poisson equations, on geometrically intricate domains. We develop a Monte Carlo method for solving such BVPs with arbitrary first-order linear boundary conditions---Dirichlet, Neumann, and Robin. Our method directly generalizes the
walk on stars (WoSt)
algorithm, which previously tackled only the first two types of boundary conditions, with a few simple modifications. Unlike conventional numerical methods, WoSt does not need finite element meshing or global solves. Similar to Monte Carlo rendering, it instead computes pointwise solution estimates by simulating random walks along star-shaped regions inside the BVP domain, using efficient ray-intersection and distance queries. To ensure WoSt produces
bounded-variance
estimates in the presence of Robin boundary conditions, we show that it is sufficient to modify how WoSt selects the size of these star-shaped regions. Our generalized WoSt algorithm reduces estimation error by orders of magnitude relative to alternative grid-free methods such as the
walk on boundary
algorithm. We also develop
bidirectional
and
boundary value caching
strategies to further reduce estimation error. Our algorithm is trivial to parallelize, scales sublinearly with increasing geometric detail, and enables progressive and view-dependent evaluation.
期刊介绍:
ACS Applied Polymer Materials is an interdisciplinary journal publishing original research covering all aspects of engineering, chemistry, physics, and biology relevant to applications of polymers.
The journal is devoted to reports of new and original experimental and theoretical research of an applied nature that integrates fundamental knowledge in the areas of materials, engineering, physics, bioscience, polymer science and chemistry into important polymer applications. The journal is specifically interested in work that addresses relationships among structure, processing, morphology, chemistry, properties, and function as well as work that provide insights into mechanisms critical to the performance of the polymer for applications.