Approximation of Parabolic Equations Using the Wasserstein Metric

David Kinderlehrer; Noel J. Walkington

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 33, Issue: 4, page 837-852
  • ISSN: 0764-583X

Abstract

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We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature of the algorithms presented here is that derivatives do not enter into the variational principles, so, for example, discontinuous approximations may be used for approximating the heat equation. We present formulae for computing a Wasserstein metric which enters into the variational formulations.

How to cite

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Kinderlehrer, David, and Walkington, Noel J.. "Approximation of Parabolic Equations Using the Wasserstein Metric." ESAIM: Mathematical Modelling and Numerical Analysis 33.4 (2010): 837-852. <http://eudml.org/doc/197465>.

@article{Kinderlehrer2010,
abstract = { We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature of the algorithms presented here is that derivatives do not enter into the variational principles, so, for example, discontinuous approximations may be used for approximating the heat equation. We present formulae for computing a Wasserstein metric which enters into the variational formulations. },
author = {Kinderlehrer, David, Walkington, Noel J.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Wasserstein metric; parabolic equations; numerical approximations.; variational method; parabolic initial-boundary value problems; time discretization; convection-diffusion equations; Fokker-Planck equation; heat equation; Stefan problem},
language = {eng},
month = {3},
number = {4},
pages = {837-852},
publisher = {EDP Sciences},
title = {Approximation of Parabolic Equations Using the Wasserstein Metric},
url = {http://eudml.org/doc/197465},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Kinderlehrer, David
AU - Walkington, Noel J.
TI - Approximation of Parabolic Equations Using the Wasserstein Metric
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 4
SP - 837
EP - 852
AB - We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature of the algorithms presented here is that derivatives do not enter into the variational principles, so, for example, discontinuous approximations may be used for approximating the heat equation. We present formulae for computing a Wasserstein metric which enters into the variational formulations.
LA - eng
KW - Wasserstein metric; parabolic equations; numerical approximations.; variational method; parabolic initial-boundary value problems; time discretization; convection-diffusion equations; Fokker-Planck equation; heat equation; Stefan problem
UR - http://eudml.org/doc/197465
ER -

Citations in EuDML Documents

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  1. Michel Chipot, Stuart Hastings, David Kinderlehrer, Transport in a molecular motor system
  2. Michael Westdickenberg, Jon Wilkening, Variational particle schemes for the porous medium equation and for the system of isentropic Euler equations
  3. Michel Chipot, Stuart Hastings, David Kinderlehrer, Transport in a molecular motor system
  4. Jean-David Benamou, Numerical resolution of an “unbalanced” mass transport problem
  5. Jean-David Benamou, Numerical resolution of an “unbalanced” mass transport problem
  6. Daniel Matthes, Horst Osberger, Convergence of a variational lagrangian scheme for a nonlinear drift diffusion equation

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