One-Dimensional Numerical Algorithms for Gradient Flows in the p-Wasserstein Spaces

We numerically approximate, on the real line, solutions to a large class of parabolic partial differential equations which are “gradient flows” of some energy functionals with respect to the L p -Wasserstein metrics for all p >1. Our method relies on variational principles involving the optimal t...

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Bibliographic Details
Published inActa applicandae mathematicae Vol. 125; no. 1; pp. 121 - 134
Main Authors Agueh, Martial, Bowles, Malcolm
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.06.2013
Springer Nature B.V
Subjects
Algorithms
Applications of Mathematics
Approximation
Calculus of Variations and Optimal Control; Optimization
Computational Mathematics and Numerical Analysis
Gradient flow
Mathematical analysis
Mathematical functions
Mathematical models
Mathematics
Mathematics and Statistics
Numerical analysis
Optimization
Partial Differential Equations
Probability Theory and Stochastic Processes
Studies
Transport
Variational principles
Numerical approximation
Parabolic partial differential equation
35K20
Wasserstein metric
Gradient flow
65M22
49M25
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ISSN0167-8019
1572-9036
DOI10.1007/s10440-012-9783-2

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Summary:We numerically approximate, on the real line, solutions to a large class of parabolic partial differential equations which are “gradient flows” of some energy functionals with respect to the L p -Wasserstein metrics for all p >1. Our method relies on variational principles involving the optimal transport problem with general strictly convex cost functions.
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ISSN:0167-8019
1572-9036
DOI:10.1007/s10440-012-9783-2