Solving parameter estimation problems with discrete adjoint exponential integrators

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.