Numerical Computation of the Gradient and the Action of the Hessian for Time-Dependent PDE-Constrained Optimization Problems

We present a systematic derivation of the algorithms required for computing the gradient and the action of the Hessian of an arbitrary misfit function for large-scale parameter estimation problems involving linear time-dependent PDEs with stationary coefficients. These algorithms are derived using the adjoint method for time-stepping schemes of arbitrary order and are therefore well-suited for distributed parameter estimation problems where the forward solution needs to be solved to high accuracy. Two examples demonstrate how specific PDEs can be prepared for use with these algorithms. A numerical example illustrates that the order of accuracy of higher-order time-stepping schemes is inherited by their corresponding adjoint time-stepping schemes and misfit gradient computations.