Browsing by Author "Liu, JG"

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    A stochastic version of Stein Variational Gradient Descent for efficient sampling
    Li, L; Li, Y; Liu, JG; Liu, Z; Lu, J
    We propose in this work RBM-SVGD, a stochastic version of Stein Variational Gradient Descent (SVGD) method for efficiently sampling from a given probability measure and thus useful for Bayesian inference. The method is to apply the Random Batch Method (RBM) for interacting particle systems proposed by Jin et al to the interacting particle systems in SVGD. While keeping the behaviors of SVGD, it reduces the computational cost, especially when the interacting kernel has long range. Numerical examples verify the efficiency of this new version of SVGD.
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    Analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit
    (SIAM Journal on Numerical Analysis, 2010-09-23) Liu, JG; Mieussens, L
    We present a mathematical analysis of the asymptotic preserving scheme proposed in [M. Lemou and L. Mieussens, SIAM J. Sci. Comput., 31 (2008), pp. 334-368] for linear transport equations in kinetic and diffusive regimes. We prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles. This property is satisfied under an explicitly given CFL condition. This condition tends to a parabolic CFL condition for small mean free paths and is close to a convection CFL condition for large mean free paths. Our analysis is based on very simple energy estimates. © 2010 Society for Industrial and Applied Mathematics.
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    Asymmetry in crystal facet dynamics of homoepitaxy by a continuum model
    (2017-04-23) Liu, JG; Lu, J; Margetis, D; Marzuola, JL
    In the absence of external material deposition, crystal surfaces usually relax to become flat by decreasing their free energy. We study an asymmetry in the relaxation of macroscopic plateaus, facets, of a periodic surface corrugation in 1+1 dimensions via a continuum model below the roughening transition temperature. The model invokes a highly degenerate parabolic partial differential equation (PDE) for surface diffusion, which is related to the weighted-$H^{-1}$ (nonlinear) gradient flow of a convex, singular surface free energy in homoepitaxy. The PDE is motivated both by an atomistic broken-bond model and a mesoscale model for steps. By constructing an explicit solution to the PDE, we demonstrate the lack of symmetry in the evolution of top and bottom facets in periodic surface profiles. Our explicit, analytical solution is compared to numerical simulations of the PDE via a regularized surface free energy.
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    Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces
    (Journal of Nonlinear Science, 2016-12-29) Gao, Y; Liu, JG; Lu, J
    This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton–Cabrera–Frank-type model following the Xiang’s work (Xiang in SIAM J Appl Math 63(1):241–258, 2002). We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first-order convergence rate.
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    Data clustering based on Langevin annealing with a self-consistent potential
    (Quarterly of Applied Mathematics, 2018-10-11) Lafata, K; Zhou, Z; Liu, JG; Yin, FF
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    Fractional Stochastic Differential Equations Satisfying Fluctuation-Dissipation Theorem
    (Journal of Statistical Physics, 2017-10-01) Li, L; Liu, JG; Lu, J
    © 2017, Springer Science+Business Media, LLC. We propose in this work a fractional stochastic differential equation (FSDE) model consistent with the over-damped limit of the generalized Langevin equation model. As a result of the ‘fluctuation-dissipation theorem’, the differential equations driven by fractional Brownian noise to model memory effects should be paired with Caputo derivatives, and this FSDE model should be understood in an integral form. We establish the existence of strong solutions for such equations and discuss the ergodicity and convergence to Gibbs measure. In the linear forcing regime, we show rigorously the algebraic convergence to Gibbs measure when the ‘fluctuation-dissipation theorem’ is satisfied, and this verifies that satisfying ‘fluctuation-dissipation theorem’ indeed leads to the correct physical behavior. We further discuss possible approaches to analyze the ergodicity and convergence to Gibbs measure in the nonlinear forcing regime, while leave the rigorous analysis for future works. The FSDE model proposed is suitable for systems in contact with heat bath with power-law kernel and subdiffusion behaviors.
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    Learning interacting particle systems: diffusion parameter estimation for aggregation equations
    (2018-02-14) Huang, H; Liu, JG; Lu, J
    In this article, we study the parameter estimation of interacting particle systems subject to the Newtonian aggregation. Specifically, we construct an estimator $\widehat{\nu}$ with partial observed data to approximate the diffusion parameter $\nu$, 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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    On the mean-field limit for the Vlasov-Poisson-Fokker-Planck system
    Huang, H; Liu, JG; Pickl, P
    We devise and study a random particle blob method for approximating the Vlasov-Poisson-Fokkker-Planck (VPFP) equations by a $N$-particle system subject to the Brownian motion in $\mathbb{R}^3$ space. More precisely, we show that maximal distance between the exact microscopic and the mean-field trajectories is bounded by $N^{-\frac{1}{3}+\varepsilon}$ ($\frac{1}{63}\leq\varepsilon<\frac{1}{36}$) for a system with blob size $N^{-\delta}$ ($\frac{1}{3}\leq\delta<\frac{19}{54}-\frac{2\varepsilon}{3}$) up to a probability $1-N^{-\alpha}$ for any $\alpha>0$, which improves the cut-off in [10]. Our result thus leads to a derivation of VPFP equations from the microscopic $N$-particle system. In particular we prove the convergence rate between the empirical measure associated to the particle system and the solution of the VPFP equations. The technical novelty of this paper is that our estimates crucially rely on the randomness coming from the initial data and from the Brownian motion.
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    Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime
    (2017-04-23) Gao, Y; Liu, JG; Lu, J
    We study in this work a continuum model derived from 1D attachment-detachment-limited (ADL) type step flow on vicinal surface, $ u_t=-u^2(u^3)_{hhhh}$, where $u$, considered as a function of step height $h$, is the step slope of the surface. We formulate a notion of weak solution to this continuum model and prove the existence of a global weak solution, which is positive almost everywhere. We also study the long time behavior of weak solution and prove it converges to a constant solution as time goes to infinity. The space-time H\"older continuity of the weak solution is also discussed as a byproduct.