 Award ID(s):
 2108628
 NSFPAR ID:
 10345334
 Date Published:
 Journal Name:
 Journal of Dynamics and Differential Equations
 ISSN:
 10407294
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
More Like this

null (Ed.)Abstract We rigorously justify the meanfield limit of an N particle system subject to Brownian motions and interacting through the Newtonian potential in $${\mathbb {R}}^3$$ R 3 . Our result leads to a derivation of the Vlasov–Poisson–Fokker–Planck (VPFP) equations from the regularized microscopic N particle system. More precisely, we show that the maximal distance between the exact microscopic trajectories and the meanfield trajectories is bounded by $$N^{\frac{1}{3}+\varepsilon }$$ N  1 3 + ε ( $$\frac{1}{63}\le \varepsilon <\frac{1}{36}$$ 1 63 ≤ ε < 1 36 ) with a blob size of $$N^{\delta }$$ N  δ ( $$\frac{1}{3}\le \delta <\frac{19}{54}\frac{2\varepsilon }{3}$$ 1 3 ≤ δ < 19 54  2 ε 3 ) up to a probability of $$1N^{\alpha }$$ 1  N  α for any $$\alpha >0$$ α > 0 . Moreover, we prove the convergence rate between the empirical measure associated to the regularized particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates rely on the randomness coming from the initial data and from the Brownian motions.more » « less

Wasserstein gradient flows provide a powerful means of understanding and solving many diffusion equations. Specifically, FokkerPlanck equations, which model the diffusion of probability measures, can be understood as gradient descent over entropy functionals in Wasserstein space. This equivalence, introduced by Jordan, Kinderlehrer and Otto, inspired the socalled JKO scheme to approximate these diffusion processes via an implicit discretization of the gradient flow in Wasserstein space. Solving the optimization problem associated with each JKO step, however, presents serious computational challenges. We introduce a scalable method to approximate Wasserstein gradient flows, targeted to machine learning applications. Our approach relies on inputconvex neural networks (ICNNs) to discretize the JKO steps, which can be optimized by stochastic gradient descent. Contrarily to previous work, our method does not require domain discretization or particle simulation. As a result, we can sample from the measure at each time step of the diffusion and compute its probability density. We demonstrate the performance of our algorithm by computing diffusions following the FokkerPlanck equation and apply it to unnormalized density sampling as well as nonlinear filtering.more » « less

We revisit the variational characterization of conservative di↵usion as entropic gra dient flow and provide for it a probabilistic interpretation based on stochastic calculus. It was shown by Jordan, Kinderlehrer, and Otto that, for diffusions of Langevin–Smoluchowski type, the Fokker–Planck probability density flow maximizes the rate of relative entropy dissipation, as mea sured by the distance traveled in the ambient space of probability measures with finite second moments, in terms of the quadratic Wasserstein metric. We obtain novel, stochasticprocess ver sions of these features, valid along almost every trajectory of the dffusive motion in the backwards direction of time, using a very direct perturbation analysis. By averaging our trajectorial results with respect to the underlying measure on path space, we establish the maximal rate of entropy dissipation along the Fokker–Planck flow and measure exactly the deviation from this maximum that corresponds to any given perturbation. A bonus of our trajectorial approach is that it derives the HWI inequality relating relative entropy (H), Wasserstein distance (W), and relative Fisher information (I).more » « less

Bressan, A ; Lewicka, M ; Wang, D. ; Zheng, Y.X. (Ed.)In this paper we review the algorithm development in high order methods for some conservation laws. The emphasis is on our recent contribution in the study of two model classes: FokkerPlancktype equations and hyperbolic conservation law systems. For the former we will review freeenergysatisfying and positivitypreserving schemes. For the later we will review the general invariantregionpreserving (IRP) limiter, and its application to high order methods for multidimensional hyperbolic systems of conservation laws.more » « less

Abstract To date, there is no consensus on the probability distribution of particle velocities during bedload transport, with some studies suggesting an exponential‐like distribution while others a Gaussian‐like distribution. Yet, the form of this distribution is key for the determination of sediment flux and the dispersion characteristics of tracers in rivers. Combining theoretical analysis of the Fokker‐Planck equation for particle motions, numerical simulations of the corresponding Langevin equation, and measurements of motion in high‐speed imagery from particle‐tracking experiments, we examine the statistics of bedload particle trajectories, revealing a two‐regime distance‐time (
L ‐ ) scaling for the particle hops (measured from start to stop). We show that particles of short hop distances scale asT _{p}L ~giving rise to the Weibull‐like front of the hop distance distribution, while particles of long hop distances transition to a different scaling regime of L ~ leading to the exponential‐like tail of the hop distance distribution. By demonstrating that the predominance of mostly long hop particles results in a Gaussian‐like velocity distribution, while a mixture of both short and long hop distance particles leads to an exponential‐like velocity distribution, we argue that the form of the probability distribution of particle velocities can depend on the physical environment within which particle transport occurs, explaining and unifying disparate views on particle velocity statistics reported in the literature.T _{p}