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ON THE SEPARABLE NONLINEAR LEAST SQUARES PROBLEMS

ISSN号：0254-9409
期刊名称：《计算数学：英文版》
时间：0
分类：O24[理学—计算数学;理学—数学]
作者机构：[1]LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Graduate University of Chinese Academy of Sciences, Beijing 100190, China, [2]LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
相关基金：Acknowledgments. This work is partially supported by Chinese NSF grant 10231060 and the CAS Knowledge Innovation Program.

关键词：非线性方程, 计算方法, 代数, 高斯-牛顿法, Separable nonlinear least squares problem, Variable projection method, Gauss-Newton method, Levenberg-Marquardt method, Trust region method, Asymptotical convergence rate, Data fitting

中文摘要：

可分离的非线性的最少的广场问题是非线性的最少的广场问题的一个特殊的类，在客观函数在变量的不同部分上线性、非线性的地方。如此的问题在实践有宽广应用。为这种问题的大多数存在算法从 Golub 和 Pereyra 建议的可变设计方法被导出，它在一个分开的框架下面利用可分性。然而，如果，方法将基于可变设计策略是无效的在那里存在一些限制到变量当真实问题总是做，就算限制简单地是球限制。我们在场由注意麻袋布的某些术语能从坡度被导出的事实基于特殊近似到麻袋布的一个新算法。我们的方法坚持说可变设计的所有优点基于方法，并且而且它能容易与信任区域方法被相结合并且能被用于一般抑制可分离的非线性的问题。我们的方法的集中分析被介绍，数字结果斧子也报导了。

英文摘要：

Separable nonlinear least squares problems are a special class of nonlinear least squares problems, where the objective functions are linear and nonlinear on different parts of variables. Such problems have broad applications in practice. Most existing algorithms for this kind of problems are derived from the variable projection method proposed by Golub and Pereyra, which utilizes the separability under a separate framework. However, the methods based on variable projection strategy would be invalid if there exist some constraints to the variables, as the real problems always do, even if the constraint is simply the ball constraint. We present a new algorithm which is based on a special approximation to the Hessian by noticing the fact that certain terms of the Hessian can be derived from the gradient. Our method maintains all the advantages of variable projection based methods, and moreover it can be combined with trust region methods easily and can be applied to general constrained separable nonlinear problems. Convergence analysis of our method is presented and numerical results are also reported.

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