模拟退火法进化玻耳兹曼机的实现策略

玻耳兹曼机是较新的并行约束网络结构,在优化技术中有着其他技术无法比拟的优势。用模拟退火法对玻耳兹曼机进行进化,改善其搜索性能,突出其优势。通过理论分析和实践,取得了较好的效果,得出了有意义的结论。     

基于玻耳兹曼熵分析的图像分割方法研究*

根据热力学玻耳兹曼熵关系式,定义基于图像灰度谱的玻耳兹曼熵谱,将图像空间局部结构隐含于灰度谱的客观事实与玻耳兹曼熵谱联系在一起。最后在像素近邻空间进行统计计算,通过识别玻耳兹曼熵谱特征,实现图像分割。实验与比较分析表明,该方法具有显著优势。     

传输方程下基于模型的光学层析图像重建

为克服光学层析图像重建的病态性,采用一种基于模型的重建方法来进行图像重建。由于广义高斯马尔可夫随机场模型具有全局平滑、边缘保留等特性,因此将其引入到服从辐射传输方程的光学层析图像重建中,并将其作为图像先验信息,同时通过最大后验概率理论,利用基于梯度的迭代优化算法来对目标函数进行优化求解。鉴于目标函数关于光学参数的梯度计算是算法中的难点,对此,提出了一种基于梯度树的梯度计算方法。实验证明:该方法与不带有先验模型的重建方法相比,不仅可进一步提高图像的重建质量,而且可降低重建病态性。     

基于梯度树的光学层析正则化重建

光学层析成像是一个病态重建过程,为降低重建过程中的病态特性,需加入合适的先验信息。目前,大多数重建都是基于扩散方程的,在某些情况下,这种重建会失败。直接基于玻耳兹曼传输模型,并以图像熵为正则化项的梯度迭代重建是一种有效的方法。该方法中,梯度计算是个难点。对此,提出一种基于梯度树的求解方法,降低光学层析图像重建的病态性,有效地重建光学层析图像。     

Normal,high order discrete velocity models of the Boltzmann equation

This paper aims at approximations of the collision operator in the Boltzmann equation. The developed framework guarantees the “normality” of the approximation, which means correct collision invariants, H-Theorem, and equilibrium solutions. It fits into the discrete velocity model framework, is given in such a way that it is understandable with undergraduate level mathematics and can be used to construct approximations with arbitrary high convergence orders. At last we give an example alongside a numerical verification. Here the convergence orders range up to $3$ ($2$) and the time complexity is given by $3+\frac{1}{2}$ ($4+\frac{2}{3}$) in $2$ ($3$) dimensions.     

Fully Lagrangian and Lattice Boltzmann Methods for the Advection-Diffusion Equation

In the last decade or so, the Lattice–Boltzmann method (LBM) has achieved great success in computational fluid dynamics. The Fully–Lagrangian method (FLM) is the generalization of LBM for conservation systems. LBM can also be developed from FLM. In this paper a FL model and a LB model are developed for D-dimensional advection-diffusion equation. The LB model can be viewed as an improved version of the FL model. Numerical results of simulation of 1-dimensional advection-diffusion equation are presented. The numerical results are found to be in good agreement with the analytic solution.     

An analysis of natural convection using the thermal finite element discrete Boltzmann equation

The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE.     

一类偏微分方程的格子Boltzmann模型

研究了对流扩散方程、Burgers方程和Modified-Burgers方程等具有相同形式的一类偏微分方程。并且构建了带修正函数项的D1Q3格子Boltzmann模型求解这类方程。为了能准确地恢复出此宏观方程,利用Chapman-Enskog展开和多尺度分析技术,推导出了各个方向的平衡态分布函数和修正函数的具体表达式。数值计算结果表明该模型是稳定、有效的。     

An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation

In this paper, we study a time discrete scheme for the initial value problem of the ES-BGK kinetic equation. Numerically solving these equations are challenging due to the nonlinear stiff collision (source) terms induced by small mean free or relaxation time. We study an implicit-explicit (IMEX) time discretization in which the convection is explicit while the relaxation term is implicit to overcome the stiffness. We first show how the implicit relaxation can be solved explicitly, and then prove asymptotically that this time discretization drives the density distribution toward the local Maxwellian when the mean free time goes to zero while the numerical time step is held fixed. This naturally imposes an asymptotic-preserving scheme in the Euler limit. The scheme so designed does not need any nonlinear iterative solver for the implicit relaxation term. Moreover, it can capture the macroscopic fluid dynamic (Euler) limit even if the small scale determined by the Knudsen number is not numerically resolved. We also show that it is consistent to the compressible Navier-Stokes equations if the viscosity and heat conductivity are numerically resolved. Several numerical examples, in both one and two space dimensions, are used to demonstrate the desired behavior of this scheme.     

Application of the Boltzmann kinetic equation to the eddy problems

The main purpose of this work is to investigate the feasibility of applying a kinetic approach to the problem of modeling turbulent and unstable flows. First, initial value problems with the Taylor–Green (TG) type and isotropic velocity conditions for compressible flow in two-dimensional (2D) and three-dimensional (3D) periodic domains are considered. Further, 3D direct numerical simulation of decaying isotropic turbulence is performed. Macroscopic flow quantities of interest are examined. The simulation is based on the direct numerical solution of the Boltzmann kinetic equation using an explicit–implicit scheme for the relaxation stage. Comparison with the solution of the Bhatnagar–Gross–Krook (BGK) model equation obtained by using an implicit scheme is carried out for the decaying isotropic turbulence problem and demonstrates a small difference. For the TG initial condition results show a fragmentation of the large initial eddies and subsequently the full damping of the system. Numerical data are close to the analytic solution of TG problem. A dependence of the kinetic energy on the wave number is obtained by means of the Fourier expansion of velocity components. A power-law exponent for the kinetic energy spectrum tends to the theoretical value “−3” for 2D turbulence in 2D case and to the famous Kolmogorov value “−5/3” in 3D case.     

A Galerkin Method for the Simulation of the Transient 2-D/2-D and 3-D/3-D Linear Boltzmann Equation

Many production steps used in the manufacturing of integrated circuits involve the deposition of material from the gas phase onto wafers. Models for these processes should account for gaseous transport in a range of flow regimes, from continuum flow to free molecular or Knudsen flow, and for chemical reactions at the wafer surface. We develop a kinetic transport and reaction model whose mathematical representation is a system of transient linear Boltzmann equations. In addition to time, a deterministic numerical solution of this system of kinetic equations requires the discretization of both position and velocity spaces, each two-dimensional for 2-D/2-D or each three-dimensional for 3-D/3-D simulations. Discretizing the velocity space by a spectral Galerkin method approximates each Boltzmann equation by a system of transient linear hyperbolic conservation laws. The classical choice of basis functions based on Hermite polynomials leads to dense coefficient matrices in this system. We use a collocation basis instead that directly yields diagonal coefficient matrices, allowing for more convenient simulations in higher dimensions. The systems of conservation laws are solved using the discontinuous Galerkin finite element method. First, we simulate chemical vapor deposition in both two and three dimensions in typical micron scale features as application example. Second, stability and convergence of the numerical method are demonstrated numerically in two and three dimensions. Third, we present parallel performance results which indicate that the implementation of the method possesses very good scalability on a distributed-memory cluster with a high-performance Myrinet interconnect.     

基于格子Boltzmann方法的一维Burgers方程的数值模拟

基于格子Boltzmann方法(LBM)的一维Burgers方程的数值解法,已有2-bit和4-bit模型。文中通过选择合适的离散速度模型构造出恰当的平衡态分布函数; 然后, 利用单松弛的格子Bhatnagar-Gross-Krook模型、Chapman-Enskog展开和多尺度技术, 提出了用于求解一维Burgers方程的3-bit的格子Boltzmann模型,即D1Q3模型,并进行了数值实验。实验结果表明,该方法的数值解与解析解吻合的程度很好,且误差比现有文献中的误差更小,从而验证了格子Boltzamnn模型的有效性。