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A flow perspective on nonlinear least-squares problems

Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß-Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the correspon...

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Bibliographic Details
Main Authors: Bock, Hans Georg (Author) , Gutekunst, Jürgen (Author) , Potschka, Andreas (Author) , Suárez Garcés, María Elena (Author)
Format: Article (Journal)
Language:English
Published: 03 October 2020
In: Vietnam journal of mathematics
Year: 2020, Volume: 48, Issue: 4, Pages: 987-1003
ISSN:2305-2228
DOI:10.1007/s10013-020-00441-z
Online Access:Verlag, kostenfrei, Volltext: https://doi.org/10.1007/s10013-020-00441-z
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Author Notes:Hans Georg Bock, Jürgen Gutekunst, Andreas Potschka, María Elena Suaréz Garcés
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Summary:Just as the damped Newton method for the numerical solution of nonlinear algebraic problems can be interpreted as a forward Euler timestepping on the Newton flow equations, the damped Gauß-Newton method for nonlinear least squares problems is equivalent to forward Euler timestepping on the corresponding Gauß-Newton flow equations. We highlight the advantages of the Gauß-Newton flow and the Gauß-Newton method from a statistical and a numerical perspective in comparison with the Newton method, steepest descent, and the Levenberg-Marquardt method, which are respectively equivalent to Newton flow forward Euler, gradient flow forward Euler, and gradient flow backward Euler. We finally show an unconditional descent property for a generalized Gauß-Newton flow, which is linked to Krylov-Gauß-Newton methods for large-scale nonlinear least squares problems. We provide numerical results for large-scale problems: An academic generalized Rosenbrock function and a real-world bundle adjustment problem from 3D reconstruction based on 2D images.
Item Description:Gesehen am 25.11.2021
Physical Description:Online Resource
ISSN:2305-2228
DOI:10.1007/s10013-020-00441-z

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