Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part II: The Lagrange--Newton Solver and Its Application to Optimal Control of Steady Viscous Flows

In part I of this article, we proposed a Lagrange--Newton--Krylov--Schur (LNKS) method for the solution of optimization problems that are constrained by partial differential equations. LNKS uses Krylov iterations to solve the linearized Karush--Kuhn--Tucker system of optimality conditions in the full space of states, adjoints, and decision variables, but invokes a preconditioner inspired by reduced space sequential quadratic programming (SQP) methods. The discussion in part I focused on the (inner, linear) Krylov solver and preconditioner. In part II, we discuss the (outer, nonlinear) Lagrange--Newton solver and address globalization, robustness, and efficiency issues, including line search methods, safeguarding Newton with quasi-Newton steps, parameter continuation, and inexact Newton ideas. We test the full LNKS method on several large-scale three-dimensional configurations of a problem of optimal boundary control of incompressible Navier--Stokes flow with a dissipation objective functional. Results of numerical experiments on up to 256 Cray T3E-900 processors demonstrate very good scalability of the new method. Moreover, LNKS is an order of magnitude faster than quasi-Newton reduced SQP, and we are able to solve previously intractable problems of up to 800,000 state and 5,000 decision variables at about 5 times the cost of a single forward flow solution.