Parallel Lagrange--Newton--Krylov--Schur Methods for PDE-Constrained Optimization. Part I: The Krylov--Schur Solver

Abstract

Large-scale optimization of systems governed by partial differential equations (PDEs) is a frontier problem in scientific computation. Reduced quasi-Newton sequential quadratic programming (SQP) methods are state-of-the-art approaches for such problems. These methods take full advantage of existing PDE solver technology and parallelize well. However, their algorithmic scalability is questionable; for certain problem classes they can be very slow to converge. In this two-part article we propose a new method for steady-state PDE-constrained optimization, based on the idea of using a full space Newton solver combined with an approximate reduced space quasi-Newton SQP preconditioner. The basic components of the method are Newton solution of the first-order optimality conditions that characterize stationarity of the Lagrangian function; Krylov solution of the Karush--Kuhn--Tucker (KKT) linear systems arising at each Newton iteration using a symmetric quasi-minimum residual method; preconditioning of the KKT system using an approximate state/decision variable decomposition that replaces the forward PDE Jacobians by their own preconditioners, and the decision space Schur complement (the reduced Hessian) by a BFGS approximation initialized by a two-step stationary method. Accordingly, we term the new method {\it Lagrange--Newton--Krylov--Schur} (LNKS). It is fully parallelizable, exploits the structure of available parallel algorithms for the PDE forward problem, and is locally quadratically convergent. In part I of this two-part article, we investigate the effectiveness of the KKT linear system solver. We test our method on two optimal control problems in which the state constraints are described by the steady-state Stokes equations. The objective is to minimize dissipation or the deviation from a given velocity field; the control variables are the boundary velocities. Numerical experiments on up to 256 Cray T3E processors and on an SGI Origin 2000 include scalability and performance assessment of the LNKS algorithm and comparisons with reduced SQP for up to $1,000,000$ state and 50,000 decision variables. In part II of the article, we address globalization and inexactness issues, and apply LNKS to the optimal control of the steady incompressible Navier--Stokes equations.