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Lagrangian schemes for Wasserstein gradient flows Düring, Bertram


A wide range of diffusion equations can be interpreted as gradient flow with respect to Wasserstein distance of an energy functional. Examples include the heat equation, the porous medium equation, and the fourth-order Derrida-Lebowitz-Speer-Spohn equation. When it comes to solving equations of gradient flow type numerically, schemes that respect the equation's special structure are of particular interest. The gradient flow structure gives rise to a variational scheme by means of the minimising movement scheme (also called JKO scheme, after the seminal work of Jordan, Kinderlehrer and Otto) which constitutes a time-discrete minimization problem for the energy. While the scheme has been used originally for analytical aspects, a number of authors have explored the numerical potential of this scheme. Such schemes often use a Lagrangian representation where instead of the density, the evolution of a time-dependent homeomorphism that describes the spatial redistribution of the density is considered.

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