Adaptive Newton-type schemes based on projections

Amrein, Mario; Hilber, Norbert

doi:10.1007/s40819-020-00868-5

Please use this identifier to cite or link to this item: https://doi.org/10.21256/zhaw-20857

Publication type:	Article in scientific journal
Type of review:	Peer review (publication)
Title:	Adaptive Newton-type schemes based on projections
Authors:	Amrein, Mario Hilber, Norbert
et. al:	No
DOI:	10.1007/s40819-020-00868-5 10.21256/zhaw-20857
Published in:	International Journal of Applied and Computational Mathematics
Volume(Issue):	6
Issue:	120
Issue Date:	2020
Publisher / Ed. Institution:	Springer
ISSN:	2199-5796 2349-5103
Language:	English
Subjects:	Newton-type method; Vector field; Adaptive root finding; Nonlinear equation; Globalization concept; Continuous Newton method
Subject (DDC):	510: Mathematics
Abstract:	In this work we present and discuss a possible globalization concept for Newton-type methods. We consider nonlinear problems f(x)=0 in Rn using the concepts from ordinary differential equations as a basis for the proposed numerical solution procedure. Thus, the starting point of our approach is within the framework of solving ordinary differential equations numerically. Accordingly, we are able to reformulate general Newton-type iteration schemes using an adaptive step size control procedure. In doing so, we derive and discuss a discrete adaptive solution scheme, thereby trying to mimic the underlying continuous problem numerically without losing the famous quadratic convergence regime of the classical Newton method in a vicinity of a regular solution. The derivation of the proposed adaptive iteration scheme relies on a simple orthogonal projection argument taking into account that, sufficiently close to regular solutions, the vector field corresponding to the Newton scheme is approximately linear. We test and exemplify our adaptive root-finding scheme using a few low-dimensional examples. Based on the presented examples, we finally show some performance data.
URI:	https://digitalcollection.zhaw.ch/handle/11475/20857
Fulltext version:	Published version
License (according to publishing contract):	CC BY 4.0: Attribution 4.0 International
Departement:	School of Management and Law
Organisational Unit:	Institute for Risk & Insurance (IRI) Institute of Wealth & Asset Management (IWA)
Appears in collections:	Publikationen School of Management and Law

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Amrein, M., & Hilber, N. (2020). Adaptive Newton-type schemes based on projections. International Journal of Applied and Computational Mathematics, 6(120). https://doi.org/10.1007/s40819-020-00868-5

Amrein, M. and Hilber, N. (2020) ‘Adaptive Newton-type schemes based on projections’, International Journal of Applied and Computational Mathematics, 6(120). Available at: https://doi.org/10.1007/s40819-020-00868-5.

M. Amrein and N. Hilber, “Adaptive Newton-type schemes based on projections,” International Journal of Applied and Computational Mathematics, vol. 6, no. 120, 2020, doi: 10.1007/s40819-020-00868-5.

AMREIN, Mario und Norbert HILBER, 2020. Adaptive Newton-type schemes based on projections. International Journal of Applied and Computational Mathematics. 2020. Bd. 6, Nr. 120. DOI 10.1007/s40819-020-00868-5

Amrein, Mario, and Norbert Hilber. 2020. “Adaptive Newton-Type Schemes Based on Projections.” International Journal of Applied and Computational Mathematics 6 (120). https://doi.org/10.1007/s40819-020-00868-5.

Amrein, Mario, and Norbert Hilber. “Adaptive Newton-Type Schemes Based on Projections.” International Journal of Applied and Computational Mathematics, vol. 6, no. 120, 2020, https://doi.org/10.1007/s40819-020-00868-5.