带状方程组二叉树MIMD算法

利用行处理法和分治策略给出一个基于分布式存储MIMD二叉树随机模型求解任意带状方程组的并行迭代算法，证明算法对相容性带状方程组收敛并分析算法的通信复杂度。     

λ算法在轴对称情况的改进

本文给出了一种改进的λ算法。该算法可以用于计算二维流动和轴对称流动。在导出了以广义黎曼变量表示的欧拉方程组后，本文用两步两点迎风格式对方程组离散求解。由于采用了广义黎曼变量表示的方程组，离散求解时充分考虑了变量的信息依赖域，所以算法稳定性好；由于仅用到待算点单侧一点的构造格式，所以算法简单，编程容易，同时也节省了计算时间。作为算例，我们对钝体超音绕流进行了广泛计算，结果表明，本算法速度快，稳定性好     

开敞式蜗壳轴流式水轮机湍流计算及性能预估

本文在进行模型水轮机试验的基础上,完成了开敞式蜗壳轴流式模型水轮机的全流道三维定常湍流计算。计算程序以雷诺时均连续方程和N-S方程作为控制方程,选用标准κ-ε双方程湍流模型使方程组封闭。数值求解通过CIMPLEC算法实现速度、压力的分离求解,离散差分格式具有二阶精度。通过计算获得了各过流部件的流动细节,预估了水轮机的能量性能,并与试验结果进行了比较。     

模拟退火算法在化学自由能模型中的应用

对化学自由能模型进行系统性的研究，着重分析化学反应动态平衡条件下粒子组分的求解方案，提出应用模拟退火算法寻找自由能密度函数极小值点的求解方案.该方案同时解决了两个难题：1）在一级相变区化学势平衡方程组可能遇到多个解而无法甄别其物理意义.通过模拟退火算法定位到自由能密度函数曲面的最低点，因而可从多个解中甄别出稳定的热力学平衡态.2）模拟退火算法用随机的"热涨落"消除初值敏感性，因而可采用同一套初值计算不同的温度密度点，为实现宽区域上大量温度密度点的连续快速计算奠定基础.作为该平衡态求解方案的应用，计算氦流体在"等离子相变"区的物态方程，揭示了丰富的"等离子相变"现象，并与第一性原理计算揭示的氢流体"液液相变"现象进行类比.     

辐射输运与电子能量强耦合源项的一种高效计算方法

本文研究辐射输运和电子能量耦合方程组的数值方法.在具有光性厚特征的应用问题中,这两类方程的耦合源项表现出强刚性,使得设计稳健高效的数值格式陷入困难.针对辐射输运多群模型和电子能量的耦合方程组的刚性源项,我们给出一种基于电子温度变化规律拟设(ansatz)的积分算法,其时间步长不受刚性源项限制,从而使得计算效率比传统显式方法或隐式非线性迭代获得本质的提高.在所基于的拟设有效时,算法确保给出具有物理意义的解,数值算例显示其给出的解具有较高的精确度.     

强爆炸火球数值模拟中的算子分裂方法

采用算子分裂方法将辐射流体力学方程组分裂为对流项和刚性源项,设计一种高效求解刚性源项方程组的数值方法.数值实验表明:该方法对时间步长不敏感,计算精度能够满足工程计算要求.在不对方程组做任何近似的情况下,数值给出了较长时间内火球冲击波阵面压力峰值及阵面位置的变化,结果与实际强爆炸中的经验公式吻合较好.     

求解刚性燃烧化学反应系统的Krylov子空间中的指数积分法

为了提高求解大时间步长的刚性燃烧化学反应方程组的效率,本文发展了一种使用Krylov子空间中的指数积分(EIKS)求解的方法。该方法基于多时间尺度法(MTS),使用指数积分格式求解线性化后的化学反应方程组,并使用Krylov子空间方法进行矩阵降维。本文使用MTS-EIKS方法模拟了H_2/CO/空气和CH4/空气的自着火过程,并引入自相关动态自适应化学(CoDAC)机理简化。结果表明,CoDAC-MTS-EIKS的联合简化加速方法可以准确、快速地模拟时间步长为10~(-6)s的自着火问题,在计算大步长刚性燃烧化学反应方程组方面有很好的应用前景。     

基于Fepg系统HTS块材磁悬浮数值计算

文中基于高温超导体临界态模型和E-J本构关系幂指数模型,借助于虚拟电导率和Maxwell微分方程组,推导出了直接求解无限长永磁轨道上方超导块材内部磁场强度H变量的二维偏微分弱解积分形式。在有限元自动生成平台Fepg系统上,结合有限元描述语言,成功地生成了二维超导块材内部电磁数值计算有限元FOR-TRAN求解代码。采用生成的有限元计算代码模拟计算出了一磁悬浮原型系统中的超导块材悬浮力-位移曲线。数值结果显示了超导块材在永磁轨道上方竖直方向移动过程中其内部磁场分布特性。     

线性方程组正交化行处理并行算法

针对求解任意线性代数方程组，本文利用正交化行处理法和分治策略给出一个基于分布式存储多指令多数据流1级q叉树树机模型的并行迭代算法并讨论算法的复杂度。     

高密度比多介质可压缩流动的PPM方法

介绍一种基于流体体积分数模型的界面捕捉方法，数值模拟高密度比及含强激波的多介质可压缩流动，将PPM方法应用到多介质体积分数形式的Euler方程组，用双波近似求解一般刚性气体状态方程Riemann问题，可方便处理界面强剪切的滑移线问题，并给出了水下激波和气泡相互作用以及多相Richtmyer-Meshkov不稳定性的计算结果。     

基于海杂波散射特性的微弱信号检测方法

从研究海杂波的电磁散射特性出发,利用随机微分理论对海杂波的物理特性进行了系统地分析.首先建立了海杂波电磁散射所满足的随机微分方程,然后利用Itô公式得到了海杂波散射信号幅度和相位的扩散过程模型,最后利用海杂波散射信号的规律给出了微弱信号检测的相关处理方法. 关键词： 随机微分方程 海杂波 散射 微弱信号     

Fan sub-equation method for Wick-type stochastic partial differential equations

An improved algorithm is devised for using Fan sub-equation method to solve Wick-type stochastic partial differential equations. Applying the improved algorithm to the Wick-type generalized stochastic KdV equation, we obtain more general Jacobi and Weierstrass elliptic function solutions, hyperbolic and trigonometric function solutions, exponential function solutions and rational solutions.     

D-leaping: Accelerating stochastic simulation algorithms for reactions with delays

We propose a novel, accelerated algorithm for the approximate stochastic simulation of biochemical systems with delays. The present work extends existing accelerated algorithms by distributing, in a time adaptive fashion, the delayed reactions so as to minimize the computational effort while preserving their accuracy. The accuracy of the present algorithm is assessed by comparing its results to those of the corresponding delay differential equations for a representative biochemical system. In addition, the fluctuations produced from the present algorithm are comparable to those from an exact stochastic simulation with delays. The algorithm is used to simulate biochemical systems that model oscillatory gene expression. The results indicate that the present algorithm is competitive with existing works for several benchmark problems while it is orders of magnitude faster for certain systems of biochemical reactions.     

Estimating parameters in stochastic systems: A variational Bayesian approach

This work is concerned with approximate inference in dynamical systems, from a variational Bayesian perspective. When modelling real world dynamical systems, stochastic differential equations appear as a natural choice, mainly because of their ability to model the noise of the system by adding a variation of some stochastic process to the deterministic dynamics. Hence, inference in such processes has drawn much attention. Here a new extended framework is derived that is based on a local polynomial approximation of a recently proposed variational Bayesian algorithm. The paper begins by showing that the new extension of this variational algorithm can be used for state estimation (smoothing) and converges to the original algorithm. However, the main focus is on estimating the (hyper-) parameters of these systems (i.e. drift parameters and diffusion coefficients). The new approach is validated on a range of different systems which vary in dimensionality and non-linearity. These are the Ornstein-Uhlenbeck process, the exact likelihood of which can be computed analytically, the univariate and highly non-linear, stochastic double well and the multivariate chaotic stochastic Lorenz ’63 (3D model). As a special case the algorithm is also applied to the 40 dimensional stochastic Lorenz ’96 system. In our investigation we compare this new approach with a variety of other well known methods, such as the hybrid Monte Carlo, dual unscented Kalman filter, full weak-constraint 4D-Var algorithm and analyse empirically their asymptotic behaviour as a function of observation density or length of time window increases. In particular we show that we are able to estimate parameters in both the drift (deterministic) and the diffusion (stochastic) part of the model evolution equations using our new methods.     

蒙特卡罗方法在解微分方程边值问题中的应用

介绍了蒙特卡罗方法的基本原理以及随机数的产生方法。基于蒙特卡罗方法的思想,结合有限差分方法,建立了求解微分方程边值问题的随机概率模型,并以第一类边界条件的拉普拉斯方程和一个给定初值及边界条件的非稳态热传导方程为数值算例,研究了蒙特卡罗方法在求解微分方程边值问题中的应用。结果表明:利用蒙特卡罗方法,不仅可以有效解决给定边界条件的微分方程,对于给定初值条件的微分方程,也可以从时域有限差分方程出发,采用蒙特卡罗方法进行求解。数值模拟和对误差的理论分析均表明,增加蒙特卡罗试验中的模拟粒子点数,可以提高计算结果的精度。     

A theory of open systems based on stochastic differential equations

For a model of an open quantum system—a concentrated ensemble consisting of similar atoms and interacting with a one-dimensional quantum vacuum environment with a zero photon density—quantum stochastic differential equations of a non-Wiener type of the general form have been obtained; based on the equations, kinetic equations describing a wide class of physical systems are derived. The distinctive feature of such systems is effects of suppression of collective spontaneous emission and stabilization of the excited state. For the open classical system exposed to the action of noise in the form of a Levy process of the general non-Gaussian kind, kinetic equations of the Fokker-Planck type with fractional derivatives have been obtained based on classical non-Wiener stochastic differential equations. This emphasizes the common base of the developed theory for different types of open systems, which is expressed in using the mathematical formalism of stochastic differential equations of the general non-Wiener type.     

Spatially adaptive stochastic numerical methods for intrinsic fluctuations in reaction–diffusion systems

Stochastic partial differential equations are introduced for the continuum concentration fields of reaction–diffusion systems. The stochastic partial differential equations account for fluctuations arising from the finite number of molecules which diffusively migrate and react. Spatially adaptive stochastic numerical methods are developed for approximation of the stochastic partial differential equations. The methods allow for adaptive meshes with multiple levels of resolution, Neumann and Dirichlet boundary conditions, and domains having geometries with curved boundaries. A key issue addressed by the methods is the formulation of consistent discretizations for the stochastic driving fields at coarse-refined interfaces of the mesh and at boundaries. Methods are also introduced for the efficient generation of the required stochastic driving fields on such meshes. As a demonstration of the methods, investigations are made of the role of fluctuations in a biological model for microorganism direction sensing based on concentration gradients. Also investigated, a mechanism for spatial pattern formation induced by fluctuations. The discretization approaches introduced for SPDEs have the potential to be widely applicable in the development of numerical methods for the study of spatially extended stochastic systems.     

Linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations

Abstract

Necessary and sufficient conditions for the linearization of one-dimensional nonautonomous jump-diffusion stochastic differential equations are given. Stochastic integrating factor is introduced to solve the linear jump- diffusion stochastic differential equations. Closed form solutions to certain linearizable jump-diffusion stochastic differential equations are obtained.     

Approximate Gaussian representation of evolution equations I. Single degree of freedom nonlinear equations

We have developed a generalization of the method of statistical linearization to enable us to describe transient and other nonstationary phenomena obeying stochastic nonlinear differential equations. This approximation technique provides an optimal Gaussian representation with time-dependent parameters. The algorithm specifies a set of ordinary differential equations for the Gaussian parameters in terms of the time-dependent average nonlinearities. We apply the general formalism developed herein for single degree of freedom dissipative systems to a particular example.     

Stationary Response of Lotka-Volterra System with Real Noises

A stochastic version of Lotka-Volterra model subjected to real noises is proposed and investigated. The approximate stationary probability densities for both predator and prey are obtained analytically. The original system is firstly transformed to a pair of It o stochastic differential equations. The Ito formula is then carried out to obtain the It o stochastic differential equation for the period orbit function. The orbit function is considered as slowly varying process under reasonable assumptions. By applying the stochastic averaging method to the orbit function in one period, the averaged Ito stochastic differential equation of the motion orbit and the corresponding Fokker-Planck equation are derived. The probability density functions of the two species are thus formulated. Finally, a classical real noise model is given as an example to show the proposed approximate method. The accuracy of the proposed procedure is verified by Monte Carlo simulation.