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Backward step control for Hilbert space problems

  • Andreas PotschkaEmail author
Original Paper
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Abstract

We analyze backward step control globalization for finding zeros of Gâteaux-differentiable functions that map from a Banach space to a Hilbert space. The results include global convergence to a distinctive solution characterized by propagating the initial guess by a generalized Newton flow with guaranteed bounds on the discrete nonlinear residual norm decrease and an (also numerically) easily controllable asymptotic linear residual convergence rate. The convergence theory can be exploited to construct efficient numerical methods, which we demonstrate for the case of a Krylov–Newton method and an approximation-by-discretization multilevel framework. Both approaches optimize the asymptotic linear residual convergence rate, either over the Krylov subspace or through adaptive discretization, which in turn yields practical and efficient stopping criteria and refinement strategies that balance the nonlinear residuals with the relative residuals of the linear systems. We apply these methods to the class of nonlinear elliptic boundary value problems and present numerical results for the Carrier equation and the minimum surface equation.

Keywords

Newton-type methods Globalization Hilbert space Backward step control 

Mathematics Subject Classification (2010)

65J15 58C15 65F08 35J66 74S05 

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Notes

Acknowledgements

The author is grateful to Felix Lenders and to Gerd Wachsmuth for comments on an earlier draft of this manuscript and to the anonymous reviewers for their fruitful comments.

Funding information

This work was funded by the European Research Council through S. Engell’s and H.G. Bock’s ERC Advanced Investigator Grant MOBOCON (291 458) and by the German Federal Ministry of Education and Research under grants 05M2013-GOSSIP and 05M2016-MOPhaPro.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Interdisciplinary Center for Scientific ComputingHeidelberg UniversityHeidelbergGermany

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