Publication: Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions
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Estimating the expansion coefficients of a geomagnetic field model using first-order derivatives of associated Legendre functions

- Article in a journal -
 

Area
Geophysics

Author(s)
H. Martin Bücker , Johannes Willkomm

Published in
Special issue of Optimization Methods & Software: Advances in Algorithmic Differentiation Optimization Methods & Software

Editor(s)
Bruce Christianson, Shaun A. Forth, Andreas Griewank

Year
2018

Publisher
Taylor & Francis

Abstract
The associated Legendre functions are defined on a closed interval. Thus, their derivatives do not exist at the endpoints of the interval. However, one-sided derivatives may exist at the endpoints if the ordinary limit is replaced by a one-sided limit. When computer models that evaluate associated Legendre functions at an endpoint are transformed by automatic differentiation, an approach is needed that is tailored to one-sided derivatives. Rather than employing a black-box approach of automatic differentiation, a hierarchical approach is introduced that is based on analytic first-order one-sided derivatives of the associated Legendre functions at the endpoints. This hierarchical approach is implemented in the automatic differentiation software ADiMat and its feasibility is demonstrated in a parameter estimation problem arising from geomagnetic field modelling.

Cross-References
Christianson2018Sio

AD Tools
ADiMat

AD Theory and Techniques
Hierarchical Approach

BibTeX
@ARTICLE{
         Bucker2018Ete,
       crossref = "Christianson2018Sio",
       author = "H. Martin B{\"u}cker and Johannes Willkomm",
       title = "Estimating the expansion coefficients of a geomagnetic field model using first-order
         derivatives of associated {L}egendre functions",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "924--944",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1448086",
       url = "https://doi.org/10.1080/10556788.2018.1448086",
       eprint = "https://doi.org/10.1080/10556788.2018.1448086",
       abstract = "The associated Legendre functions are defined on a closed interval. Thus, their
         derivatives do not exist at the endpoints of the interval. However, one-sided derivatives may exist
         at the endpoints if the ordinary limit is replaced by a one-sided limit. When computer models that
         evaluate associated Legendre functions at an endpoint are transformed by automatic differentiation,
         an approach is needed that is tailored to one-sided derivatives. Rather than employing a black-box
         approach of automatic differentiation, a hierarchical approach is introduced that is based on
         analytic first-order one-sided derivatives of the associated Legendre functions at the endpoints.
         This hierarchical approach is implemented in the automatic differentiation software ADiMat and its
         feasibility is demonstrated in a parameter estimation problem arising from geomagnetic field
         modelling.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank",
       ad_area = "Geophysics",
       ad_tools = "ADiMat",
       ad_theotech = "Hierarchical Approach"
}


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