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Accuracy of parameter sensitivities of DAE systems using finite difference methods

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Area
Differential-Algebraic Equation

Author(s)
Atiyah Elsheikh , Wolfgang Wiechert

Published in
MATHMOD 2012: The 7th Vienna International Conference on Mathematical Modelling

Editor(s)
Inge Troch, Felix Breitenecker

Year
2012

Publisher
IFAC-PapersOnLine.net

Abstract
This work evaluates the accuracy of Finite Difference (FD) methods when computing parameter sensitivities of badly-scaled DAE systems. Due to their simple implementation, FD methods are commonly favoured especially when the underlying mathematical model is a hard-coded sophisticated simulator. Nevertheless, FD methods may impose serious numerical problems even if FD step sizes and solver tolerances w.r.t. the order of the FD scheme are ideally selected. Judging the precision of the resulting parameter sensitivities is practically difficult. With the availability of powerful Automatic Differentiation (ad) tools for equation-based simulation languages like ADModelica, there is a new possibility to examine step sizes of various FD schemes, solver tolerances and the resulting precision for realistic large scale examples. This can be done by comparing numerical parameter sensitivities with highly precise analytical solutions using direct integration of sensitivity equation systems generated by ad techniques. It is shown with a realistically sized example that FD methods are actually more critical than usually assumed.

AD Tools
ADModelica

AD Theory and Techniques
Forward Mode

BibTeX
@INPROCEEDINGS{
         Elsheikh2012Aop,
       author = "Atiyah Elsheikh and Wolfgang Wiechert",
       title = "Accuracy of parameter sensitivities of {DAE} systems using finite difference methods",
       booktitle = "MATHMOD 2012: The 7th Vienna International Conference on Mathematical Modelling",
       year = "2012",
       editor = "Troch, Inge and Breitenecker, Felix",
       volume = "7",
       number = "1",
       series = "Mathematical Modelling",
       pages = "136--142",
       address = "Vienna, Austria",
       month = "Feb",
       publisher = "IFAC-PapersOnLine.net",
       abstract = "This work evaluates the accuracy of Finite Difference (FD) methods when computing
         parameter sensitivities of badly-scaled DAE systems. Due to their simple implementation, FD methods
         are commonly favoured especially when the underlying mathematical model is a hard-coded
         sophisticated simulator. Nevertheless, FD methods may impose serious numerical problems even if FD
         step sizes and solver tolerances w.r.t. the order of the FD scheme are ideally selected. Judging the
         precision of the resulting parameter sensitivities is practically difficult. With the availability
         of powerful Automatic Differentiation (AD) tools for equation-based simulation languages like
         ADModelica, there is a new possibility to examine step sizes of various FD schemes, solver
         tolerances and the resulting precision for realistic large scale examples. This can be done by
         comparing numerical parameter sensitivities with highly precise analytical solutions using direct
         integration of sensitivity equation systems generated by AD techniques. It is shown with a
         realistically sized example that FD methods are actually more critical than usually assumed.",
       booktitle2 = "Proceeding of the 7th Vienna International Conference on Mathematical Modelling",
       doi = "10.3182/20120215-3-AT-3016.00024",
       keywords = "finite difference methods,differential algebraic equations,sensitivity
         analysis,automatic differentiation,accuracy,simulation languages",
       owner = "ElsheikhA",
       timestamp = "2012.11.10",
       ad_area = "Differential-Algebraic Equation",
       ad_tools = "ADModelica",
       ad_theotech = "Forward Mode"
}

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