<i>Γ</i>-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

Fleißner, Florentine

doi:10.1051/cocv/2017035

Cited by 9 publications

(10 citation statements)

References 15 publications

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“…Conditions which ensure the convergence of the discrete solutions to a curve of maximal slope for the Γ-limit of the energies, as τ and ε tend to zero, was exhibited in particular cases, for instance by Sandier and Serfaty in [13] and Colombo and Gobbino in [8]. While a wider treatment has been given by Braides, Colombo, Gobbino, and Solci in [7], or by Fleissner in [10]. The general case may present different limits, corresponding to the relation between the two small parameters ε and τ , as shown by Braides in [6], and precised for oscillating potentials by Ansini, Braides and Zimmer in [2] (see also [1]).…”

Section: Introduction the Methods Of Minimizing Movements Was Introdumentioning

confidence: 99%

Perturbations of minimizing movements and curves of maximal slope

Tribuzio¹

2018

Networks &Amp; Heterogeneous Media

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We modify the De Giorgi's minimizing movements scheme for a functional φ, by perturbing the dissipation term, and find a condition on the perturbations which ensures the convergence of the scheme to an absolutely continuous perturbed minimizing movement. The perturbations produce a variation of the metric derivative of the minimizing movement. This process is formalized by the introduction of the notion of curve of maximal slope for φ with a given rate. We show that if we relax the condition on the perturbations we may have many different meaningful effects; in particular, some perturbed minimizing movements may explore different potential wells.

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Section: Introduction the Methods Of Minimizing Movements Was Introdumentioning

confidence: 99%

Perturbations of minimizing movements and curves of maximal slope

Tribuzio¹

2018

Networks &Amp; Heterogeneous Media

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“…Proof of Theorem 2.4. Taking the limit in the energy estimate (12), thanks to the conditions of Colombo-Gobbino, and inequalities (15) and (16), we have…”

Section: Remarkmentioning

confidence: 99%

“…In the present paper we offer a contribution to the general problem of understanding the interaction between energy and dissipation terms in variational approaches to gradient-flow type evolutions from the standpoint of minimizing movements (see also e.g. [11,12,13] for related work).…”

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confidence: 99%

Perturbed minimizing movements of families of functionals

Braides¹,

Tribuzio²

2021

Discrete &Amp; Continuous Dynamical Systems - S

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We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a (Γ-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters ε and τ , governing energy and time scales, respectively. We characterize the extreme cases when ε/τ and τ /ε converges to 0 sufficiently fast, and exhibit a sufficient condition that guarantees that the limit is indeed independent of ε and τ . We give examples showing that this in general is not the case, and apply this approach to study some discrete approximations, the homogenization of wiggly energies and geometric crystalline flows obtained as limits of ferromagnetic energies.

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“…Implementing the splitting scheme for the regular energy functional F m (ρ) −´ρ gives a sequence ρ h,m , and we shall prove below that ρ h,m converges to a solution of the limiting gradient flow as m → ∞ and h → 0. However, it is known [17] that the limit depends in general on the interplay between the time-step h and the regularization parameter (m → ∞ here), and for technical reasons we shall enforce the condition mh → 0 as m → ∞ and h → 0.…”

Section: Application To a Tumor Growth Model With Very Degenerate Enerymentioning

confidence: 99%

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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Γ-convergence and relaxations for gradient flows in metric spaces: a minimizing movement approach

Cited by 9 publications

References 15 publications

Perturbations of minimizing movements and curves of maximal slope

Perturbations of minimizing movements and curves of maximal slope

Perturbed minimizing movements of families of functionals

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

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