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.
A2757optimization problem to an unconstrained problem by introducing some Lagrangian multipliers and a Lagrangian functional. Then, an inexact Newton method is employed to solve a large sparse nonlinear system of equations derived from the first-order optimality condition. From the algorithmic viewpoint, although the inexact Newton method enjoys fast convergence, the full-space method poses some computational challenges. First, at each Newton iteration, a large sparse saddle point system needs to be solved, which is often highly ill-conditioned [5]. Second, the convergence of the inexact Newton method is often problematic. Numerical pieces of evidence [11,28,30] suggest that the slow convergence is often determined by a small subset of equations in the system with the highest nonlinearities. The family of continuation methods, such as the grid-sequencing approach [2,34] or the parameter continuation approach [3,32], is quite robust for these difficult nonlinear problems but not efficient; see their applications for PDE-constrained optimization problems [7,48,49,51]. Alternatively, we extend some nonlinear preconditioning techniques [29,30], previously developed for the system of nonlinear equations, to PDE-constrained optimization problems. The fast convergence of the inexact Newton method can be restored when it is employed in conjunction with the nonlinear preconditioning.Similar to the linear case, a nonlinear preconditioner can be applied on the right or on the left of a nonlinear function. For the left nonlinear preconditioner, the algorithm reformulates the original nonlinear function implicitly into a more balanced function. The additive Schwarz preconditioned inexact Newton algorithm (ASPIN) [11,28] belongs to this class, where the nonlinearly preconditioned function is defined based on the additive Schwarz framework and the new system is solved by an inexact Newton algorithm. ASPIN has been successfully applied for some nonlinear systems of equati...