Multigrid solution of distributed optimal control problems constrained by semilinear elliptic PDEs

Optimization constrained by partial differential equations (PDEs) is a research area in which the scientific and engineering communities have seen a growing interest over the last decade. The recent rise in interest was fostered by the tremendous increase in computing power over the last twenty years. This can be attributed both to the tremendous advances in high-performance computing technologies and to its wide range of applicability. However, just growth in computing power is insufficient for tackling PDE-constrained optimization problems and there is always a need for ever-increasing efficient algorithms. The objective of this dissertation is to develop, analyze and implement multigrid preconditioners for the linear systems arising in the Newton-Krylov solution process for the nonlinear PDE constraints. Our main focus is on semilinear elliptic constraints with no additional control or state constraints. We analyze the preconditioners both theoretically and numerically. In this work we show that the multigrid preconditioners are of optimal order (p = 2) for piecewise linear finite element approximations. We also study control-constrained optimal control problems constrained by linear-elliptic equations. This problem is non-trivial from the optimization point of view as the KKT system is now a complementarity system. We employ semismooth Newton methods (SSNMs) to solve this problem efficiently. The multigrid preconditioners discussed here are extensions of preconditioners developed previously for the unconstrained case. We present some new results and techniques that yield optimal order multigrid preconditioners, at least for the case when the control is discretized using piecewise-constant finite elements