Kybernetika 46 no. 2, 341-360, 2010

All-at-once preconditioning in PDE-constrained optimization

Tyrone Rees, Martin Stoll and Andy Wathen


The optimization of functions subject to partial differential equations (PDE) plays an important role in many areas of science and industry. In this paper we introduce the basic concepts of PDE-constrained optimization and show how the all-at-once approach will lead to linear systems in saddle point form. We will discuss implementation details and different boundary conditions. We then show how these system can be solved efficiently and discuss methods and preconditioners also in the case when bound constraints for the control are introduced. Numerical results will illustrate the competitiveness of our techniques.


partial differential equations, optimal control, preconditioning


65F10, 65N22, 65F50, 76D07