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1.
基于混沌粒子群算法的Tikhonov正则化参数选取 总被引:2,自引:0,他引:2
Tikhonov正则化方法是求解不适定问题最为有效的方法之一,而正则化参数的最优选取是其关键.本文将混沌粒子群优化算法与Tikhonov正则化方法相结合,基于Morozov偏差原理设计粒子群的适应度函数,利用混沌粒子群优化算法的优点,为正则化参数的选取提供了一条有效的途径.数值实验结果表明,本文方法能有效地处理不适定问题,是一种实用有效的方法. 相似文献
2.
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. 相似文献
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1 前言 数学物理反问题是应用数学领域中成长和发展最快的领域之一.反问题大多是不适定的.对于不适定问题的解法已有不少的学者进行探索和研究,Tikhonov正则化方法是一种理论上最完备而在实践上行之有效的方法(参见[5,6,7,8,13]). 相似文献
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余瑞艳 《数学的实践与认识》2014,(10)
为克服Landweber迭代正则化方法在求解大规模不适定问题时收敛速度慢的不足,将埃特金加速技巧与不动点迭代相结合,构建了能快速收敛的改进Landweber迭代正则化方法.数值实验结果表明:改进的迭代正则化方法在稳定求解不适定问题时,能够快速地收敛至问题的最优解,较Landweber迭代正则化方法大大提高了收敛速度. 相似文献
6.
贺国强 《数学物理学报(A辑)》1996,16(2):154-161
该文考虑在某种Hilbert尺度上求解不适定问题的ТИФОНОВ正则化方法,讨论了对应的正则化解的收敛特征,并用此文的结果分析求解解析延拓问题的n阶ТИФОНОВ正则化方法的性质。 相似文献
7.
应用正则化子建立求解不适定问题的正则化方法的探讨 总被引:9,自引:0,他引:9
根据紧算子的奇异系统理论,提出一种新的正则化子进而建立了一类新的求解不适定问题的正则化方法。分别通过正则参数的先验选取和后验确定方法,证明了正则解的收敛性并得到了其最优的渐近收敛阶;验证了应用Newton迭代法计算最佳参数的可行性。最后建立了当算子与右端均有扰动时相应的正则化求解策略。文中所述方法完善了一般优化正则化策略的构造理论。 相似文献
8.
贺国强 《数学物理学报(A辑)》1996,(2)
该文考虑在某种Hilbert尺度上求解不适定问题的Tихонов正则化方法,讨论了对应的正则化解的收敛特征,并用此文的结果分析求解解析延拓问题的n阶Tихонов正则化方法的性质. 相似文献
9.
对非线性不适定算子方程,引入一种双参数正则化方法求解,讨论了这种正则化方法解的存在性、稳定性和收敛性. 相似文献
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Tikhonov regularization is a popular method for the solution of linear discrete ill-posed problems with error-contaminated data. Nonstationary iterated Tikhonov regularization is known to be able to determine approximate solutions of higher quality than standard Tikhonov regularization. We investigate the choice of solution subspace in iterative methods for nonstationary iterated Tikhonov regularization of large-scale problems. Generalized Krylov subspaces are compared with Krylov subspaces that are generated by Golub–Kahan bidiagonalization and the Arnoldi process. Numerical examples illustrate the effectiveness of the methods. 相似文献
12.
Michiel E. Hochstenbach Lothar Reichel 《Journal of Computational and Applied Mathematics》2012,236(8):2179-2185
Linear discrete ill-posed problems of small to medium size are commonly solved by first computing the singular value decomposition of the matrix and then determining an approximate solution by one of several available numerical methods, such as the truncated singular value decomposition or Tikhonov regularization. The determination of an approximate solution is relatively inexpensive once the singular value decomposition is available. This paper proposes to compute several approximate solutions by standard methods and then extract a new candidate solution from the linear subspace spanned by the available approximate solutions. We also describe how the method may be used for large-scale problems. 相似文献
13.
Tikhonov regularization is one of the most popular methods for solving linear systems of equations or linear least-squares
problems with a severely ill-conditioned matrix A. This method replaces the given problem by a penalized least-squares problem. The present paper discusses measuring the residual
error (discrepancy) in Tikhonov regularization with a seminorm that uses a fractional power of the Moore-Penrose pseudoinverse
of AA
T
as weighting matrix. Properties of this regularization method are discussed. Numerical examples illustrate that the proposed
scheme for a suitable fractional power may give approximate solutions of higher quality than standard Tikhonov regularization. 相似文献
14.
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. 相似文献
15.
Tikhonov Regularization of Large Linear Problems 总被引:1,自引:0,他引:1
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. 相似文献
16.
Copositive optimization problems are particular conic programs: optimize linear forms over the copositive cone subject to
linear constraints. Every quadratic program with linear constraints can be formulated as a copositive program, even if some
of the variables are binary. So this is an NP-hard problem class. While most methods try to approximate the copositive cone
from within, we propose a method which approximates this cone from outside. This is achieved by passing to the dual problem,
where the feasible set is an affine subspace intersected with the cone of completely positive matrices, and this cone is approximated
from within. We consider feasible descent directions in the completely positive cone, and regularized strictly convex subproblems.
In essence, we replace the intractable completely positive cone with a nonnegative cone, at the cost of a series of nonconvex
quadratic subproblems. Proper adjustment of the regularization parameter results in short steps for the nonconvex quadratic
programs. This suggests to approximate their solution by standard linearization techniques. Preliminary numerical results
on three different classes of test problems are quite promising. 相似文献
17.
Alessandro Buccini Marco Donatelli Lothar Reichel 《Numerical Linear Algebra with Applications》2017,24(4)
Tikhonov regularization is one of the most popular approaches to solving linear discrete ill‐posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method. 相似文献
18.
This paper discusses the solution of large-scale linear discrete ill-posed problems with a noise-contaminated right-hand side. Tikhonov regularization is used to reduce the influence of the noise on the computed approximate solution. We consider problems in which the coefficient matrix is the sum of Kronecker products of matrices and present a generalized global Arnoldi method, that respects the structure of the equation, for the solution of the regularized problem. Theoretical properties of the method are shown and applications to image deblurring are described. 相似文献
19.
Marco Donatelli 《Numerical Algorithms》2012,60(4):651-668
Nonstationary iterated Tikhonov is an iterative regularization method that requires a strategy for defining the Tikhonov regularization parameter at each iteration and an early termination of the iterative process. A classical choice for the regularization parameters is a decreasing geometric sequence which leads to a linear convergence rate. The early iterations compute quickly a good approximation of the true solution, but the main drawback of this choice is a rapid growth of the error for later iterations. This implies that a stopping criteria, e.g. the discrepancy principle, could fail in computing a good approximation. In this paper we show by a filter factor analysis that a nondecreasing sequence of regularization parameters can provide a rapid and stable convergence. Hence, a reliable stopping criteria is no longer necessary. A geometric nondecreasing sequence of the Tikhonov regularization parameters into a fixed interval is proposed and numerically validated for deblurring problems. 相似文献
20.
In this paper, we consider large-scale linear discrete ill-posed problems where the right-hand side contains noise. Regularization techniques such as Tikhonov regularization are needed to control the effect of the noise on the solution. In many applications such as in image restoration the coefficient matrix is given as a Kronecker product of two matrices and then Tikhonov regularization problem leads to the generalized Sylvester matrix equation. For large-scale problems, we use the global-GMRES method which is an orthogonal projection method onto a matrix Krylov subspace. We present some theoretical results and give numerical tests in image restoration. 相似文献