带复数核的第一类Fredholm积分方程的正则化方法及其应用

本文推广了Ｔｉｋｈｏｎｏｖ正则化方法，导出了带复数核的第一类Ｆｒｅｄｈｏｌｍ积分方程的正则解应满足的正则积分微分方程，并讨论了正则解的收敛性·作为这一方法的应用，数值求解了与二维摇板造波问题相应的一类逆问题，并给出了选择最佳正则参数的一个实用的方法     

A Modified Tikhonov Regularization for Linear Operator Equations

We construct with the aid of regularizing filters a new class of improved regularization methods, called modified Tikhonov regularization (MTR), for solving ill-posed linear operator equations. Regularizing properties and asymptotic order of the regularized solutions are analyzed in the presence of noisy data and perturbation error in the operator. With some accurate estimates in the solution errors, optimal convergence order of the regularized solutions is obtained by a priori choice of the regularization parameter. Furthermore, numerical results are given for several ill-posed integral equations, which not only roughly coincide with the theoretical results but also show that MTR can be more accurate than ordinary Tikhonov regularization (OTR).     

求解第一类Fredholm积分方程的一种新的正则化算法

应用一种新的正则化方法建立了一类新的求解第一类Fredholm积分方程的正则化算法, 并借助Matlab软件给出了数值算例.数值结果与理论分析基本一致,而且表明文中建立的正则化比通常的Tikhonov正则化更精确.     

解第一类算子方程的一种迭代正则化方法

提出了一种求解第一类算子方程的新的迭代正则化方法,并依据广义Arcangeli方法选取正则参数,建立了正则解的收敛性.与通常的Tikhonov正则化方法相比较,提高了正则解的渐近阶估计.     

Examples of Saturated Convergence Rates for Tikhonov Regularization

Tikhonov regularization is one of the most popular methods for solving linear operator equations of the first kind Au = f with bounded operator, which are ill-posed in general (Fredholm's integral equation of the first kind is a typical example). For problems with inexact data (both the operator and the right-hand side) the rate of convergence of regularized solutions to the generalised solution u ₊ (i.e.the minimal-norm least-squares solution) can be estimated under the condition that this solution has the source form: u ₊ im(A*A). It is well known that for Tikhonov regularization the highest-possible worst-case convergence rates increase with only for some values of , in general not greater than one. This phenomenon is called the saturation of convergence rate. In this article the analysis of this property of the method with a criterion of a priori regularization parameter choice is presented and illustrated by examples constructed for equations with compact operators.This revised version was published online in October 2005 with corrections to the Cover Date.     

The tensor Golub–Kahan–Tikhonov method applied to the solution of ill-posed problems with a t-product structure

This paper discusses an application of partial tensor Golub–Kahan bidiagonalization to the solution of large-scale linear discrete ill-posed problems based on the t-product formalism for third-order tensors proposed by Kilmer and Martin (M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641-658). The solution methods presented first reduce a given (large-scale) problem to a problem of small size by application of a few steps of tensor Golub–Kahan bidiagonalization and then regularize the reduced problem by Tikhonov's method. The regularization operator is a third-order tensor, and the data may be represented by a matrix, that is, a tensor slice, or by a general third-order tensor. A regularization parameter is determined by the discrepancy principle. This results in fully automatic solution methods that neither require a user to choose the number of bidiagonalization steps nor the regularization parameter. The methods presented extend available methods for the solution for linear discrete ill-posed problems defined by a matrix operator to linear discrete ill-posed problems defined by a third-order tensor operator. An interlacing property of singular tubes for third-order tensors is shown and applied. Several algorithms are presented. Computed examples illustrate the advantage of the tensor t-product approach, in comparison with solution methods that are based on matricization of the tensor equation.     

解带有扰动数据的第一类Volterra积分方程的谱正则化方法

该文的主要目的是通过使用Legendre配置方法和正则化策略来求解带有噪声数据的第一类Volterra积分方程,并给出该方法收敛性分析的严格数学证明.数值实验表明了该方法的有效性.     

第一类弱奇异核Fredholm积分方程的数值求解

第一类弱奇异核Fredholm积分方程由于奇异及本质的不适定性,给求解带来很大难度．本文首先利用克雷斯变换将方程转化,并对转化后的方程进行高斯一勒让德离散,得到一离散不适定的线性方程组,结合正则化方法对该类问题进行数值求解．最后给出了数值模拟,验证了本文方法的可行性及有效性．     

A Method for Solving Fredholm Integral Equations of the First Kind Based on Chebyshev Wavelets

In this paper, we suggest a method for solving Fredholm integral equation of the first kind based on wavelet basis. The continuous Legendre and Chebyshev wavelets of the first, second, third and fourth kind on [0,1] are used and are utilized as a basis in Galerkin method to approximate the solution of integral equations. Then, in some examples the mentioned wavelets are compared with each other.     

球面上第二类Fredholm积分方程配置方法

球面上第二类 Fredholm积分方程经球坐标变换可化为矩形域 H0 上的问题求解 .用有限元法构造H0 上的插值函数 ,它必须满足在 H0 的左、右两边连续 ,然后用配置方程求方程的近似解     

双调和方程柯西问题的修正的Tikhonov正则化方法

本文研究了双调和方程柯西问题,这类是不适定的,即问题的解(如果存在)不连续依赖于测量数据.首先在精确解的先验假设下给出问题的条件稳定性结果.接着利用修正的Tikhonov正则化方法求解此不适定问题.在先验和后验正则化参数选取规则下,给出正则解和精确解之间的误差估计式.最后给出几个数值例子验证此正则化方法求解此类反问题的有效性.     

Tikhonov Regularization of Large Linear Problems

Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a new numerical method, based on Lanczos bidiagonalization and Gauss quadrature, for Tikhonov regularization of large-scale problems. An estimate of the norm of the error in the data is assumed to be available. This allows the value of the regularization parameter to be determined by the discrepancy principle.     

Wavelet Collocation Methods for a First Kind Boundary Integral Equation in Acoustic Scattering

In this paper we consider a wavelet algorithm for the piecewise constant collocation method applied to the boundary element solution of a first kind integral equation arising in acoustic scattering. The conventional stiffness matrix is transformed into the corresponding matrix with respect to wavelet bases, and it is approximated by a compressed matrix. Finally, the stiffness matrix is multiplied by diagonal preconditioners such that the resulting matrix of the system of linear equations is well conditioned and sparse. Using this matrix, the boundary integral equation can be solved effectively.     

解第一类算子方程的一种新的正则化方法

对算子与右端都为近似给定的第一类算子方程提出一种新的正则化方法，依据广义Ａｒｃａｎｇｅｌｉ方法选取正则参数，建立了正则解的收敛性。这种新的正则化方法与通常的Ｔｉｋｈｏｎｏｖ正则化方法相比较，提高了正则解的渐近阶估计。     

Computation of Control for Linear Approximately Controllable System Using Tikhonov Regularization

Computation of control for a controlled partial differential equation is a di?cult task, especially when the control problem is ill posed. In this paper, we propose a method of computing the regularized control of a diffusion control system using Tikhonov regularization approach when the system is approximately controllable. The method proposed here for choosing regularization parameter guarantees the convergence of the proposed control.     

含第一类临界参数抽象动力方程边值问题之积分算子求解法

该文讨论了一类抽象Volterra型积分算子，利用此获得了含第一类临界参数的抽象动力方程边值问题的解。这种新的求解法我们称为积分算子求解法。     

第一类弱奇异Volterra积分方程解的渐近展开式

针对核函数和自由项代数且对数奇异的第一类线性Volterra积分方程,通过Laplace变换导出这类方程的解在零点的渐近展开式,对于方程解的奇异性质给出准确刻画.对于核函数仅代数奇异的情形,还得到方程的解在无穷远点的渐近展开式.这些展开式可以分别作为当自变量变小或变大时方程的近似解.最后,给出实例说明展开式的正确性及有...     

Stability of Fredholm Integral Equation of the First Kind in Reproducing Kernel Space

It is well known that the problem on the stability of the solutions for Fredholm integral equation of the first kind is an ill-posed problem in C[a,b] or L2[a,b].In this paper,the representation of the...     

复双球面上变系数奇异积分方程的正则化

利用复双球面上的立体角系数的方法和置换公式,讨论复双球垒域上变系数奇异积分方程的正则化问题,推广了复超球面上变系数奇异积分方程的结论．