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

TL;DR

This paper develops algorithms for efficiently computing gradients and Hessian actions in large-scale, time-dependent PDE-constrained optimization, enabling high-accuracy parameter estimation.

Contribution

It introduces a systematic derivation of adjoint-based algorithms for gradient and Hessian computations applicable to arbitrary order time-stepping schemes in PDE-constrained optimization.

Findings

Algorithms are suitable for distributed parameter estimation.

Higher-order time-stepping schemes inherit accuracy in adjoint computations.

Numerical examples validate the effectiveness of the proposed methods.

Abstract

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.