Discretization of flux-limited gradient flows: $\Gamma $-convergence and numerical schemes

Abstract

We study a discretization in space and time for a class of nonlinear diffusion equations with flux limitation. That class contains the so-called relativistic heat equation, as well as other gradient flows of Renyi entropies with respect to transportation metrics with finite maximal velocity. Discretization in time is performed with the JKO method, thus preserving the variational structure of the gradient flow. This is combined with an entropic regularization of the transport distance, which allows for an efficient numerical calculation of the JKO minimizers. Solutions to the fully discrete equations are entropy dissipating, mass conserving, and respect the finite speed of propagation of support.First, we give a proof of Γ-convergence of the infinite chain of JKO steps in the joint limit of infinitely refined spatial discretization and vanishing entropic regularization. The singularity of the cost function makes the construction of the recovery sequence significantly more difficult than in the L p -Wasserstein case. Second, we define a practical numerical method by combining the JKO time discretization with a "light speed" solver for the spatially discrete minimization problem using Dykstra's algorithm, and demonstrate its efficiency in a series of experiments.

Operator-splitting schemes for degenerate, non-local, conservative-dissipative systems

“…where we have combined (47) with Lemma 3.10 to conclude (40) on the left hand side of (48), multiplying through by −1, delivers the sought result.…”

“…In recent years, this regularisation technique has found applications in a variety of domains such as machine learning, image processing, graphics and biology. In particular, several works have developed entropic regularisation schemes for solving evolutionary equations, such as nonlinear diffusion equations [12,44], flux-limited gradient flows [40] and a tumour growth model of Hele-Shaw type [17]. We refer the reader to the recent monograph [45] for a great detailed account of the entropic regularisation technique.…”

Entropic regularisation of non-gradient systems

“…As in general entropic regularisation techniques for optimal transport problems, the regularised scheme leverages the reformulation of this smooth optimisation problem as a Kullback-Leibler projection and makes use of Dykstra's algorithm to attain a fast and convergent numerical scheme [11,49]. Similar ideas have been applied to other evolutionary equations such as flux-limited gradient flows [42] and a tumour growth model of Hele-Shaw type [23].…”