A new MOX Report entitled “Uniform Convergence of the Schwarz Alternating Method for Optimal Control Problems” by Ciaramella, G.; Gong, W.; Kwok, F., Tan, Z. has appeared in the MOX Report Collection.
Check it out here: https://www.mate.polimi.it/biblioteca/add/qmox/63-2026.pdf
Abstract: In this paper, we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems, which is equivalent to the corresponding method for the associated saddle-point systems. A distinctive feature in this setting is that the local error propagation operators are not necessarily nonexpansive in the energy norm, which stands in marked contrast to the standard elliptic boundary-value case. We develop a rigorous uniform convergence theory in the continuous setting and then extend the analysis to finite difference discretizations. In both formulations, we prove that the Schwarz iteration converges whenever its counterpart for the underlying elliptic equation is convergent. Furthermore, we show that the contraction factor for the auxiliary elliptic equation in the maximum norm provides a uniform upper bound, which is independent of the regularization parameter $\alpha$, for the contraction factor of the optimal control ! iteration in the same norm. The theoretical framework is also extended to cover one-level alternating Schwarz and parallel Schwarz variants. Numerical experiments are presented to validate the theoretical results.
