On an Elliptical Trust-Region Procedure for Ill-Posed Nonlinear Least-Squares Problems

In this paper, we address the stable numerical solution of ill-posed nonlinear least-squares problems with small residual. We propose an elliptical trust-region reformulation of a Levenberg–Marquardt procedure. Thanks to an appropriate choice of the trust-region radius, the proposed procedure guaran...

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Bibliographic Details
Published in:Journal of optimization theory and applications 2018-09, Vol.178 (3), p.824-859
Main Authors: Bellavia, Stefania, Riccietti, Elisa
Format: Article
Language:English
Subjects:
Applications of Mathematics
Calculus of Variations and Optimal Control
Optimization
Engineering
Ill posed problems
Least squares
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Regularization
Theory of Computation
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Summary:In this paper, we address the stable numerical solution of ill-posed nonlinear least-squares problems with small residual. We propose an elliptical trust-region reformulation of a Levenberg–Marquardt procedure. Thanks to an appropriate choice of the trust-region radius, the proposed procedure guarantees an automatic choice of the free regularization parameters that, together with a suitable stopping criterion, ensures regularizing properties to the method. Specifically, the proposed procedure generates a sequence that even in case of noisy data has the potential to approach a solution of the unperturbed problem. The case of constrained problems is considered, too. The effectiveness of the procedure is shown on several examples of ill-posed least-squares problems.
ISSN:0022-3239
1573-2878
DOI:10.1007/s10957-018-1318-1