Approximation of the Newton Step by a Defect Correction Process

Abstract. In this paper, an optimal control problem governed by a partial differential equation is considered. TheNewton step for this system can be computed by solving a coupled system of equations. To do this efficiently with aniterative defect correction process, a modifying operator is introduced into the system. This operator is motivated bylocal mode analysis. The operator can be used also for preconditioning in GMRES. We give a detailed convergenceanalysis for the defect correction process and show the derivation of the modifying operator. Numerical tests aredone on the small disturbance shape optimization problem in two dimensions for the defect correction process and forGMRES.Key words, optimal control governed by PDEs, iterative methods, defect correction, GMRES, preconditioning,Newton step, SQESubject classification. Applied and Numerical Mathematics1. Introduction. Many optimization problems can be formulated as equality constrained problems with a specialstructure. If one considers optimal control or optimal design problems, the variables are partitioned into the state andcontrol or design variables which we denote by _ and u, respectively. This leads to the following problem formulationrain .Y'(4),u) s.t. h(_b, u) = O.(_,u)If one is interested in algorithms with a fast rate of convergence, one would tend to use Newton's method for theseproblems. Note that this method can be applied in two different ways. Under appropriate assumptions, see Section 2.1,one can solve for each control variable u the system equation h(_b(u), u) = 0 to obtain a state q_(u) which depends onu. This is typical, when h represents a boundary value problem where the control variable is on the right hand side ofthe differential equation. Then one can apply Newton's method to the unconstrained minimization problem