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The implementation of the discrete adjoint method for exponential timedifferencing (ETD) schemes is considered. This is important for parameterestimation problems that are constrained by stiff time-dependent PDEs when thediscretized PDE system is solved using an exponential integrator. We alsodiscuss the closely related topic of computing the action of the sensitivitymatrix on a vector, which is required when performing a sensitivity analysis.The PDE system is assumed to be semi-linear and can be the result of alinearization of a nonlinear PDE, leading to exponential Rosenbrock-typemethods. We discuss the computation of the derivatives of the$\varphi$-functions that are used by ETD schemes and find that the derivativesstrongly depend on the way the $\varphi$-functions are evaluated numerically. Ageneral adjoint exponential integration method, required when computing thegradients, is developed and its implementation is illustrated by applying it tothe Krogstad scheme. The applicability of the methods developed here to patternformation problems is demonstrated using the Swift-Hohenberg model.

The Discrete Adjoint Method for Exponential Integration