Tikhonov 正则化的饱和性与逆结果及后验参数选取 *

对解非线性不适定问题的Tikhonov正则化证明了饱和性与一些逆结果 ,并考虑了正则化参数的最优后验选取 .     

L-曲线估计确定正则参数的双网格迭代法

本文考虑对不适定问题离散化得到的大规模不适定线性方程组进行Tiknonov正则化,然后用双网格迭代法求解得到的Tikhonov正则化方程组,并用L-曲线估计法来确定正则参数.试验问题的数值结果表明双网格迭代法求解正则化后的对称正定线性方程组效果很好,且L-曲线估计法确定正则参数计算量很小.     

一类逆时反问题的改进正则化方法的收敛性

1引 言 非线性反问题广泛地存在于许多科学和工程问题中,反问题求解的主要困难在于问题的不适定性,即待求函数或参量不连续依赖于观测数据.用来求解非线性不适定问题的方法主要有Tikhonov正则化方法和迭代正则化方法[1,2,3,4].Tikhonov正则化方法是通过引入正则化参数及稳定泛函,将目标泛函离散化,从而得到解的一个稳定近似,即正则化解.     

一类修正Helmholtz方程柯西问题的条件稳定性及正则化方法[英文]

本文研究带非齐次Dirichlet及Neumann数据的一类修正Helmholtz方程柯西问题. 该问题是不适定的, 需要借助一些正则化方法恢复其数值稳定性. 文章在解的先验假设下给出问题的条件稳定性; 构造一种广义-分数Tikhonov正则化方法处理这一问题, 并结合正则化参数的先验与后验选取规则获得该方法的收敛性估计; 用一些数值实验结果验证我们的方法是满意可行的.     

约束Fractional Tikhonov正则化的模迭代方法

反问题是现在数学物理研究中的一个热点问题,而反问题求解面临的一个本质性困难是不适定性。求解不适定问题的普遍方法是:用与原不适定问题相“邻近”的适定问题的解去逼近原问题的解,这种方法称为正则化方法.如何建立有效的正则化方法是反问题领域中不适定问题研究的重要内容.当前,最为流行的正则化方法有基于变分原理的Tikhonov正则化及其改进方法,此类方法是求解不适定问题的较为有效的方法,在各类反问题的研究中被广泛采用,并得到深入研究.     

基于改进正则化蝙蝠算法求解第一类Fredholm积分方程

本文提出了一种改进正则化蝙蝠算法来求解第一类Fredholm积分方程.对蝙蝠算法的速度惯性系数做出调整以增加种群多样性,添加高斯扰动来进一步优化集群,并采用Tikhonov正则化方法解决不适定性.计算实例表明:改进正则化蝙蝠算法的收敛速度和精度都优于传统正则化蝙蝠算法,并解决了严重偏离点的问题.     

Laplace方程的Cauchy问题的一种正则化方法

本文主要研究Laplace方程的Cauchy问题,该问题在很多领域有广泛的应用.众所周知,Laplace方程的Cauchy问题是严重不适定问题,即其解不连续依赖于所给的Cauchy数据.本文应用一个高阶Tikhonov正则化方法求解矩形区域上的Laplace方程的Cauchy问题,在对精确解的适当的先验界假设和正则化参数选取下,得到了相应的收敛性估计,数值结果表明所提的方法是高效稳定的.     

非线性不适定问题的双参数正则化

在Banach空间中, 研究一类非线性不适定问题的正则化. 所研究的算子是多值的, 且是近似的. 假定原始问题是可解的, 利用带双参数的Tikhonov正则化方法构造出强收敛的逼近步骤. 所得结论是前人结论的推广和延拓.     

确定热方程未知源问题的超阶正则化方法

该文研究了热传导方程中未知源的确定问题.针对问题的不适定性,提出了一种结合超阶惩罚项的Tikhonov正则化方法.在由偏差原理选取正则化参数情况下,方法能够在不同光滑条件下获得最优收敛阶.计算过程不需要事先知道光滑度和精确解的先验界.数值试验表明,该方法是有效和稳定的.     

不适定问题的迭代Tikhonov正则化方法

Tikhonov正则化方法是研究不适定问题最重要的正则化方法之一,但由于这种方法的饱和效应,使得不可能随着解的光滑性假设的提高而提高收敛率,即不能使正则解与准确解的误差估计达到阶数最优．本文所讨论的迭代的Tikhonov正则化方法对此进行了改进,保证了误差估计总可以达到阶数最优．数值试验结果表明计算效果良好．     

A new variant of L-curve for Tikhonov regularization

In this paper we introduce a new variant of L-curve to estimate the Tikhonov regularization parameter for the regularization of discrete ill-posed problems. This method uses the solution norm versus the regularization parameter. The numerical efficiency of this new method is also discussed by considering some test problems.     

Regularization Using QR Factorization and the Estimation of the Optimal Parameter

In this paper we propose a direct regularization method using QR factorization for solving linear discrete ill-posed problems. The decomposition of the coefficient matrix requires less computational cost than the singular value decomposition which is usually used for Tikhonov regularization. This method requires a parameter which is similar to the regularization parameter of Tikhonov's method. In order to estimate the optimal parameter, we apply three well-known parameter choice methods for Tikhonov regularization.This revised version was published online in October 2005 with corrections to the Cover Date.     

Short-term cascaded hydroelectric system scheduling based on chaotic particle swarm optimization using improved logistic map

In order to solve the model of short-term cascaded hydroelectric system scheduling, a novel chaotic particle swarm optimization (CPSO) algorithm using improved logistic map is introduced, which uses the water discharge as the decision variables combined with the death penalty function. According to the principle of maximum power generation, the proposed approach makes use of the ergodicity, symmetry and stochastic property of improved logistic chaotic map for enhancing the performance of particle swarm optimization (PSO) algorithm. The new hybrid method has been examined and tested on two test functions and a practical cascaded hydroelectric system. The experimental results show that the effectiveness and robustness of the proposed CPSO algorithm in comparison with other traditional algorithms.     

Regularization of backward heat conduction problem

We study the backward heat conduction problem in an unbounded region. The problem is ill-posed, in the sense that the solution if it exists, does not depend continuously on the data. Continuous dependence of the data is restored by cutting-off high frequencies in Fourier domain. The cut-off parameter acts as a regularization parameter. The discrepancy principle, for choosing the regularization parameter and double exponential transformation methods for numerical implementation of regularization method have been used. An example is presented to illustrate applicability and accuracy of the proposed method.     

A hybrid simplex search and particle swarm optimization for unconstrained optimization

This paper proposes the hybrid NM-PSO algorithm based on the Nelder–Mead (NM) simplex search method and particle swarm optimization (PSO) for unconstrained optimization. NM-PSO is very easy to implement in practice since it does not require gradient computation. The modification of both the Nelder–Mead simplex search method and particle swarm optimization intends to produce faster and more accurate convergence. The main purpose of the paper is to demonstrate how the standard particle swarm optimizers can be improved by incorporating a hybridization strategy. In a suite of 20 test function problems taken from the literature, computational results via a comprehensive experimental study, preceded by the investigation of parameter selection, show that the hybrid NM-PSO approach outperforms other three relevant search techniques (i.e., the original NM simplex search method, the original PSO and the guaranteed convergence particle swarm optimization (GCPSO)) in terms of solution quality and convergence rate. In a later part of the comparative experiment, the NM-PSO algorithm is compared to various most up-to-date cooperative PSO (CPSO) procedures appearing in the literature. The comparison report still largely favors the NM-PSO algorithm in the performance of accuracy, robustness and function evaluation. As evidenced by the overall assessment based on two kinds of computational experience, the new algorithm has demonstrated to be extremely effective and efficient at locating best-practice optimal solutions for unconstrained optimization.     

Error estimates for large-scale ill-posed problems

The computation of an approximate solution of linear discrete ill-posed problems with contaminated data is delicate due to the possibility of severe error propagation. Tikhonov regularization seeks to reduce the sensitivity of the computed solution to errors in the data by replacing the given ill-posed problem by a nearby problem, whose solution is less sensitive to perturbation. This regularization method requires that a suitable value of the regularization parameter be chosen. Recently, Brezinski et al. (Numer Algorithms 49, 2008) described new approaches to estimate the error in approximate solutions of linear systems of equations and applied these estimates to determine a suitable value of the regularization parameter in Tikhonov regularization when the approximate solution is computed with the aid of the singular value decomposition. This paper discusses applications of these and related error estimates to the solution of large-scale ill-posed problems when approximate solutions are computed by Tikhonov regularization based on partial Lanczos bidiagonalization of the matrix. The connection between partial Lanczos bidiagonalization and Gauss quadrature is utilized to determine inexpensive bounds for a family of error estimates. In memory of Gene H. Golub. This work was supported by MIUR under the PRIN grant no. 2006017542-003 and by the University of Cagliari.     

Large-scale Tikhonov regularization via reduction by orthogonal projection

This paper presents a new approach to computing an approximate solution of Tikhonov-regularized large-scale ill-posed least-squares problems with a general regularization matrix. The iterative method applies a sequence of projections onto generalized Krylov subspaces. A suitable value of the regularization parameter is determined by the discrepancy principle.     

求解离散不适定问题的正则化GMERR方法

迭代极小残差方法是求解大型线性方程组的常用方法, 通常用残差范数控制迭代过程.但对于不适定问题, 即使残差范数下降, 误差范数未必下降. 对大型离散不适定问题,组合广义最小误差(GMERR)方法和截断奇异值分解(TSVD)正则化方法, 并利用广义交叉校验准则(GCV)确定正则化参数,提出了求解大型不适定问题的正则化GMERR方法.数值结果表明, 正则化GMERR方法优于正则化GMRES方法.     

THE QUASI-BOUNDARY VALUE METHOD FOR IDENTIFYING THE INITIAL VALUE OF THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION

In this article, we consider to solve the inverse initial value problem for an inhomogeneous space-time fractional diffusion equation. This problem is ill-posed and the quasi-boundary value method is proposed to deal with this inverse problem and obtain the series expression of the regularized solution for the inverse initial value problem. We prove the error estimates between the regularization solution and the exact solution by using an a priori regularization parameter and an a posteriori regularization parameter choice rule. Some numerical results in one-dimensional case and two-dimensional case show that our method is effcient and stable.     

Central difference regularization method for inverse source problem on the Poisson equation

In this paper, we consider an inverse problem of determining an unknown source for the Poisson equation. Since this problem is mildly ill-posed, we apply a central difference regularization method to solve this problem. Furthermore, the convergence estimate is established under a priori choice of the regularization parameter. Some numerical results verify that the proposed method is stable and effective.