Parallel generalized Lagrange–Newton method for fully coupled solution of PDE-constrained optimization problems with bound-constraints

Abstract

In large-scale simulations of optimization problems constrained by partial differential equations (PDEs), the class of fully coupled methods is attracting great attention due to their impressive robustness and scalability. In this study, we introduce and investigate a parallel generalized Lagrange–Newton solver, which is based on the Lagrange active–set reduced–space (LASRS) method and the domain decomposition technique, for solving a family of PDE-constrained optimization problems with inequality constraints, i.e., the distributed control problem with bound-constraints. In the approach, a Lagrangian functional is constructed and the corresponding first-order optimality conditions are derived. After that, the resultant nonlinear algebraic system is solved by the active–set reduced–space method to guarantee the nonlinear consistency of the fully coupled system in a monolithic way. For the linear system arising at each generalized Newton iteration, the overlapping additive Schwarz preconditioners are employed to enhance the convergence of the linear iterations and the scalability of the large-scale simulations. Numerical results on the Tianhe supercomputer show that the fully coupled methods are robust and effective for some two- and three-dimensional distributed control problems with bound-constraints.