An unbalanced optimal transport splitting scheme for general advection-reaction-diffusion problems

Abstract

In this paper, we show that unbalanced optimal transport provides a convenient framework to handle reaction and diffusion processes in a unified metric framework. We use a constructive method, alternating minimizing movements for the Wasserstein distance and for the Fisher-Rao distance, and prove existence of weak solutions for general scalar reaction-diffusion-advection equations. We extend the approach to systems of multiple interacting species, and also consider an application to a very degenerate diffusion problem involving a Gamma-limit. Moreover, some numerical simulations are included.and the next Fisher-Rao JKO stephandles the reaction part of the evolution. As already mentioned, the WFR metric will allow to suitable control both steps in a unified metric framework. We will first state a general convergence result for scalar reaction-diffusion equations, and then illustrate on a few particular examples how the general idea can be adapted to treat e.g. prey-predator systems or very degenerate Hele-Shaw diffusion problems. In this work we do not focus on optimal results and do not seek full generality, but rather wish to illustrate the efficiency of the general approach. Another advantage of the splitting scheme is that is well adapted to existing Monge/Kantorovich/Wasserstein numerical solvers, and the Fisher-Rao step turns out to be a simple pointwise convex problem which can be implemented in a very simple way. See also [10,13] for a more direct numerical approach by entropic regularization. Throughout the paper we will illustrate the theoretical results with a few numerical tests. All the numerical simulations were implemented with the augmented Lagrangian ALG2-JKO scheme from [6] for the Wasserstein step, and we used a classical Newton algorithm for the Fisher-Rao step.The paper is organized as follows. In section 2 we recall the basic definitions and useful properties of the Wasserstein-Fisher-Rao distance WFR. Section 3 contains the precise description of the splitting scheme and a detailed convergence analysis for a broad class of reaction-diffusion equations. In section 4 we present an extension to some prey-predator multicomponent systems with nonlocal interactions. In section 5 we extend the general result from section 3 to a very degenerate tumor growth model studied in [34], corresponding to a pure WFR gradient flow: we show that the splitting scheme captures fine properties of the model, particularly the Γ-convergence of discrete gradient flows as the degenerate diffusion parameter of Porous Medium type m → ∞. The last section 6 contains an extension to a tumor-growth model coupled with an evolution equation for the nutrients. PreliminariesLet us first fix some notations. Throughout the whole paper, Ω denotes a possibly unbounded convex subset of R d , Q T represents the product space [0, T ] × Ω, for T > 0, and we write M + = M + (Ω) for the set of nonnegative finite Radon measures on Ω. We say that a curve of measures t → ρ t ∈ C w ([0, 1]; M + ) is narrowly continuous if it is continuous wi...

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“…and it should be no surprise that these WF R κ , F R κ , W Γ , WΩ distances will appear frequently in this work. We refer to [28,29,21,29,8,7,26,15] and references therein and thereof for a detailed account of the unbalanced theory and various applications [20,22,24,23,25,13,14,16] (see also [17] for the so-called unnormalized optimal transport). For the sake of completeness let us also cite [35,36,12,37] for related generalized Wasserstein distances allowing for unequal masses, and [5,11] for partial optimal transport where only a given fraction of the prescribed marginals is moved.…”