Restricted Power Diagrams on the GPU

We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions. We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to...

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Vydané v:	Computer graphics forum Ročník 40; číslo 2; s. 1 - 12
Hlavní autori:	Basselin, J., Alonso, L., Ray, N., Sokolov, D., Lefebvre, S., Lévy, B.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Oxford Blackwell Publishing Ltd 01.05.2021 Wiley
Predmet:	Algorithms CCS Concepts Computational fluid dynamics Computational Geometry Computer memory Computer Science Computing methodologies → Parallel algorithms Data Structures and Algorithms Decomposition Distributed, Parallel, and Cluster Computing Domains Fluid flow Graphics processing units Incompressible flow Incompressible fluids Integrals Operators (mathematics) Tessellation Theory of computation → Computational geometry Triangulation Voronoi graphs Power Diagram GPU Programming Voronoi Diagram
ISSN:	0167-7055, 1467-8659
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Abstract	We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions. We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to compute many decompositions, for instance, centroidal Voronoi tesselation algorithms and incompressible fluid dynamics simulations. We propose an efficient solution that directly evaluates the integrals over every cell without computing the power diagram explicitly and without intersecting it with a tetrahedralization of the domain. Most computations are performed on the fly, without storing the power diagram. We manipulate a triangulation of the boundary of the domain (instead of tetrahedralizing the domain) to speed up the process. Moreover, the cells are treated independently one from another, making it possible to trivially scale up on a parallel architecture. Despite recent Voronoi diagram generation methods optimized for the GPU, computing integrals over restricted power diagrams still poses significant challenges; the restriction to a complex simulation domain is difficult and likely to be slow. It is not trivial to determine when a cell of a power diagram is completely computed, and the resulting integrals (e.g. the weighted Laplacian operator matrix) do not fit into fast (shared) GPU memory. We address all these issues and boost the performance of the state‐of‐the‐art algorithms by a factor 2 to 3 for (unrestricted) Voronoi diagrams and a ×50 speed‐up with respect to CPU implementations for restricted power diagrams. An essential ingredient to achieve this is our new scheduling strategy that allows us to treat each Voronoi/power diagram cell with optimal settings and to benefit from the fast memory.
AbstractList	We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions. We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to compute many decompositions, for instance, centroidal Voronoi tesselation algorithms and incompressible fluid dynamics simulations. We propose an efficient solution that directly evaluates the integrals over every cell without computing the power diagram explicitly and without intersecting it with a tetrahedralization of the domain. Most computations are performed on the fly, without storing the power diagram. We manipulate a triangulation of the boundary of the domain (instead of tetrahedralizing the domain) to speed up the process. Moreover, the cells are treated independently one from another, making it possible to trivially scale up on a parallel architecture. Despite recent Voronoi diagram generation methods optimized for the GPU, computing integrals over restricted power diagrams still poses significant challenges; the restriction to a complex simulation domain is difficult and likely to be slow. It is not trivial to determine when a cell of a power diagram is completely computed, and the resulting integrals (e.g. the weighted Laplacian operator matrix) do not fit into fast (shared) GPU memory. We address all these issues and boost the performance of the state-of-the-art algorithms by a factor 2 to 3 for (unrestricted) Voronoi diagrams and a ×50 speed-up with respect to CPU implementations for restricted power diagrams. An essential ingredient to achieve this is our new scheduling strategy that allows us to treat each Voronoi/power diagram cell with optimal settings and to benefit from the fast memory. We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions. We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to compute many decompositions, for instance, centroidal Voronoi tesselation algorithms and incompressible fluid dynamics simulations. We propose an efficient solution that directly evaluates the integrals over every cell without computing the power diagram explicitly and without intersecting it with a tetrahedralization of the domain. Most computations are performed on the fly, without storing the power diagram. We manipulate a triangulation of the boundary of the domain (instead of tetrahedralizing the domain) to speed up the process. Moreover, the cells are treated independently one from another, making it possible to trivially scale up on a parallel architecture. Despite recent Voronoi diagram generation methods optimized for the GPU, computing integrals over restricted power diagrams still poses significant challenges; the restriction to a complex simulation domain is difficult and likely to be slow. It is not trivial to determine when a cell of a power diagram is completely computed, and the resulting integrals (e.g. the weighted Laplacian operator matrix) do not fit into fast (shared) GPU memory. We address all these issues and boost the performance of the state‐of‐the‐art algorithms by a factor 2 to 3 for (unrestricted) Voronoi diagrams and a ×50 speed‐up with respect to CPU implementations for restricted power diagrams. An essential ingredient to achieve this is our new scheduling strategy that allows us to treat each Voronoi/power diagram cell with optimal settings and to benefit from the fast memory.
Author	Sokolov, D. Basselin, J. Ray, N. Alonso, L. Lévy, B. Lefebvre, S.
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SubjectTerms	Algorithms CCS Concepts Computational fluid dynamics Computational Geometry Computer memory Computer Science Computing methodologies → Parallel algorithms Data Structures and Algorithms Decomposition Distributed, Parallel, and Cluster Computing Domains Fluid flow Graphics processing units Incompressible flow Incompressible fluids Integrals Operators (mathematics) Tessellation Theory of computation → Computational geometry Triangulation Voronoi graphs
Title	Restricted Power Diagrams on the GPU
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