Convergence analysis of the Schwarz alternating method for unconstrained elliptic optimal control problems

TL;DR

This paper analyzes the convergence of the Schwarz alternating method for unconstrained elliptic optimal control problems, establishing conditions for convergence and how the rate depends on the regularization parameter.

Contribution

It extends convergence analysis of the Schwarz method from elliptic equations to optimal control problems, providing bounds and numerical validation.

Findings

Convergence of the Schwarz method for control problems follows from that for elliptic equations.

Convergence rate improves as the regularization parameter decreases.

Numerical results confirm theoretical convergence and rate dependence on regularization.

Abstract

In this paper we analyze the Schwarz alternating method for unconstrained elliptic optimal control problems. We discuss the convergence properties of the method in the continuous case first and then apply the arguments to the finite difference discretization case. In both cases, we prove that the Schwarz alternating method is convergent if its counterpart for an elliptic equation is convergent. Meanwhile, the convergence rate of the method for the elliptic equation under the maximum norm also gives a uniform upper bound (with respect to the regularization parameter $\alpha$) of the convergence rate of the method for the optimal control problem under the maximum norm of proper error merit functions in the continuous case or vectors in the discrete case. Our numerical results corroborate our theoretical results and show that with $\alpha$ decreasing to zero, the method will converge faster. We also give some exposition of this phenomenon.