Nonlinear Preconditioning Techniques for Full-Space Lagrange--Newton Solution of PDE-Constrained Optimization Problems

The full-space Lagrange--Newton algorithm is one of the numerical algorithms for solving problems arising from optimization problems constrained by nonlinear partial differential equations. Newton-type methods enjoy fast convergence when the nonlinearity in the system is well-balanced; however, for some problems, such as the control of incompressible flows, even linear convergence is difficult to achieve and a long stagnation period often appears in the iteration history. In this work, we introduce a nonlinearly preconditioned inexact Newton algorithm for the boundary control of incompressible flows. The system has nine field variables, and each field variable plays a different role in the nonlinearity of the system. The nonlinear preconditioner approximately removes some of the field variables, and as a result, the nonlinearity is balanced and inexact Newton converges much faster when compared to the unpreconditioned inexact Newton method or its two-grid version. Some numerical results are presented to demonstrate the robustness and efficiency of the algorithm.