Notions of optimal transport theory and how to implement them on a computer

Autor: Erica L. Schwindt, Bruno Levy
Přispěvatelé: Geometry and Lighting (ALICE), Inria Nancy - Grand Est, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Department of Algorithms, Computation, Image and Geometry (LORIA - ALGO), Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Laboratoire Lorrain de Recherche en Informatique et ses Applications (LORIA), Institut National de Recherche en Informatique et en Automatique (Inria)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)-Université de Lorraine (UL)-Centre National de la Recherche Scientifique (CNRS)
Jazyk: angličtina
Rok vydání: 2017
Předmět:
shape interpolation
Theoretical computer science
010103 numerical & computational mathematics
02 engineering and technology
Computational algorithm
fluid dynamics
01 natural sciences
Mathematics - Analysis of PDEs
49M15
35J96
65D18
FOS: Mathematics
0202 electrical engineering
electronic engineering
information engineering
Optimal transport
[INFO]Computer Science [cs]
Mathematics - Numerical Analysis
0101 mathematics
[MATH]Mathematics [math]
Physics
GRASP
General Engineering
020207 software engineering
Transport theory
Numerical Analysis (math.NA)
Geometry processing
Computer Graphics and Computer-Aided Design
[INFO.INFO-MO]Computer Science [cs]/Modeling and Simulation
Human-Computer Interaction
Mathematical theory
Scientific domain
Laguerre polynomials
Voronoi diagram
Mathematics
Analysis of PDEs (math.AP)
Zdroj: Computers and Graphics
Computers and Graphics, Elsevier, 2018, pp.1-22. ⟨10.1016/j.cag.2018.01.009⟩
Computers and Graphics, 2018, pp.1-22. ⟨10.1016/j.cag.2018.01.009⟩
ISSN: 0097-8493
DOI: 10.1016/j.cag.2018.01.009⟩
Popis: This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.
32 pages, 17 figures
Databáze: OpenAIRE
Externí odkaz: