Efficient Algorithms for Solution of Regularized Total Least Squares
Efficient Algorithms for Solution of Regularized Total Least Squares
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DOI：
10.1137/s0895479802419889
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发表时间：
2005-02
期刊：
影响因子：
--
通讯作者：
R. Renaut;Hongbin Guo
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文献类型：
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作者：
R. Renaut;Hongbin Guo
考虑错误污染的系统$Ax \approx B$，其中A是病态的。这样的系统可以使用Tikhonov类正则化总最小二乘（RTLS）方法来求解。Golub，汉森和O 'Leary [SIAM J. Matrix Anal.应用程序、21（1999），pp. 185- 194]提出了一种用于求解RTLS问题的增广拉格朗日公式的参数相关直接算法，以及Sima、货车Huelman和Golub [Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers，Tech.报告SCCM-03-03，SCCM，斯坦福大学，斯坦福大学，CA，2003]介绍了一种基于二次特征值问题的求解技术RTLSQEP。Guo和Renaut [A regularized total least squares algorithm，in Total Least Squares and Errors-in-Variables Modeling：Analysis，Algorithms and Applications，S.货车休伊和P. Lemmerling编辑，Kluwer Academic Publishers，多尔德雷赫特，荷兰，2002，pp. [57- 66]导出了RTLS的特征值问题，该问题可用迭代逆幂法求解。在这里，我们提出了一种替代推导的特征值问题的约束TLS通过增广拉格朗日约束归一化残差。这扩展了特征问题的分析，并导致推导出更有效的算法相比，原来的配方。提出了基于对分搜索和标准L曲线方法的附加算法。这些算法根据需要规定的参数而变化。数值和收敛性结果支持不同的版本和对比RTLSQEP。
Error-contaminated systems $Ax \approx b$, for which A is ill-conditioned, are considered. Such systems may be solved using Tikhonov-like regularized total least squares (RTLS) methods. Golub, Hansen, and O'Leary [SIAM J. Matrix Anal. Appl., 21 (1999), pp. 185--194] presented a parameter-dependent direct algorithm for the solution of the augmented Lagrange formulation for the RTLS problem, and Sima, Van Huffel, and Golub [Regularized Total Least Squares Based on Quadratic Eigenvalue Problem Solvers, Tech. Report SCCM-03-03, SCCM, Stanford University, Stanford, CA, 2003] have introduced a technique for solution based on a quadratic eigenvalue problem, RTLSQEP. Guo and Renaut [A regularized total least squares algorithm, in Total Least Squares and Errors-in-Variables Modeling: Analysis, Algorithms and Applications, S. Van Huffel and P. Lemmerling, eds., Kluwer Academic Publishers, Dordrecht, The Netherlands, 2002, pp. 57--66] derived an eigenproblem for the RTLS which can be solved using the iterative inverse power method. Here we present an alternative derivation of the eigenproblem for constrained TLS through the augmented Lagrangian for the constrained normalized residual. This extends the analysis of the eigenproblem and leads to derivation of more efficient algorithms compared to the original formulation. Additional algorithms based on bisection search and a standard L-curve approach are presented. These algorithms vary with respect to the parameters that need to be prescribed. Numerical and convergence results supporting the different versions and contrasting with RTLSQEP are presented.