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Regularization methods for uniformly rank-deficient nonlinear least-squares problems
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
Umeå University, Faculty of Science and Technology, Departement of Computing Science.
2005 (English)In: Journal of Optimization Theory and Applications, Vol. 127, no 1, 1-26 p.Article in journal (Refereed) Published
Abstract [en]

In solving the nonlinear least-squares problem of minimizing parallel to f(x)parallel to(2)(2), difficulties arise with standard approaches, such as the Levenberg-Marquardt approach, when the Jacobian of f is rank-deficient or very ill-conditioned at the solution. To handle this difficulty, we study a special class of least-squares problems that are uniformly rank-deficient, i.e., the Jacobian of f has the same deficient rank in the neighborhood of a solution. For such problems, the solution is not locally unique. We present two solution tecniques: (i) finding a minimum-norm solution to the basic problem, (ii) using a Tikhonov regularization. Optimality conditions and algorithms are given for both of these strategies. Asymptotical convergence properties of the algorithms are derived and confirmed by numerical experiments. Extensions of the presented ideas make it possible to solve more general nonlinear least-squares problems in which the Jacobian of f at the solution is rank-deficient or ill-conditioned.

Place, publisher, year, edition, pages
2005. Vol. 127, no 1, 1-26 p.
Identifiers
URN: urn:nbn:se:umu:diva-21929ISBN: 0022-3239 OAI: oai:DiVA.org:umu-21929DiVA: diva2:212184
Available from: 2009-04-21 Created: 2009-04-21 Last updated: 2009-04-21

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