Non-commutative Discretize-then-Optimize Algorithms for Elliptic PDE-Constrained Optimal Control Problems
For researchers in PDE-constrained optimization, this work relaxes a well-accepted design criterion, potentially simplifying algorithm development.
The paper shows that commutativity between discretize-then-optimize and optimize-then-discretize algorithms is not necessary for convergence in elliptic PDE-constrained optimal control, and proposes H1 semi-norm regularization to recover convergence when commutativity is lost. Numerical experiments validate the approach.
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.