# Solving Parameter Estimation Problems with Discrete Adjoint Exponential Integrators

## TL;DR

This paper develops discrete adjoint formulas for exponential integrators used in inverse problems, enabling efficient gradient computation and optimization for complex PDE models.

## Contribution

It derives the discrete adjoint formulas for EPIRK-W exponential integrators, facilitating their use in inverse problems with efficient adjoint code generation.

## Key findings

- Validated adjoint formulas with Lorenz-96 model
- Demonstrated efficiency in inverse problem solving
- Enabled Jacobian approximation independence from model state

## 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 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.