Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares

Authors:	Jerry Eriksson Må rten E. Gulliksson.

Affiliation:	Department of Computing Science, Umeå, Sweden ; Department of Engineering, Physics, and Mathematics, Mid-Sweden University, Sundsvall, Sweden

Abstract:	A nonlinear least squares problem with nonlinear constraints may be ill posed or even rank-deficient in two ways. Considering the problem formulated as $min_{x} 1/2Vert f_{2}(x) Vert _{2}^{2}$ subject to the constraints $f_{1}(x) = 0$ , the Jacobian $J_{1} = partial f_{1}/ partial x$ and/or the Jacobian , $f = [f_{1};f_{2}]$ , may be ill conditioned at the solution. We analyze the important special case when $J_{1}$ and/or do not have full rank at the solution. In order to solve such a problem, we formulate a nonlinear least norm problem. Next we describe a truncated Gauss-Newton method. We show that the local convergence rate is determined by the maximum of three independent Rayleigh quotients related to three different spaces in $mathbb{R} ^{n}$ . Another way of solving an ill-posed nonlinear least squares problem is to regularize the problem with some parameter that is reduced as the iterates converge to the minimum. Our approach is a Tikhonov based local linear regularization that converges to a minimum norm problem. This approach may be used both for almost and rank-deficient Jacobians. Finally we present computational tests on constructed problems verifying the local analysis.

Keywords:	Nonlinear least squares nonlinear constraints optimization regularization Gauss-Newton method

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