Title: Non-commutative Discretize-then-Optimize Algorithms for Elliptic PDE-Constrained Optimal Control Problems,
Abstract: In this paper, we analyze the convergence of several discretize-then-optimize
algorithms, based on either a second-order or a fourth-order finite difference
discretization, for solving elliptic PDE-constrained optimization or optimal
control problems. To ensure the convergence of a discretize-then-optimize
algorithm, one well-accepted criterion is to choose or redesign the
discretization scheme such that the resultant discretize-then-optimize
algorithm commutes with the corresponding optimize-then-discretize algorithm.
In other words, both types of algorithms would give rise to exactly the same
discrete optimality system. However, such an approach is not trivial. In this
work, by investigating a simple distributed elliptic optimal control problem,
we first show that enforcing such a stringent condition of commutative property
is only sufficient but not necessary for achieving the desired convergence. We
then propose to add some suitable $H_1$ semi-norm penalty/regularization terms
to recover the lost convergence due to the inconsistency caused by the loss of
commutativity. Numerical experiments are carried out to verify our theoretical
analysis and also validate the effectiveness of our proposed regularization
techniques.