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。     

求解Richardson迭代方程的快速配置法

本文依据多尺度快速配置法求解第一类Fredholm积分方程的Richardson迭代正则化方程.该方法得到了离散Richardson迭代正则化方程的快速解,在积分算子是弱扇形紧算子时,利用改进的迭代停止准则,给出了Richardson迭代正则化方法所得近似解的收敛率.最后,数值例子说明了算法的有效性.     

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Gmres方法

在处理具有线性的、空间位移不变的成像系统所成的图像恢复问题时,提出了一种基于Krylov向量完全正交化的正则化Gmres方法。该算法考虑了图像恢复中的不适定性及计算时的复杂性两个方面,将正则化算法与广义极小残余算法相结合,通过正则化方法将模型离散后的积分方程转化为一适定问题,然后利用广义极小残余算法得到结果。在数值模拟时,对不同的方法进行了对比分析,结果表明所选的方法能够明显改善图像恢复的质量。     

Boltzmann方程各向异性矩模型的数值方法

本文发展了Boltzmann方程各向异性正则化矩模型(AHME)的数值方法,将其简记为ANRxx算法.AHME由樊玉伟等~([1])作为各向同性正则化矩模型(HME)的推广提出,而本文实现的ANRxx算法亦可视为HME离散得到的NRxx算法~([2])间的推广.本文对ANRxx算法进行了代表性的数值实验.数值结果表明,ANRxx算法作为Boltzmann方程的数值矩方法有明显的收敛性,同时在各向异性较明显的例子上有优于NRxx算法的表现.     

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.     

求解二维波动方程系数逆问题的时域卷积迭代法

在本文中,作者提出了一种求解二维波动方程的时域卷积迭代法(简称TCIM)。在每次迭代中,需解波方程初、边值问题及卷积型第一类积分方程,并引用[4]作为人工边界条件,方法简单,具有通用性,可推广到三维及弹性波方程。技巧地应用了正则法,结果是稳定精确的,数值结果见文[8]。     

基于正则代数重建算法的二维温度场重建

在使用代数重建算法(Algebraic Reconstruction Technique,ART)对二维非均匀温度场进行重建时,离散误差和投影噪声会随着迭代修正被引入,为了平衡离散误差,减小算法对噪声的敏感度,在ART中引入了正则化项,并使用留一交叉验证法对单位正则化参数进行了选取,根据投影穿过待测区域路径的长度和单位正则化参数动态调整每条投影的正则化权重,实现了对每条投影离散误差和噪声水平的衡量。在不同的投影分布情况下,使用该算法对高斯单峰对称和高斯单峰偏置温度场进行了仿真重建,重建结果表明相比于传统迭代算法,该算法可有效提高温度场的重建精度,并且具有较好的稳定性。     

求解二维椭圆型界面问题的外推完全多重网格法

针对二维椭圆型界面问题的离散化方程,应用外推插值技巧和样条插值方法在细网格层上构造合适的迭代初始值,加快V型多重网格法求解离散化系统的速度,设计了外推完全多重网格(EXFMG)法.数值实验表明新算法有效降低了迭代次数,计算量更少.     

基于MG-GMRES算法的图像超分辨率重建

提出了一种基于多层网格(MG)和广义极小残余(GMRES)算法相结合的图像超分辨率重建快速算法.首先采用正则化方法给出图像超分辨率重建模型;然后在系统介绍MG和GMRES算法的基础上,针对图像超分辨率重建中非对称线性稀疏方程的求解,提出多层网格-广义极小残余(MG-GMRES)算法;详细讨论了MG-GMRES算法的光滑、限制、插值操作以及计算复杂度.实验研究表明该算法的重建结果相当有效,与MG、GMRES和Richrdson迭代相比,具有更快的收敛速度.     

A Nyström method for Fredholm integral equations with right-hand sides having isolated singularities

Fredholm integral equations on the interval [?1,1] with right-hand sides having isolated singularities are considered. The original equation is reduced to an equivalent system of Fredholm integral equations with smooth input functions. The Nyström method is applied to the system after a polynomial regularization. The convergence, stability and well conditioning of the method is proved in spaces of weighted continuous functions. The special case of the weakly singular and symmetric kernel is also investigated. Several numerical tests are included.     

Numerical solution of nonlinear two-dimensional Fredholm integral equations of the second kind using Sinc Nyström method

In this paper, a double-exponential (DE) Sinc Nyström method is utilized to solve nonlinear two-dimensional Fredholm integral equations of the second kind. Using the DE transformation, the Sinc quadrature rule for a definite integral is extended to double integral over a rectangular region. Therefore, a nonlinear Fredholm integral equation is reduced to a system of nonlinear algebraic equations, which is solved using the Newton iteration method. Convergence analysis shows that the proposed method can converge exponentially. Several numerical examples are provided to demonstrate the high efficiency and accuracy of the proposed method.     

The backward problem for a time-fractional diffusion-wave equation in a bounded domain

This paper is devoted to solve the backward problem for a time-fractional diffusion-wave equation in a bounded domain. Based on the series expression of the solution for the direct problem, the backward problem for searching the initial data is converted into solving the Fredholm integral equation of the first kind. The existence, uniqueness and conditional stability for the backward problem are investigated. We use the Tikhonov regularization method to deal with the integral equation and obtain the series expression of the regularized solution for the backward problem. Furthermore, the convergence rate for the regularized solution can be proved by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical results for five examples in one-dimensional case and two-dimensional case show that the proposed method is efficient and stable.     

A weighted H1 seminorm regularization method for Fredholm integral equations of the first kind

Many problems in mathematics and engineering lead to Fredholm integral equations of the first kind, e.g. signal and image processing. These kinds of equations are difficult to solve numerically since they are ill-posed. Therefore, regularization is required to obtain a reasonable approximate solution. This paper presents a new regularization method based on a weighted H¹ seminorm. Details of numerical implementation are given. Numerical examples, including one-dimensional and two-dimensional integral equations, are presented to illustrate the efficiency of the proposed approach. Numerical results show that the proposed regularization method can restore edges as well as details.     

Kernel-Splitting Technique for Enclosing the Solution of Fredholm Equations of the First Kind

We derive a numerical method for solving linear Fredholm integral equations of the first kind. Based on series expansion techniques, the kernel of the corresponding integral equation is splitted into a finite rank degenerate part and an infinite dimensional, normwise small remainder. By enclosing the remainder term, the original problem, is transformed into a degenerate set-valued problem. For this problem, we derive a numerical method that provides a rigorous control of approximation and roundoff errors. We show that this approach provides a regularization scheme.     

Numerical solution of laplace's equation on non-simply connected regions on R 2

We consider the interior Dirichlet problem for Laplace's equation on a non-simply connected two-dimensional regions with smooth boundaries.The solution is sought as the real part of a holomorphic function on the region, given as Cauchy-type integral.The approximate double layer density function is found by solving a system of Fredholm integral equations of second kind.Because of the non-uniqueness of the solution of the system we solve it using a technique based on the solution of the “Modified Dirichlet problem”.The Nystrom's method coupled with the trapezoidal rule is used as numerical integration scheme.The linear system derived from the integral equation is solved using the conjugate gradient applied to the normal equation.Theoretical and computational details of the method are presented.     

The regularization method for Fredholm integral equations of the first kind

In this work, the Fredholm integral equations of the first kind will be examined. The regularization method combined with the existing techniques are applied to handle the ill-posed Fredholm problems. Examples will be used to highlight the reliability of the regularization method.     

Numerical results for linear Fredholm integral equations of the first kind over surfaces in three dimensions

Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method – the Tikhonov method – to deal with this ill-conditioning and to obtain accurate numerical results.