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 con. gurations 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.