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"
}