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Gamma-convergence of the Discrete Internal Energy and Application to Gradient Flows

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We want to approximate diffusion equations with finite numbers of particles. As the 2-Wasserstein energy functional for these equations is not defined for point masses, we spread uniformly the mass of each particle in some ball around it. This “tessellation” gives rise to a discrete energy functional well defined on point masses, which we prove Gamma-converges in the 2-Wasserstein topology to its continuum version as the number of particles increases. For the linear diffusion case we use this result to show the convergence of the resulting discrete gradient flow to the heat equation. By a JKO scheme on the discrete gradient flow, we show numerical simulations. This is a joint work with J. A. Carrillo, Y. Huang, P. Sternberg and G. Wolansky.

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