The back-and-forth method for Wasserstein gradient flows

Jacobs, Matt; Lee, Wonjun; Léger, Flavien

doi:10.1051/cocv/2021029

“…As we mentioned above, we obtain this result via a variational approximation scheme, which is a simplified version of the more general scheme introduced in [JKT21]. Although there are many different ways to establish the well-posedness of the system (such as the degenerate diffusion approach considered in [PQV14] and [GKM22]), we emphasize our variational scheme, as it has a very efficient numerical implementation via the Back-and-Forth method [JL20,JLL21]. For instance, the images displayed in Figure 1 are computed on a high-resolution 1024 × 1024 grid.…”

Section: Summary Of Main Resultsmentioning

confidence: 99%

Tumor Growth with Nutrients: Regularity and Stability

Jacobs¹,

Kim²,

Tong³

2022

Preprint

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In this paper we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the contact inhibition among the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics. In particular, we provide contraction estimates, exponential rate of asymptotic convergence, and boundary regularity of the tumor patch. These results are in sharp contrast to the models either with nutrient diffusion or with death rate in tumor cells.

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“…As we mentioned above, we obtain this result via a variational approximation scheme, which is a simplified version of the more general scheme introduced in [JKT21]. Although there are many different ways to establish the well-posedness of the system (such as the degenerate diffusion approach considered in [PQV14] and [GKM22]), we emphasize our variational scheme, as it has a very efficient numerical implementation via the Back-and-Forth method [JL20,JLL21]. For instance, the images displayed in Figure 1 are computed on a high-resolution 1024 × 1024 grid.…”

Section: 𝜓(𝑛 − 𝑏)𝜌 𝑑𝑥 𝑑𝑡;mentioning

confidence: 96%

Tumor growth with nutrients: Regularity and stability

Jacobs

¹

,

Kim

²

,

Tong

³

2023

Comm. Amer. Math. Soc.

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4

0

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In this paper, we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the incompressibility of the tumor cells. We show that when the nutrients do not diffuse and the cells do not die, the tumor density exhibits regularizing dynamics thanks to an unexpected comparison principle. Using the comparison principle, we provide quantitative L 1 L^1 -contraction estimates and establish the C 1 , α C^{1,\alpha } -boundary regularity of the tumor patch. Furthermore, whenever the initial nutrient n 0 n_0 either lies entirely above or entirely below the critical value n 0 = 1 n_0=1 , we are able to give a complete characterization of the long-time behavior of the system. When n 0 n_0 is constant, we can even describe the dynamics of the full system in terms of some simpler nutrient-free and parameter-free model problems. These results are in sharp contrast to the observed behavior of the models either with nutrient diffusion or with death rate in tumor cells.

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“…The advantage of this perspective is that our scheme can approximate any flow of the form (P ). Furthermore, the dual problem associated to our scheme has a very efficient numerical implementation using the recently introduced backand-forth method [JL19,JWL]. In particular, the numerical implementation via the back-andforth method does not require introducing an additional time dimension, which allows for a faster computation time than schemes based around the Benamou-Brenier formula.…”

Section: Introductionmentioning

confidence: 99%

Darcy's Law with a Source term

Jacobs,

Kim,

Tong

2020

Preprint

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0

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We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source term, our scheme is still able to use the standard Wasserstein-2-metric even though the total mass changes over time. Leveraging the dual formulation of our scheme, we show that the discrete-intime approximations satisfy many useful properties expected for the continuum solutions, such as a comparison principle and uniform L 1 -equicontinuity. Many of these properties are new even in the well-understood case where the growth term is absent. Finally, we show that our discrete approximations converge to a solution of the corresponding PDE system, including a tumor growth model with a general nonlinear source term.

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The back-and-forth method for Wasserstein gradient flows

Cited by 9 publications

References 21 publications

Tumor Growth with Nutrients: Regularity and Stability

Tumor Growth with Nutrients: Regularity and Stability

Tumor growth with nutrients: Regularity and stability

Darcy's Law with a Source term

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