  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 discretizations for
evolutionary partial differential equations. In order to allow the use of these
discretizations in the context of inverse problems adjoints of exponential
integrators are required. This work derives the discrete adjoint formulae for a
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, and allows efficient
adjoint code generation via algorithmic differentiation. We use the discrete
EPIRK-W adjoints to solve inverse problems with the Lorenz-96 model and a
computational magnetics benchmark test. Numerical results validate our
theoretical derivations.
