全变差正则化在抛物型方程初始条件反问题的应用

当反问题反演的函数不连续时,一般的正则化算法反演效果不令人满意,用全变差正则化方法对抛物型方程初始条件反问题进行求解,并进行了数值分析和数值模拟,结果显示数值解与真解吻合较好,表明该方法对于不连续函数求解具有高效、稳定等优点.     

热传导方程反问题的Tikhonov正则化法

为了分析热传导方程反问题所涉及的初始条件.论文把这一类问题转化成第一类Fredholm积分方程,运用Tikhonov正则化的反演法和牛顿法获取正则化参数,得到这一问题的数值解.通过数值实验,验证了这一算法在实际应用中的有效性.     

用演化算法求解抛物型方程扩散系数的识别问题

基于演化算法给出了一类求解参数识别反问题的一般方法,该方法表明只要找到好的、求解相应的正问题的数值方法,演化算法就可以用于求解此类反问题。设计有效的求解反问题的演化算法的关键是寻找一种适合反问题的解空间的编码表示形式、适当的适应值函数形式以及有效的计算正问题的数值方法。该文结合算法、传统的求解反问题的工方法和正则化技术,设计了一类求解参数识别反问题的方法。为验证此类方法,将其用于求解一维扩散方程的     

正则化方法在线性SFS问题中的应用

线性ShapefromShading问题(LSFS)是一类特殊的ShapefromShading问题,此时反射图是表面梯度分量的线性组合的函数。文章在待恢复表面平滑和已知光照方向等假定下,把线性SFS问题正则化,并利用Kaczmarz算法求解得到线性方程组。此种方法能处理区域形状不规则和边界条件不完备的情况。文章用Kaczmarz算法给出了一种从不可积向量场求得最接近的可积向量场的方法,该方法能处理区域形状不规则的情况。     

带微调参量的正则化方法及其在降质图像恢复问题中的应用

1 引言降质图像恢复问题就是图像处理领域里一类反问题。降质图像恢复中的解通常是病态的,利用正则化方法恢复图像取得了较好效果。然而,传统的正则化方法中正则逆算子只含正则化参数,它不能充分地融合其它信息,得到的正则解逼近真解的效果不很理想。为了取得好的恢复效果,各种各样融合其它信息的方法提出来了,以使正则化方法的恢复效果更好。例如,使用局部正则化参数,图像的边缘和纹理区域使用较小的正则化参数,平滑的区域使用较大的正则化参数,局部方差较小的区域正则化参数较大,局部方差较大的区域正则化参数较小,产生自适应正则化参数的正则化方法。人们根据图像的能量、导数的二次平均、曲率的二次平均等设计出各种各样的正则化算子,不同的正则化算子将导致不同的     

以密度为导向的条纹正则优化方法

在光学干涉测量技术中,准确地实现条纹正则化是提取条纹图中的相位信息的前提.文中基于双重正交带通滤波器的正则优化法,提出了以条纹密度信息为质量评估标准来引导条纹正则化的优化方法,包含条纹图局部优化与整体优化两部分,是对双重正交带通滤波器的正则化方法的改进.首先以条纹背景强度和振幅具有的局部线性与连续性特征设计能量函数,并通过最小化能量函数获取背景强度和条纹振幅的值;然后根据高低密度区域条纹的分析,提出密度对正则化过程的导向作用以及局部优化的可行性.实验结果表明,以密度为导向的条纹正则优化法具有均方误差低、抗噪性高等优点;提出的条纹正则优化方法在操作上不需要对噪声严重的条纹图进行除噪预处理,对背景强度与振幅的线性与非线性变化可同时处理,具有较高的条纹处理效率和准确性,可应用于复杂的条纹图正则化处理.     

超声逆散射图象重建问题中截断奇异值分解正则化方法研究

为解决超声逆散射成像问题中的非线性性,人们需要反复地求解前向散射方程和逆散射方程,以达到对全场和未知函数的精确近似,从而根据这一未知函数的精确近似,较好地重建物体内部的断层图象.前向散射方程是一个适定的方程组,可以采用通常的方法进行求解;而逆散射方程则是一个不适定性的方程组,即使数据中存在一个微小的误差,都可能引起解的较大偏离,因此,对这个不适定方程组的求解问题是整个迭代算法成功的关键.而在不适定性问题的求解过程中,正则化参数的选取又是非常重要的.求解不适定性方程的传统方法是Tikhonov正则化方法,这一方法的实质是在传统最小二乘方法上加上一个小于1的滤波因子,对于超声逆散射成像问题来说,效果并不太好.本文将截断奇异值分解正则化方法应用于逆散射方程的求解问题中,并对正则化参数的选取方法进行修正.数值仿真结果表明,这一方法配合适当的正则化参数选取,可以更好地滤除噪声,提高重建图象的质量与可信度,同时还可以减小迭代过程中的计算量.     

基于正则化方法的阶梯边界条件反问题研究

针对复杂的第三类周期性阶梯型边界条件,基于Tikhonov正则化方法,通过求解极坐标下的二维导热反问题,建立了计算空心圆柱体钢坯表面换热系数分布的热物理模型.考虑计算结果的精确性与稳定性,提出'斜率法'以选取正则化系数.利用数值仿真验证了斜率法的适用性,计算结果中R2值为0.998;将斜率法应用于实验数据中,计算结果与预期相符合.结果 表明,斜率法很好的平衡了解的精确性与稳定性,可以有效解决导热反问题的不适定性问题.     

热—板传输系统的稳定性分析

本文研究由处于相邻区域的板方程和热方程构成的耦合系统的稳定性质, 其中耦合来自两个区域的交界 面上的传输边界条件. 在该传输系统中, 热方程起着控制器的作用, 且耗散通过交界面传输并影响板方程. 文献[1] 证明了在板方程上施加额外的控制器时, 该二维传输系统的能量呈指数衰减. 通过应用频域方法, 椭圆方程的正则 性理论等, 可以得到: 仅由热方程的耗散即可使得闭环系统指数稳定. 这一指数稳定的结论与相应的一维传输系统 的性质吻合. 最后, 文章还分析了不同传输边界条件下的板–热耦合系统的稳定性.     

电阻抗成像中混合罚函数正则化算法的仿真研究

该文将变差函数作为罚函数引入到电阻抗成像的正则化重构算法中，从而提出了一种新的电阻抗成像算法，文中称为混合罚函数正则化算法。与常规Tikhonov正则化算法相比，该算法的突出优点是：在确保重构解适定的同时，提高重构图像的对比度和锐度，且计算量增加不大。仿真对比实验结果显示，新算法所得重构图像目标区域与背景区域之间的边界清晰，定位更加准确，与真实医学图像更加符合，这对EIT重构成像技术早日走上实用化有积极的意义。     

Saulyev'S Techniques For Solving A Parabolic Equation With A Non Linear Boundary Specification

In this paper a two-dimensional heat equation is considered. The problem has both Neumann and Dirichlet boundary conditions and one non-local condition in which an integral of the unknown solution u occurs. The Dirichlet boundary condition contains an additional unknown function \mu (t) . In this paper the numerical solution of this equation is treated. Due to the structure of the boundary conditions a reduced one-dimensional heat equation for the new unknown v(\hskip1pty, t) = \vint u(x, y, t)\,\hbox{d}x can be formulated. The resulting problem has a non-local boundary condition. This one-dimensional heat equation is solved by Saulyev's formula. From the solution of this one-dimensional problem an approximation of the function \mu (t) is obtained. Once this approximation is known, the given two-dimensional problem reduces to a standard heat equation with the usual Neumann's boundary conditions. This equation is solved by an extension of the Saulyev's techniques. Results of numerical experiments are presented.     

Wavelets Galerkin method for solving stochastic heat equation

In this paper, a new computational method based on the second kind Chebyshev wavelets (SKCWs) together with the Galerkin method is proposed for solving a class of stochastic heat equation. For this purpose, a new stochastic operational matrix for the SKCWs is derived. A collocation method based on block pulse functions is employed to derive a general procedure for forming this matrix. The SKCWs and their operational matrices of integration and stochastic Itô-integration are used to transform the under consideration problem into the corresponding linear system of algebraic equations which can be simply solved to achieve the solution of the problem. The proposed method is very convenient for solving such problems, since the initial and boundary conditions are taken into account automatically. Moreover, the efficiency of the proposed method is shown for some concrete examples. The results reveal that the proposed method is very accurate and efficient.     

A Numerical Method to Solve Identification Problem for the Lower Coefficient and the Source in the Convection–Reaction Equation

We consider the problem of identifying simultaneously the kinetic reaction coefficient and source function depending only on a spatial variable in one-dimensional linear convection–reaction equation. As additional conditions, a non-local integral condition for the solution of the equation and condition of final overdetermination are given. This problem belongs to the class of combined inverse problems. By integrating the equation with the use of additional integral condition, the problem is transformed to a coefficient inverse problem with local conditions. The derivative with respect to the spatial variable is discretized and a special representation is proposed to solve the resultant semi-discrete problem. As a result, for each discrete value of the spatial variable, the semi-discrete problem splits into two parts: a Cauchy problem and a linear equation with respect to the approximate value of the unknown kinetic coefficient. To determine the source function, an explicit formula is also obtained. The numerical solution of the Cauchy problem uses the implicit Euler method. Numerical experiments are carried out on the basis of the proposed method.     

Application of the method of fundamental solutions and radial basis functions for inverse transient heat source problem

The paper considers the determination of heat sources in unsteady 2-D heat conduction problem. The determination of the strength of a heat source is achieved by using the boundary condition, initial condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the θ-method with the method of fundamental solution and radial basis functions is proposed. Due to ill conditioning of the inverse transient heat conduction problem the Tikhonov regularization method based on SVD decomposition was used. In order to determine the optimum value of the regularization parameter the L-curve criterion was used. For testing purposes of the proposed algorithm the 2-D inverse boundary-initial-value problems in square region Ω with the known analytical solutions are considered. The numerical results show that the proposed method is easy to implement and pretty accurate. Moreover the accuracy of the results does not depend on the value of the θ parameter and is greater in the case of the identification of the temperature field than in the case of the identification of the heat sources function.     

流曲线概念及造型方法

定义了一种可用于造型、具有明确流体力学意义的功能曲线——流曲线,将直匀流绕物面作二维无旋不可压缩定常流动时的流函数表达为曲面后．利用流函数曲面控制边界与流曲线的对应关系采用拉普拉斯方程第一边值逆问题的数值解法，提出了来流为直匀流拘流曲线造型方法并给出了实例,结果表明：流曲线定义及造型方法可用于与流体有关的平面曲线造型,可反映流场信息并对曲线的流体力学性能进行初步评估.     

A numerical method for solving a nonlinear 2-D optimal control problem with the classical diffusion equation

This paper presents a numerical solution for solving a nonlinear 2-D optimal control problem (2DOP). The performance index of a nonlinear 2DOP is described with a state and a control function. Furthermore, dynamic constraint of the system is given by a classical diffusion equation. It is preferred to use the Ritz method for finding the numerical solution of the problem. The method is based upon the Legendre polynomial basis. By using this method, the given optimisation nonlinear 2DOP reduces to the problem of solving a system of algebraic equations. The benefit of the method is that it provides greater flexibility in which the given initial and boundary conditions of the problem are imposed. Moreover, compared with the eigenfunction method, the satisfactory results are obtained only in a small number of polynomials order. This numerical approach is applicable and effective for such a kind of nonlinear 2DOP. The convergence of the method is extensively discussed and finally two illustrative examples are included to observe the validity and applicability of the new technique developed in the current work.     

Heat transfer process in an elliptical channel

This paper addresses the problem of heat transport in an elliptical channel in the presence of a temperature gradient parallel to its axis. The Williams equation is used as the basic equation describing the kinetics of the process and a model of diffusive reflection is used as the boundary conditions on the channel wall. The deviation of the gas condition from the equilibrium is assumed to be small. In order to find a linear correction to the local equilibrium function of distribution, a boundary problem consisting of a linear homogeneous partial differential equation of the first order with a homogeneous boundary condition has been built. The solution of the built boundary value problem has been found by the method of characteristics. The value of the heat flow through the cross section of the channel is found by using numerical procedures implemented by the computer algebra Maple 17 system. The results were compared with the analogous results found in the open press.     

Numerical procedure for the determination of an unknown parameter in a parabolic equation

A numerical procedure for an inverse problem of determining unknown source parameter of one-dimensional parabolic equation subject to the specification of the solution at internal point along with the usual initial boundary conditions is considered. By using some transformation the problem is reformulated to a nonlocal parabolic problem. Some numerical examples using the proposed numerical procedure are presented.     

Die Lösung des Anfangs-Randwertproblems für die Wärmeleitungsgleichung im ℝ3 mit einer Integralgleichungs-methode nach dem Rotheverfahrenmit einer Integralgleichungs-methode nach dem Rotheverfahren

We are looking for a solution of the initial boundary value problem for the threedimensional heat equation in a compact domain with a boundary of continous curvature. We use Rothe's line method, which works by discretisation of the time variable. For every time step there remains an elliptic boundary value problem, which is solved by means of an integral equation. The so obtained approximate solutions converge to the exact solution of the original problem. In case of a sphere we find a simple error estimate for the approximation. For two initial conditions the practical computations show, that the integral equations method yields useful results with relative small effort.