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.     

图像恢复的正则化混合GMRES(m)方法

为了充分利用广义极小化残量方法在处理大规模线性问题时的优势,将其同正则化技术相结合应用于图像恢复领域,提出了一种新的图像恢复方法。该方法基于Arnoldi过程,用一系列规模远小于原不适定问题的最小二乘问题来逼近原问题,并应用截断奇异值分解正则化技术保证稳定求解这些最小二乘问题。其中,根据图像恢复问题的具体特点,在确定截断奇异值分解的截断次数时,对传统的L-曲线准则进行了少许修改。数值试验结果表明,试验数据与肉眼观察恢复图像的清晰程度相吻合,说明新方法是有效的。     

GMRES算法求解烟雾仿真N-S方程

烟雾在大规模战场仿真和复杂环境仿真中扮演着重要角色,因此研究烟雾仿真具有重大意义。提出用广义极小残差算法(GMRES)来求解烟雾仿真中的N-S方程。首先给出GMRES算法的计算原理;其次用GMRES算法对烟雾仿真N-S方程进行求解,并对求解结果进行收敛性分析,分析结果表明GMRES算法可以对烟雾仿真N-S方程进行求解,结果收敛;最后运用GMRES算法通过计算机技术对烟雾进行可视化仿真,仿真结果表明,采用GMRES求解算法的烟雾仿真效果比较真实,基本符合现实中的烟雾。     

基于解空间分解的GMRES算法及其在图像处理中的应用

提出一种基于解空间分解的加速GMRES算法来求解不适定问题,该算法将解空间分解为Krylov子空间和一个辅助子空间,其中一部分解用一种加速GMRES法迭代得到,另一部分解用直接求解的方法得到。数值实验和分析表明这种算法是行之有效的,在达到相同的估计精度的条件下,迭代速度大大提高,求解时间只有普通GMRES算法的五分之一,甚至更少;而且在迭代次数相同的情况下,解的精度更高,如解的均方误差平均是普通GMRES算法的五分之三。最后将该方法应用到光学图像复原,实验结果表明该方法能够明显改善光学图像的质量。     

应用粒子群优化算法选择正则化参数

对正则化方法中正则参数的选择进行了研究;提出了利用粒子群优化算法获取正则参数的方法;通过数值模拟实验;对比了该方法与遗传算法;通过图像恢复实验;比较了传统正则化滤波方法和所提出的方法;实验结果表明;所提出的方法在处理不适定问题时更具有优越性;是一种实用有效的方法。     

求解Symm积分方程的信赖域Lanczos法

Symm积分方程在位势理论中具有重要的作用,它是Hadamard意义下的不适定问题.本文基于信赖域算法并结合Lanczos迭代方法,得到了Symm积分方程的数值解法,与通常的正则化方法相比,在数据出现噪声的情况下,该方法克服了正则化方法中正则参数选取的困难,同时具有较高的精度.     

正则化HSS预处理鞍点矩阵的多尺度算法

针对鞍点求解结果收敛速度慢、CPU消耗时间较长等问题,提出一种正则化HSS预处理鞍点矩阵的多尺度算法.运用最优正则化方法确定正则参数,得到计算最优正则参数公式;通过HSS方法完成系数矩阵预处理,得到新的预处理子NHSS;为了更加具体地分析预处理后的鞍点矩阵多尺度算法特征值分布形态,择优选取预处理子参数,确保算法收敛速率...     

基于Tikhonov正则化的WSN多边定位算法研究

节点定位是无线传感器网络实现监测和跟踪的一个重要前提.针对多边定位中的不适定问题.提出了一种基于Tikhonov正则化方法的定位算法,研究了定位模型的建立、正则化参数的选取方法以及最优定位参考点数的选取等问题.实验结果表明本算法与典型的极大似然估计法相比,较大幅度地提高了定位精度,当a值选取600,采用5个参考节点时,定位精度可达到1米.     

求解二维Fredholm积分方程的参数化信赖域方法

给出了求解二维第一类Fredholm积分方程信赖域方法。通过引入正则化参数将离散后的Fredholm积分方程转化带参数的最优化问题,借助于KKT条件将二次信赖域子问题参数化,并进行分析求解,最后给出了数值模拟。     

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.     

GRAPES模式中Helmhothz方程两种求解方法的对比研究

GRAPES是中国气象局自主研发的一个全球/区域分析预报系统。其模式计算方程组经过离散化之后,积分求解过程最终归结为对一个椭圆方程或Helmholtz(赫姆霍兹)方程的求解,这个求解是整个动力框架计算的核心。在目前GRAPES全球模式的准业务计算中,对于分辨率为0.5o的系统,Helmholtz方程的求解时间占到了整个模式计算时间的三分之一强。而且随着未来高分辨率模式的进一步加细,以及模式计算精度的提高,方程求解计算总量更是呈指数式增长。为此,本文分析了GRAPES模式中求解Helmholtz方程所采用的广义共轭余差法(GCR),并对比给出了利用PETSC函数库中提供的GMRES方法求解Helmholtz方程的一些初步测试结果。结果表明,采用高精度的GMRES方法可以减少模式预报偏差,改善模式预报准确度,在大规模并行计算时具有更好的可扩展性能。     

Extrapolation of Nystrom Solution for Two-Dimensional Nonlinear Fredholm Integral Equations

In this paper, we analyze the existence of asymptotic error expansion of Nystrom solution for two-dimensional nonlinear Fredholm integral of the second kind. We show that the Nystrom solution admits an error expansion in powers of the step-size h and the step-size k. For a special choice of the numerical quadrature, the leading terms in the error expansion for the Nystrom solution contain only even powers of h and k, beginning with terms h ^2p and k ^2q . These expansions are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. Numerical examples show that how Richardson extrapolation gives a remarkable increase of precision, in addition to faster convergence.     

The use of Sherman-Morrison formula in the solution of Fredholm integral equation of second kind

We consider a constructive method for the solution of Fredholm integral equations of second kind. This method is based on a simple generalization of the well-known Sherman-Morrison formula to the infinite dimensional case. In particular, this method constructs a sequence of functions, that converges to the exact solution of the integral equation under consideration. A formal proof of this convergence result is provided for the case of Fredholm integral equations with integral kernel. Finally, a boundary value problem for the Laplace equation is considered as an example of the application of the proposed method.     

The efficient solution of direct medium problems by using translation techniques

We consider a Fredholm integral equation arising from a time-harmonic electromagnetic scattering problem for inhomogeneous media. The discretization of this equation usually produces a large dense linear system that must be solved by iterative methods. To speed up these methods we propose an efficient computation of the action of the corresponding coefficient matrix on a generic vector. This computation is mainly based on the well known addition formula for the Hankel functions and a simple translation argument. We present some numerical examples to show the efficiency of the proposed method.     

Application of Sinc-collocation method for solving a class of nonlinear Fredholm integral equations

In this paper, the numerical solution of nonlinear Fredholm integral equations of the second kind is considered by two methods. The methods are developed by means of the Sinc approximation with the single exponential (SE) and double exponential (DE) transformations. These numerical methods combine a Sinc collocation method with the Newton iterative process that involves solving a nonlinear system of equations. We provide an error analysis for the methods. So far approximate solutions with polynomial convergence have been reported for this equation. These methods improve conventional results and achieve exponential convergence. Some numerical examples are given to confirm the accuracy and ease of implementation of the methods.     

A hybrid algorithm for approximate optimal control of nonlinear Fredholm integral equations

In this paper, a novel hybrid method based on two approaches, evolutionary algorithms and an iterative scheme, for obtaining the approximate solution of optimal control governed by nonlinear Fredholm integral equations is presented. By converting the problem to a discretized form, it is considered as a quasi-assignment problem and then an iterative method is applied to find an approximate solution for the discretized form of the integral equation. An analysis for convergence of the proposed iterative method and its implementation for numerical examples are also given.     

Iterative solvers for Tikhonov regularization of dense inverse problems

According to the special demands arising from the development of science and technology, in the last decades appeared a special class of problems that are inverse to the classical direct ones. Such an inverse problem is concerned with the opposite way, usually followed by a direct one: finding the cause of a given effect or finding the law of evolution given the cause and effect. Very frequently, such inverse problems are modelled by Fredholm first-kind integral equations that give rise after discretization to (very) ill-conditioned linear systems, in classical or least squares formulation. Then, an efficient numerical solution can be obtained by using the Tikhonov regularization technique. In this respect, in the present paper, we propose three Kovarik-like algorithms for numerical solution of the regularized problem. We prove convergence for all three methods and present numerical experiments on a mathematical model of an inverse problem concerned with the determination of charge distribution generating a given electric field.     

A meshless approximate solution of mixed Volterra–Fredholm integral equations

This paper presents a meshless method using a radial basis function collocation scheme for numerical solution of mixed Volterra–Fredholm integral equations, where the region of integration is a non-rectangular domain. We will show that this method requires only a scattered data of nodes in the domain. It is shown that the proposed scheme is simple and computationally attractive. Applications of the method are also demonstrated through illustrative examples.