Approximate Gauss-Newton methods for nonlinear least squares problems
Approximate Gauss-Newton methods for nonlinear least squares problems
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DOI：
10.1137/050624935
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发表时间：
2007-01-01
影响因子：
3.1
通讯作者：
Nichols, N. K.
中科院分区：
文献类型：
--
作者：
Gratton, S.;Lawless, A. S.;Nichols, N. K.
Gauss-Newton算法是一种常用于求解非线性最小二乘问题的迭代方法。它特别适合于处理大气和海洋预报中出现的超大规模变分数据同化问题。该程序由一系列的线性最小二乘近似的非线性问题，其中每一个是解决了一个“内部”的直接或迭代过程。与牛顿法及其变体相比，该算法是有吸引力的，因为它不需要评估目标函数的Hessian中的二阶导数。在实际应用中，精确的高斯-牛顿方法在气象预报中的操作成本太高，为了减少计算成本并在真实的时间内解决问题，进行了各种近似。本文研究了资料同化中常用的两种近似方法对高斯-牛顿法收敛性的影响。首先，我们研究“截断”高斯-牛顿方法的内部线性最小二乘问题没有得到精确解决，第二，我们研究“扰动”高斯-牛顿方法，其中真正的线性化内部问题是近似的简化，或扰动，线性最小二乘问题。我们给出的条件，确保截断和扰动高斯-牛顿方法收敛，并得出收敛速度的迭代。通过一个简单的数值例子说明了结果。本文介绍了一个典型气象系统资料同化问题的实际应用。
The Gauss-Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an "inner" direct or iterative process. In comparison with Newton's method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss-Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss - Newton method of two types of approximation used commonly in data assimilation. First, we examine "truncated" Gauss-Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine "perturbed" Gauss-Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss-Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.