A JKO Splitting Scheme for Kantorovich--Fisher--Rao Gradient Flows

Gallouët, Thierry; Monsaingeon, Léonard

doi:10.1137/16m106666x

Cited by 35 publications

(39 citation statements)

References 40 publications

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“…in a bounded domain Ω ⊂ R d with Neumann boundary condition and suitable initial conditions. Our goal is to extend to the case F 1 = F 2 , V 1 = V 2 the method initially introduced in [18] for variational WFR-gradient flows, i-e (3.1) with F 1 = F 2 and V 1 = V 2 . We assume for simplicity that F 1 : R → R is given by , (3.2) and F 2 : R → R is given by F 2 (z) = 1 m 2 − 1 z m2 , for some m 2 > 1.…”

Section: An Existence Results For General Parabolic Equationsmentioning

confidence: 99%

“…Our splitting scheme is a variant of that originally introduced in [18], and can be viewed as an operator splitting method: each part of the PDE above is discretized (in time) in its own W, FR metric, and corresponds respectively to a W/transport/diffusion step and to a FR/reaction step. More precisely, let h > 0 be a small time step.…”

Section: An Existence Results For General Parabolic Equationsmentioning

confidence: 99%

“…We use the augmented Lagrangian method ALG-JKO from [6] to solve the Wasserstein step, and the Fisher-Rao step is just a convex pointwise minimization problem. Indeed, it is known [18,27] that FR 2 (ρ, µ) = 4 √ ρ − √ µ 2 L 2 , hence the Fisher-Rao step in (4.6) is a mere convex pointwise minimization problem of the form: for all x ∈ Ω (and omitting all indexes ρ i,h ),…”

Section: Numerical Application: Prey-predator Systemsmentioning

confidence: 99%

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

2019

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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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Section: An Existence Results For General Parabolic Equationsmentioning

confidence: 99%

Section: An Existence Results For General Parabolic Equationsmentioning

confidence: 99%

Section: Numerical Application: Prey-predator Systemsmentioning

confidence: 99%

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

2019

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“…This very interesting distance has been independently introduced by three different groups around the year 2015, and we refer to [31,32,60,63,64] for the different presentations and the different considerations which have been investigated so far. We give here only a very short description of this distance; the names chosen for it were of course different according to the different groups, and we decided to stick to the terminology chosen in [44] where it is called Kantorovich-Fisher-Rao distance to take into account all the contributions.…”

Section: In [84])mentioning

confidence: 99%

“…in [44] a splitting scheme is proposed. More precisely, one can define the following iterative algorithm:…”

Section: In [84])mentioning

confidence: 99%

{Euclidean, metric, and Wasserstein} gradient flows: an overview

2017

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This is an expository paper on the theory of gradient flows, and in particular of those PDEs which can be interpreted as gradient flows for the Wasserstein metric on the space of probability measures (a distance induced by optimal transport). The starting point is the Euclidean theory, and then its generalization to metric spaces, according to the work of Ambrosio, Gigli and Savaré. Then comes an independent exposition of the Wasserstein theory, with a short introduction to the optimal transport tools that are needed and to the notion of geodesic convexity, followed by a precise description of the Jordan-Kinderlehrer-Otto scheme and a sketch of proof to obtain its convergence in the easiest cases. A discussion of which equations are gradient flows PDEs and of numerical methods based on these ideas is also provided. The paper ends with a new, theoretical, development, due to Ambrosio, Gigli, Savaré, Kuwada and Ohta: the study of the heat flow in metric measure spaces.

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A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

2021

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A multi-step Lagrangian scheme at discrete times is proposed for the approximation of a nonlinear continuity equation arising as a mean-field limit of spatially inhomogeneous evolutionary games, describing the evolution of a system of spatially distributed agents with strategies, or labels, whose payoff depends also on the current position of the agents. The scheme is Lagrangian, as it traces the evolution of position and labels along characteristics, and is a multi-step scheme, as it develops on the following two stages: First, the distribution of strategies or labels is updated according to a best performance criterion, and then, this is used by the agents to evolve their position. A general convergence result is provided in the space of probability measures. In the special cases of replicator-type systems and reversible Markov chains, variants of the scheme, where the explicit step in the evolution of the labels is replaced by an implicit one, are also considered and convergence results are provided.

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A JKO Splitting Scheme for Kantorovich--Fisher--Rao Gradient Flows

Cited by 35 publications

References 40 publications

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

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

{Euclidean, metric, and Wasserstein} gradient flows: an overview

A multi-step Lagrangian scheme for spatially inhomogeneous evolutionary games

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