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Haijun Yu

Publications and source records attributed to Haijun Yu.

46 records · Page 3Linked to original sources
arXiv2019

Application of Bounded Total Variation Denoising in Urban Traffic Analysis

While it is believed that denoising is not always necessary in many big data applications, we show in this paper that denoising is helpful in urban traffic analysis by applying the method of bounded total variation denoising to the urban road traffic prediction and clustering problem. We propose two easy-to-implement methods to estimate the noise strength parameter in the denoising algorithm, and apply the denoising algorithm to GPS-based traffic data from Beijing taxi system. For the traffic prediction problem, we combine neural network and history matching method for roads randomly chosen from an urban area of Beijing. Numerical experiments show that the predicting accuracy is improved significantly by applying the proposed bounded total variation denoising algorithm. We also test the algorithm on clustering problem, where a recently developed clustering analysis method is applied to more than one hundred urban road segments in Beijing based on their velocity profiles. Better clustering result is obtained after denoising.

cs.LG↗
arXiv2018

Energy Stable Second Order Linear Schemes for the Allen-Cahn Phase-Field Equation

Phase-field model is a powerful mathematical tool to study the dynamics of interface and morphology changes in fluid mechanics and material sciences. However, numerically solving a phase field model for a real problem is a challenge task due to the non-convexity of the bulk energy and the small interface thickness parameter in the equation. In this paper, we propose two stabilized second order semi-implicit linear schemes for the Allen-Cahn phase-field equation based on backward differentiation formula and Crank-Nicolson method, respectively. In both schemes, the nonlinear bulk force is treated explicitly with two second-order stabilization terms, which make the schemes unconditional energy stable and numerically efficient. By using a known result of the spectrum estimate of the linearized Allen-Cahn operator and some regularity estimate of the exact solution, we obtain an optimal second order convergence in time with a prefactor depending on the inverse of the characteristic interface thickness only in some lower polynomial order. Both 2-dimensional and 3-dimensional numerical results are presented to verify the accuracy and efficiency of proposed schemes.

math.NA↗
arXiv2018

Quasi-potential Calculation and Minimum Action Method for Limit Cycle

We study the noise-induced escape from a stable limit cycle of a non-gradient dynamical system driven by a small additive noise. The fact that the optimal transition path in this case is infinitely long imposes a severe numerical challenge to resolve it in the minimum action method. We first consider the landscape of the quasi-potential near the limit cycle, which characterizes the minimal cost of the noise to drive the system far away form the limit cycle. We derive and compute the quadratic approximation of this quasi-potential near the limit cy- cle in the form of a positive definite solution to a matrix-valued periodic Riccati differential equation on the limit cycle. We then combine this local approxima- tion in the neighbourhood of the limit cycle with the minimum action method applied outside of the neighbourhood. The neighbourhood size is selected to be compatible with the path discretization error. By several numerical examples, we show that this strategy effectively improve the minimum action method to compute the spiral optimal escape path from limit cycles in various systems.

math.OC↗
arXiv2018

On Efficient Second Order Stabilized Semi-Implicit Schemes for the Cahn-Hilliard Phase-Field Equation

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank-Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.

math.NA↗
arXiv2018

Convergence analysis of a finite element approximation of minimum action methods

In this work, we address the convergence of a finite element approximation of the minimizer of the Freidlin-Wentzell (F-W) action functional for non-gradient dynamical systems perturbed by small noise. The F-W theory of large deviations is a rigorous mathematical tool to study small-noise-induced transitions in a dynamical system. The central task in the application of F-W theory of large deviations is to seek the minimizer and minimum of the F-W action functional. We discretize the F-W action functional using linear finite elements, and establish the convergence of {the approximation} through $Γ$-convergence.

math.NA↗
arXiv2017

Sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact lines

The sharp-interface limits of a phase-field model with a generalized Navier slip boundary condition for moving contact line problem are studied by asymptotic analysis and numerical simulations. The effects of the {mobility} number as well as a phenomenological relaxation parameter in the boundary condition are considered. In asymptotic analysis, we focus on the case that the {mobility} number is the same order of the Cahn number and derive the sharp-interface limits for several setups of the boundary relaxation parameter. It is shown that the sharp interface limit of the phase field model is the standard two-phase incompressible Navier-Stokes equations coupled with several different slip boundary conditions. Numerical results are consistent with the analysis results and also illustrate the different convergence rates of the sharp-interface limits for different scalings of the two parameters.

math.AP↗
arXiv2017

A Laguerre homotopy method for optimal control of nonlinear systems in semi-infinite interval

This paper presents a Laguerre homotopy method for optimal control problems in semi-infinite intervals (LaHOC), with particular interests given to nonlinear interconnected large-scale dynamic systems. In LaHOC, spectral homotopy analysis method is used to derive an iterative solver for the nonlinear two-point boundary value problem derived from Pontryagins maximum principle. A proof of local convergence of the LaHOC is provided. Numerical comparisons are made between the LaHOC, Matlab BVP5C generated results and results from literature for two nonlinear optimal control problems. The results show that LaHOC is superior in both accuracy and efficiency.

math.NA↗
arXiv2017

Convergence Analysis of an Unconditionally Energy Stable Linear Crank-Nicolson Scheme for the Cahn-Hilliard Equation

Efficient and unconditionally stable high order time marching schemes are very important but not easy to construct for nonlinear phase dynamics. In this paper, we propose and analysis an efficient stabilized linear Crank-Nicolson scheme for the Cahn-Hilliard equation with provable unconditional stability. In this scheme the nonlinear bulk force are treated explicitly with two second-order linear stabilization terms. The semi-discretized equation is a linear elliptic system with constant coefficients, thus robust and efficient solution procedures are guaranteed. Rigorous error analysis show that, when the time step-size is small enough, the scheme is second order accurate in time with aprefactor controlled by some lower degree polynomial of $1/\varepsilon$. Here $\varepsilon$ is the interface thickness parameter. Numerical results are presented to verify the accuracy and efficiency of the scheme.

math.NA↗
arXiv2017

Numerical Approximations for a Phase-Field Moving Contact Line Model with Variable Densities and Viscosities

We consider the numerical approximations of a two-phase hydrodynamics coupled phase-field model that incorporates the variable densities, viscosities and moving contact line boundary conditions. The model is a nonlinear, coupled system that consists of incompressible Navier--Stokes equations with the generalized Navier boundary condition, and the Cahn--Hilliard equations with moving contact line boundary conditions. By some subtle explicit--implicit treatments to nonlinear terms, we develop two efficient, unconditionally energy stable numerical schemes, in particular, a linear decoupled energy stable scheme for the system with static contact line condition, and a nonlinear energy stable scheme for the system with dynamic contact line condition. An efficient spectral-Galerkin spatial discretization is implemented to verify the accuracy and efficiency of proposed schemes. Various numerical results show that the proposed schemes are efficient and accurate.

math.NA↗