A flow perspective on nonlinear least-squares problems

• Verfasst von: Bock, Hans Georg [VerfasserIn]
• Verfasst von: Gutekunst, Jürgen [VerfasserIn]
• Verfasst von: Potschka, Andreas [VerfasserIn]
• Verfasst von: Suárez Garcés, María Elena [VerfasserIn]
• Titel: A flow perspective on nonlinear least-squares problems
• Verf.angabe: Hans Georg Bock, Jürgen Gutekunst, Andreas Potschka, María Elena Suaréz Garcés
• E-Jahr: 2020
• Jahr: 03 October 2020
• Umfang: 17 S.
• Fussnoten: Gesehen am 25.11.2021
• Titel Quelle: Enthalten in: Vietnam journal of mathematics
• Ort Quelle: Singapore : Springer, 1999
• Jahr Quelle: 2020
• Band/Heft Quelle: 48(2020), 4, Seite 987-1003
• ISSN Quelle: 2305-2228
• DOI: doi:10.1007/s10013-020-00441-z
• Datenträger: Online-Ressource
• Sprache: eng
• K10plus-PPN: 1779072058
• K10plus-PPN:
  Universitätsbibliothek
• Lokale URL UB: Zum Volltext

Abstract

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.