Error estimate of a random particle blob method for the Keller-Segel equation

Huang, Hui; Liu, Jian Guo

doi:10.1090/mcom/3174

Cited by 25 publications

(41 citation statements)

References 19 publications

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“…Thus, to show the convergence result in Theorem 1, it suffices to estimate the distance between the regularized empirical measureρ h (x, t) of the self-consistent system and the regularized empirical measure ρ h,δ h (x, t) of the regularized interacting particle system. Fortunately, the results recently proved in [14] give us exactly the same estimation as in (14), between of the l p h distance betweenX h,i (s) and X h,i,δ h (s), when the function F 0 is Coulomb or Newton Interactions. Thus exactly the same argument as in Section 2 will finish the proof for those systems.…”

Section: Newton and Coulomb Interactionssupporting

confidence: 61%

“…Moreover, according to exactly the same argument, see Remark 1.1 in [14], let δ h be the same as defined in (18), there is a p > d/(1 − κ), and h sufficiently small, there exists two positive constants C and C ′ depending on T max , p, d and ρ 0 and the diameter of D. The following estimate holds true:…”

Section: Newton and Coulomb Interactionsmentioning

confidence: 92%

“…Noting that the inequality above has the same form as (14), then the argument in Section 2 gives the estimation betweenX h,i (s) and X h,i,δ h (s) and gives us the following corollary on the convergence of system with Newton interaction:…”

Section: Newton and Coulomb Interactionsmentioning

confidence: 99%

“…On the other hand, in [5], where Chorin first introduced the vortex method in 1973, initial positions of the vertices are assumed to be on the lattice points hi ∈ R 2 with a weight function ρ 0 (hi)h 2 determined by the initial density. This way of sampling has been used in [22] and more recently, in [14], and it will be the initial condition we use in this paper.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of stochastic interacting particle systems in probability under a Sobolev norm

Liu

Zhang

2016

Annals of Mathematical Sciences and Applications

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In this paper, we consider particle systems with interaction and Brownian motion. We prove that when the initial data is from the sampling of Chorin's method, i.e., the initial vertices are on lattice points hi ∈ R d with mass ρ 0 (hi)h d , where ρ 0 is some initial density function, then the regularized empirical measure of the interacting particle system converges in probability to the corresponding mean-field partial differential equation with initial density ρ 0 , under the Sobolev norm of L ∞ (L 2 ) ∩ L 2 (H 1 ). Our result is true for all those systems when the interacting function is bounded, Lipschitz continuous and satisfies certain regular condition. And if we further regularize the interacting particle system, it also holds for some of the most important systems of which the interacting functions are not. For systems with repulsive Coulomb interaction, this convergence holds globally on any interval [0, t]. And for systems with attractive Newton force as interacting function, we have convergence within the largest existence time of the regular solution of the corresponding Keller-Segel equation.

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Section: Newton and Coulomb Interactionssupporting

confidence: 61%

Section: Newton and Coulomb Interactionsmentioning

confidence: 92%

Section: Newton and Coulomb Interactionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convergence of stochastic interacting particle systems in probability under a Sobolev norm

Liu

Zhang

2016

Annals of Mathematical Sciences and Applications

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“…The last few years have seen great progress in mean-field limits for singular forces by treating them with an N -dependent cut-off. In particular, the mean-field limit for the Keller-Segel model has been rigorously proved in [21,23,24,30,31]. And the deterministic particle method for aggregation equations can be found in [11,13].…”

Section: Introductionmentioning

confidence: 99%

Learning interacting particle systems: Diffusion parameter estimation for aggregation equations

Huang

Liu

2019

Math. Models Methods Appl. Sci.

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In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation and Brownian diffusion. Specifically, we construct an estimator ν with partial observed data to approximate the diffusion parameter ν, and the estimation error is achieved. Furthermore, we extend this result to general aggregation equations with a bounded Lipschitz interaction field.

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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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Error estimate of a random particle blob method for the Keller-Segel equation

Cited by 25 publications

References 19 publications

Convergence of stochastic interacting particle systems in probability under a Sobolev norm

Convergence of stochastic interacting particle systems in probability under a Sobolev norm

Learning interacting particle systems: Diffusion parameter estimation for aggregation equations

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

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