Parallel Multilevel Restricted Schwarz Preconditioners with Pollution Removing for PDE-Constrained Optimization

We develop a class of V-cycle type multilevel restricted additive Schwarz (RAS) methods and study the numerical and parallel performance of the new fully coupled methods for solving large sparse Jacobian systems arising from the discretization of some optimization problems constrained by nonlinear partial differential equations. Straightforward extensions of the one-level RAS to multilevel do not work due to the pollution effects of the coarse interpolation. We then introduce, in this paper, a pollution removing coarse-to-fine interpolation scheme for one of the components of the multi-component linear system, and show numerically that the combination of the new interpolation scheme with the RAS smoothed multigrid method provides an effective family of techniques for solving rather difficult PDE-constrained optimization problems. Numerical examples involving the boundary control of incompressible Navier-Stokes flows are presented in detail.