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

Understand

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.