积分梯度法的两种格式

在二维柱坐标系下Lagrange流体力学的计算中,积分梯度法是动量方程的一种有效离散方法.积分梯度法中,IGT(Integral Gradient Total)格式不能保持柱几何下一维球对称性;IGA(Integral Gradient Average)格式可以保持一维球对称性,但当相邻网格质量相差比较大时,会得到远远脱离真实物理现象的加速度.深入研究IGA和IGT格式发现,当相邻网格边界压力取为质量加权时,即使相邻网格质量相差较大,对于一维平面和一维柱问题,IGT与IGA等价;在二维情形下,可以缩小IGT和IGA之间的差异.理论证明,IGA格式不能保持系统的动量守恒,IGT格式能保持系统的动量守恒性.数值模拟结果进一步显示了这两个格式的优缺点.     

A Weak Form of the Conjugate Gradient FFT Method for Two-Dimensional Elastodynamics

The problem of two-dimensional scattering of elastic waves by an elastic inclusion can be formulated in terms of a domain integral equation, in which the grad-div operator acts on a vector potential. The vector potential is the spatial convolution of a Green's function with the product of the density and the displacement over the domain of interest. A weak form of the integral equation for the unknown displacement is obtained by testing it with rooftop functions. This method shows excellent numerical performance.     

An Extended Row and Column Method for Solving Linear Systems on a Quantum Computer

International Journal of Theoretical Physics - To solve the linear systems of equations Ax = b on a quantum computer, Shao and Xiang proposed a quantum version of row and column methods by...     

无需线搜索的并行非线性共轭梯度法

给出了一类无需线搜索的无约束最优化并行算法--并行非线性共轭梯度法(NLS-PNCG),用一个固定的公式来计算搜索步长,较常用的共轭梯度法计算量小.并且证明了目的的全局收敛性,给出了数值试验,结果显示NLS-PNCG优于线搜索的非线性共轭梯度法(NCG).     

共轭梯度法求解双曲传热多宗量反问题

提出双曲传热反问题热物性参数和边界条件多宗量联合反演的一般数值求解模式,考虑了非均质和分布参数的影响,时域上采用时域精细算法进行离散,建立了便于敏度分析的有限元正演模型.由最小二乘原理建立反演模型,应用共轭梯度法进行求解.探讨了时间步长和测量误差对反演结果的影响,并进行了数值验证.     

共轭梯度法求解多宗量瞬态热传导反问题

提出瞬态热传导反问题中热物性参数和边界条件多宗量反演的一般数值求解模式,导出了敏度计算公式,并用共轭梯度法求解,探讨了测点数目、测量误差和变量初值对反演结果的影响,并进行了数值验证.     

Conjugate Gradient method for finding fundamental solitary waves

The Conjugate Gradient method (CGM) is known to be the fastest generic iterative method for solving linear systems with symmetric sign definite matrices. In this paper, we modify this method so that it can find fundamental solitary waves of nonlinear Hamiltonian equations. The main obstacle that such a modified CGM overcomes is that the operator of the equation linearized about a solitary wave is not sign definite. Instead, it has a finite number of eigenvalues on the opposite side of zero than the rest of its spectrum. We present versions of the modified CGM that can find solitary waves with prescribed values of either the propagation constant or power. We also extend these methods to handle multi-component nonlinear wave equations. Convergence conditions of the proposed methods are given, and their practical implications are discussed. We demonstrate that our modified CGMs converge much faster than, say, Petviashvili’s or similar methods, especially when the latter converge slowly.     

Recent Advances in the Theoretical Methods and Computational Schemes for Investigations of Resonances in Few-Body Atomic Systems

This paper describes the research works performed in my group in recent years in the theoretical methods and computational schemes for investigations of resonances in few-body atomic systems. These methods include the complex absorbing potential method, the stabilization method, and the method of complex scaling. In addition to atomic resonances, this paper also describes some of our recent works on investigations of Borromean bindings in three-body atomic systems. By systematically changing the two-body interaction from the long range Coulomb potential to the short range screened Coulomb (Yukawa) potential, we have investigated the Borromean bindings for hydrogen molecule ion H ₂ ⁺ and the muonic molecular ions.     

Method for Solving Physical Problems Described by Linear Differential Equations

Russian Physics Journal - A method for solving physical problems is suggested in which the general solution of a differential equation in partial derivatives is written in the form of decomposition...     

An Efficient Computational Algorithm for Implementing the Maximum Entropy Method in Deconvolution Inversion Problems

We discuss a theoretical informational approach to solving ill-posed problems of function recovery based on the use of the maximum entropy principle. On this basis, we propose an efficient computational algorithm for implementing the procedure of solving the function-recovery problem and the method of regularization of the problem of function recovery from the convolution. In many cases, the proposed regularization method ensures high recovery quality. If the recovery accuracy is insufficient, then the Largange multipliers obtained by this method can be the best initial approximation (in the sense of the least squares) for their iterative refinement.     

基于概率加权共轭梯度算法的混凝土超声波层析成像

在混凝土层析成像中,为了提高反演的准确性和计算效率,针对共轭梯度算法提出一种加权算法——概率加权共轭梯度算法.新算法不同于常规的加权算法,权重是加在成像单元上而不是方程上.为取得较好的权重因子和较好的迭代初始值,采用IART算法的权重和迭代初始值的选取方法.模拟算例和混凝土试验均表明这种加权算法的可行性.     

Application of Homotopy Analysis Method for Solving Systems of Volterra Integral Equations

In this paper, we prove the convergence of homotopy analysis method (HAM). We also apply the homotopy analysis method to obtain approximate analytical solutions of systems of the second kind Volterra integral equations. The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown that the solutions obtained by the homotopy-perturbation method (HPM) are only special cases of the HAM solutions. Several examples are given to illustrate the efficiency and implementation of the method.     

在数字化MRI谱仪系统设计中消除梯度抖动的方法

提出了一种在磁共振成像谱仪系统中消除由于数字化设计而引起的梯度抖动的方法．该方法利用梯度波形发生器的内部时钟控制脉冲序列的启动,使得它们真正实现同步．本文除详细介绍该方法的原理之外,还通过实验证明了它的有效性．     

高能X光照相中密度重建的共轭梯度算法

通过对高能X光照相过程的物理分析,得到了辐射照相的点扩展函数,给出了估价函数的表达式.并以典型的法国试验客体模型为例,将共轭梯度算法应用到高能X光照相的密度重建中,得到了较满意的结果.这种方法从模拟辐射照相过程出发,将辐射照相的模拟结果与实际测量结果做比较并找出最大可能解.可在一定程度上克服其它密度重建算法通过反卷积过程消除模糊效应所引起的不确定性.     

Efficient Proximal Gradient Algorithms for Joint Graphical Lasso

We consider learning as an undirected graphical model from sparse data. While several efficient algorithms have been proposed for graphical lasso (GL), the alternating direction method of multipliers (ADMM) is the main approach taken concerning joint graphical lasso (JGL). We propose proximal gradient procedures with and without a backtracking option for the JGL. These procedures are first-order methods and relatively simple, and the subproblems are solved efficiently in closed form. We further show the boundedness for the solution of the JGL problem and the iterates in the algorithms. The numerical results indicate that the proposed algorithms can achieve high accuracy and precision, and their efficiency is competitive with state-of-the-art algorithms.     

A Robust and Efficient Method for Steady State Patterns in Reaction-Diffusion Systems

An inhomogeneous steady state pattern of nonlinear reaction-diffusion equations with no-flux boundary conditions is usually computed by solving the corresponding time-dependent reaction-diffusion equations using temporal schemes. Nonlinear solvers (e.g., Newton's method) take less CPU time in direct computation for the steady state; however, their convergence is sensitive to the initial guess, often leading to divergence or convergence to spatially homogeneous solution. Systematically numerical exploration of spatial patterns of reaction-diffusion equations under different parameter regimes requires that the numerical method be efficient and robust to initial condition or initial guess, with better likelihood of convergence to an inhomogeneous pattern. Here, a new approach that combines the advantages of temporal schemes in robustness and Newton's method in fast convergence in solving steady states of reaction-diffusion equations is proposed. In particular, an adaptive implicit Euler with inexact solver (AIIE) method is found to be much more efficient than temporal schemes and more robust in convergence than typical nonlinear solvers (e.g., Newton's method) in finding the inhomogeneous pattern. Application of this new approach to two reaction-diffusion equations in one, two, and three spatial dimensions, along with direct comparisons to several other existing methods, demonstrates that AIIE is a more desirable method for searching inhomogeneous spatial patterns of reaction-diffusion equations in a large parameter space.     

Schemes for Generating Cluster States via Cavity Systems

We propose a scheme for generating an N-atom cluster state via cavity quantum electrodynamics （ CQED）. In our scheme, there is no transfer of quantum information between the atoms and the cavity, i.e., the cavity is always in the vacuum state, so the cavity decay can be suppressed. Also, the generated cluster state is the entanglement of the ground states, so the atomic spontaneous emission can be avoided. Therefore, the cluster state generated in our scheme has a longer lifetime. Furthermore, the requirement on the quality factor of the cavity greatly loosened for the cavity is only virtually excited.     

Fractional Generalization of Gradient Systems

We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems.Mathematics Subject Classification (2000). 426A33, 70G60     

Experimental Investigation of Heat Transfer Coefficient from the Impingement of a Slot Jet Using Conjugate Gradient Method with Adjoint Equation

The convective heat transfer coefficients resulting from a slot jet on a plane surface have been estimated by using the real measured temperatures in the plate and inverse method. In this study, the inverse method used the conjugate gradient method with an adjoint equation. Distributions of the local heat transfer coefficient on the impingement surface were determined for various Re and H/Dh. It was found the heat transfer coefficients generally tended to decrease with increasing separation distance and to increase with an increase in Reynolds number. This presented method is able to estimate the variation of the local Nusselt number with time.     

Generalized Lagrange Jacobi Gauss-Lobatto (GLJGL) Collocation Method for Solving Linear and Nonlinear Fokker-Planck Equations

In this study, we have constructed a new numerical approach for solving the time-dependent linear and nonlinear Fokker-Planck equations. In fact, we have discretized the time variable with Crank-Nicolson method and for the space variable, a numerical method based on Generalized Lagrange Jacobi Gauss-Lobatto(GLJGL) collocation method is applied. It leads to in solving the equation in a series of time steps and at each time step, the problem is reduced to a problem consisting of a system of algebraic equations that greatly simplifies the problem. One can observe that the proposed method is simple and accurate. Indeed, one of its merits is that it is derivative-free and by proposing a formula for derivative matrices, the difficulty aroused in calculation is overcome, along with that it does not need to calculate the General Lagrange basis and matrices; they have Kronecker property. Linear and nonlinear Fokker-Planck equations are given as examples and the results amply demonstrate that the presented method is very valid, effective,reliable and does not require any restrictive assumptions for nonlinear terms.