miun.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
Local results for the Gauss-Newton method on constrained rank-deficient nonlinear least squares
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering, Physics and Mathematics.
2004 (English)In: Mathematics of Computation, ISSN 0025-5718, E-ISSN 1088-6842, Vol. 73, no 248, 1865-1883 p.Article in journal (Refereed) Published
Abstract [en]

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 $J = partial f/ partial x$, $f = [f_{1};f_{2}]$, may be ill conditioned at the solution. We analyze the important special case when $J_{1}$ and/or $J$ 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.

Place, publisher, year, edition, pages
2004. Vol. 73, no 248, 1865-1883 p.
Keyword [en]
Nonlinear least squares, nonlinear constraints, optimization, regularization, Gauss-Newton method
National Category
Computer Science
Identifiers
URN: urn:nbn:se:miun:diva-3968DOI: 10.1090/S0025-5718-03-01611-9ISI: 000222002400014Scopus ID: 2-s2.0-5744234026Local ID: 4383OAI: oai:DiVA.org:miun-3968DiVA: diva2:29000
Available from: 2008-09-30 Created: 2008-09-30 Last updated: 2016-09-23Bibliographically approved

Open Access in DiVA

No full text

Other links

Publisher's full textScopus

Search in DiVA

By author/editor
Gulliksson, Mårten E.
By organisation
Department of Engineering, Physics and Mathematics
In the same journal
Mathematics of Computation
Computer Science

Search outside of DiVA

GoogleGoogle Scholar

Altmetric score

Total: 47 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf