A Practical Factorization of a Schur Complement for PDE-Constrained Distributed Optimal Control

A distributed optimal control problem with the constraint of a linear elliptic partial differential equation is considered. A necessary optimality condition for this problem forms a saddle point system, the efficient and accurate solution of which is crucial. A new factorization of the Schur complement for such a system is proposed and its characteristics discussed. The factorization introduces two complex factors that are complex conjugate to each other. The proposed solution methodology involves the application of a parallel linear domain decomposition solver—FETI-DPH— for the solution of the subproblems with the complex factors. Numerical properties of FETI-DPH in this context are demonstrated, including numerical and parallel scalability and regularization dependence. The new factorization can be used to solve Schur complement systems arising in both range-space and full-space formulations. In both cases, numerical results indicate that the complex factorization is promising. Especially, in the full-space method with the new factorization, the number of iterations required for convergence is independent of regularization parameter values.