Convergence of a linearly transformed particle method for aggregation equations

Pinto, Martin Campos; Carrillo, José A.; Charles, Frédérique; Choi, Young-Pil

doi:10.1007/s00211-018-0958-2

Cited by 10 publications

(10 citation statements)

References 74 publications

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“…Another common approach is to leverage structural similarities between (3) and equations from fluid dynamics to develop particle methods [14,27,30,36,43,48,57,60,88,92]. Until recently, the key limitation of such methods has been developing approaches to incorporate diffusion.…”

Section: Classical Numerical Pde Methodsmentioning

confidence: 99%

Primal Dual Methods for Wasserstein Gradient Flows

et al. 2021

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Combining the classical theory of optimal transport with modern operator splitting techniques, we develop a new numerical method for nonlinear, nonlocal partial differential equations, arising in models of porous media, materials science, and biological swarming. Our method proceeds as follows: first, we discretize in time, either via the classical JKO scheme or via a novel Crank–Nicolson-type method we introduce. Next, we use the Benamou–Brenier dynamical characterization of the Wasserstein distance to reduce computing the solution of the discrete time equations to solving fully discrete minimization problems, with strictly convex objective functions and linear constraints. Third, we compute the minimizers by applying a recently introduced, provably convergent primal dual splitting scheme for three operators (Yan in J Sci Comput 1–20, 2018). By leveraging the PDEs’ underlying variational structure, our method overcomes stability issues present in previous numerical work built on explicit time discretizations, which suffer due to the equations’ strong nonlinearities and degeneracies. Our method is also naturally positivity and mass preserving and, in the case of the JKO scheme, energy decreasing. We prove that minimizers of the fully discrete problem converge to minimizers of the spatially continuous, discrete time problem as the spatial discretization is refined. We conclude with simulations of nonlinear PDEs and Wasserstein geodesics in one and two dimensions that illustrate the key properties of our approach, including higher-order convergence our novel Crank–Nicolson-type method, when compared to the classical JKO method.

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Section: Classical Numerical Pde Methodsmentioning

confidence: 99%

Primal Dual Methods for Wasserstein Gradient Flows

et al. 2021

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“…In order to develop methods with higher-order accuracy and capture competing effects in repulsive-attractive systems, recent work has considered enhancements of standard particle methods inspired by techniques from classical fluid dynamics, including vortex blob methods and linearly transformed particle methods [45,79,95]. Bertozzi and the second author's blob method for the aggregation equation obtained a higher order accurate method for singular interaction potentials W by convolving W with a mollifier ϕ ε (x) = ϕ(x/ε)/ε d , ε > 0.…”

Section: Methodsmentioning

confidence: 99%

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Carrillo

Craig

Yao

2019

Modeling and Simulation in Science, Engineering and Technology

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Given a large ensemble of interacting particles, driven by nonlocal interactions and localized repulsion, the mean-field limit leads to a class of nonlocal, nonlinear partial differential equations known as aggregation-diffusion equations. Over the past fifteen years, aggregation-diffusion equations have become widespread in biological applications and have also attracted significant mathematical interest, due to their competing forces at different length scales. These competing forces lead to rich dynamics, including symmetrization, stabilization, and metastability, as well as sharp dichotomies separating well-posedness from finite time blowup. In the present work, we review known analytical results for aggregation-diffusion equations and consider singular limits of these equations, including the slow diffusion limit, which leads to the constrained aggregation equation, and localized aggregation and vanishing diffusion limits, which lead to metastability behavior. We also review the range of numerical methods available for simulating solutions, with special attention devoted to recent advances in deterministic particle methods. We close by applying such a method -the blob method for diffusion -to showcase key properties of the dynamics of aggregation-diffusion equations and related singular limits.Aggregation-diffusion equations: dynamics, asymptotics, and singular limits 5 2 Well-posedness, steady states, and dynamicsWe now give a brief overview of analytical results for aggregation diffusion equations (4), describing conditions that ensure well-posedness or finite-time blow-up of solutions , existence or non-existence of steady states, and long time behavior of solutions. We begin in section 2.1 by considering the classical two-dimensional Keller-Segel equation, where the interaction kernel in equation (4) is given by the Newtonian potential, W (x) = 1 2π ln |x|. The analysis in this particular case will serve as a guideline to understand the equation's behavior for more general interaction kernels. In section 2.2, we review results classifying well-posedness and finite blow-up for general interaction kernels. Finally, in section 2.3, we discuss the steady states and dynamics of solutions in the diffusion-dominated regime. ρ λ (x) := λ 2 ρ(λ x), λ 1.Then, E[ρ λ ] and E[ρ] are related in the following way:

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“…These schemes are reminiscent of the convex splitting ideas in the variational L 2 framework, as developed in [55]; however, they are not directly applicable in the setting of gradient flows with respect to measures. Other numerical schemes used for nonlinear Fokker-Planck equations include finite element schemes [15], particle/blob methods [40,19,23], and methods based on the gradient flow formulation in terms of steepest descents with euclidean transport distances [10,35,51,47,36,28].…”

Section: Introductionmentioning

confidence: 99%

Fully Discrete Positivity-Preserving and Energy-Dissipating Schemes for Aggregation-Diffusion Equations with a Gradient Flow Structure

Bailo,

Carrillo,

2018

Preprint

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We propose fully discrete, implicit-in-time finite volume schemes for general nonlinear nonlocal Fokker-Planck type equations with a gradient flow structure, usually referred to as aggregation-diffusion equations, in any dimension. The schemes enjoy the positivitypreserving and energy-decaying properties, essential for their practical use. The first-order in time and space scheme unconditionally verifies these properties for general nonlinear diffusion and interaction potentials while the second-order scheme does so provided a CFL condition holds. Dimensional splitting allows for the construction of these schemes in higher dimensions with the same properties and a reduced computational cost. Numerical experiments validate the schemes and show their ability to handle complicated phenomena in aggregation-diffusion equations such as free boundaries, metastability, merging and phase transitions.

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Convergence of a linearly transformed particle method for aggregation equations

Cited by 10 publications

References 74 publications

Primal Dual Methods for Wasserstein Gradient Flows

Primal Dual Methods for Wasserstein Gradient Flows

Aggregation-Diffusion Equations: Dynamics, Asymptotics, and Singular Limits

Fully Discrete Positivity-Preserving and Energy-Dissipating Schemes for Aggregation-Diffusion Equations with a Gradient Flow Structure

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