Publication: The independent set perturbation adjoint method: A new method of differentiating mesh-based fluids models
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The independent set perturbation adjoint method: A new method of differentiating mesh-based fluids models

- Article in a journal -
 

Author(s)
F. Fang , C. C. Pain , I. M. Navon , G. J. Gorman , M. D. Piggott , P. A. Allison

Published in
International Journal for Numerical Methods in Fluids

Year
2011

Publisher
John Wiley & Sons, Ltd.

Abstract
A new scheme for differentiating complex mesh-based numerical models (e.g. finite element models), the Independent Set Perturbation Adjoint method (ISP-Adjoint), is presented. Differentiation of the matrices and source terms making up the discrete forward model is realized by a graph coloring approach (forming independent sets of variables) combined with a perturbation method to obtain gradients in numerical discretizations. This information is then convolved with the ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward model. The adjoint code is simple to implement even with complex governing equations, discretization methods and non-linear parameterizations. Importantly, the adjoint code is independent of the implementation of the forward code. This greatly reduces the effort required to implement the adjoint model and maintain it as the forward model continues to be developed; as compared with more traditional approaches such as applying automatic differentiation tools. The approach can be readily extended to reduced-order models. The method is applied to a one-dimensional Burgers' equation problem, with a highly non-linear high-resolution discretization method, and to a two-dimensional, non-linear, reduced-order model of an idealized ocean gyre.

AD Theory and Techniques
Adjoint

BibTeX
@ARTICLE{
         Fang2011Tis,
       author = "Fang, F. and Pain, C. C. and Navon, I. M. and Gorman, G. J. and Piggott, M. D. and
         Allison, P. A.",
       title = "The independent set perturbation adjoint method: A new method of differentiating
         mesh-based fluids models",
       journal = "International Journal for Numerical Methods in Fluids",
       volume = "66",
       number = "8",
       publisher = "John Wiley \& Sons, Ltd.",
       issn = "1097-0363",
       url = "http://dx.doi.org/10.1002/fld.2297",
       doi = "10.1002/fld.2297",
       pages = "976--999",
       keywords = "automatic differentiation, optimization, adjoint, finite element, reduced-order
         models",
       year = "2011",
       ad_theotech = "Adjoint",
       abstract = "A new scheme for differentiating complex mesh-based numerical models (e.g. finite
         element models), the Independent Set Perturbation Adjoint method (ISP-Adjoint), is presented.
         Differentiation of the matrices and source terms making up the discrete forward model is realized by
         a graph coloring approach (forming independent sets of variables) combined with a perturbation
         method to obtain gradients in numerical discretizations. This information is then convolved with the
         ‘mathematical adjoint’, which uses the transpose matrix of the discrete forward
         model. The adjoint code is simple to implement even with complex governing equations, discretization
         methods and non-linear parameterizations. Importantly, the adjoint code is independent of the
         implementation of the forward code. This greatly reduces the effort required to implement the
         adjoint model and maintain it as the forward model continues to be developed; as compared with more
         traditional approaches such as applying automatic differentiation tools. The approach can be readily
         extended to reduced-order models. The method is applied to a one-dimensional Burgers' equation
         problem, with a highly non-linear high-resolution discretization method, and to a two-dimensional,
         non-linear, reduced-order model of an idealized ocean gyre."
}


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