Publication: Solving parameter estimation problems with discrete adjoint exponential integrators
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Solving parameter estimation problems with discrete adjoint exponential integrators

- Article in a journal -
 

Author(s)
Ulrich Römer , Mahesh Narayanamurthi , Adrian Sandu

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 solution of inverse problems in a variational setting finds best estimates of the model parameters by minimizing a cost function that penalizes the mismatch between model outputs and observations. The gradients required by the numerical optimization process are computed using adjoint models. Exponential integrators are a promising family of time discretization schemes for evolutionary partial differential equations. In order to allow the use of these discretization schemes in the context of inverse problems, adjoints of exponential integrators are required. This work derives the discrete adjoint formulae for W-type exponential propagation iterative methods of Runge–Kutta type (EPIRK-W). These methods allow arbitrary approximations of the Jacobian while maintaining the overall accuracy of the forward integration. The use of Jacobian approximation matrices that do not depend on the model state avoids the complex calculation of Hessians in the discrete adjoint formulae. The adjoint code itself is generated efficiently via algorithmic differentiation and used to solve inverse problems with the Lorenz-96 model and a model from computational magnetics. Numerical results are encouraging and indicate the suitability of exponential integrators for this class of problems.

Cross-References
Christianson2018Sio

BibTeX
@ARTICLE{
         Romer2018Spe,
       crossref = "Christianson2018Sio",
       author = "Ulrich R{\"o}mer and Mahesh Narayanamurthi and Adrian Sandu",
       title = "Solving parameter estimation problems with discrete adjoint exponential integrators",
       journal = "Optimization Methods \& Software",
       volume = "33",
       number = "4--6",
       pages = "750--770",
       year = "2018",
       publisher = "Taylor \& Francis",
       doi = "10.1080/10556788.2018.1448087",
       url = "https://doi.org/10.1080/10556788.2018.1448087",
       eprint = "https://doi.org/10.1080/10556788.2018.1448087",
       abstract = "The solution of inverse problems in a variational setting finds best estimates of
         the model parameters by minimizing a cost function that penalizes the mismatch between model outputs
         and observations. The gradients required by the numerical optimization process are computed using
         adjoint models. Exponential integrators are a promising family of time discretization schemes for
         evolutionary partial differential equations. In order to allow the use of these discretization
         schemes in the context of inverse problems, adjoints of exponential integrators are required. This
         work derives the discrete adjoint formulae for W-type exponential propagation iterative methods of
         Runge–Kutta type (EPIRK-W). These methods allow arbitrary approximations of the Jacobian
         while maintaining the overall accuracy of the forward integration. The use of Jacobian approximation
         matrices that do not depend on the model state avoids the complex calculation of Hessians in the
         discrete adjoint formulae. The adjoint code itself is generated efficiently via algorithmic
         differentiation and used to solve inverse problems with the Lorenz-96 model and a model from
         computational magnetics. Numerical results are encouraging and indicate the suitability of
         exponential integrators for this class of problems.",
       booktitle = "Special issue of Optimization Methods \& Software: Advances in
         Algorithmic Differentiation",
       editor = "Bruce Christianson and Shaun A. Forth and Andreas Griewank"
}


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