# All-at-Once Solution of Periodic Optimal Control Problems

### Researcher:

- Martin Stoll

Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg,

Computational Methods in Systems and Control Theory,

Tel: +49 (0)391-6110-805

E-mail: stollm@mpi-magdeburg.mpg.de

### Project Description

Many processes in engineering and across the sciences are described by partial differential equations. For the past decades the solution of the discretized equations using iterative techniques has been and still is at the heart of the numerical analysis community. Advances in algorithms and computing technology has enabled scientist to investigate inverse problems where the PDE is typically a constraint for the minimization of an objective function. In this project we focus on the efficient solution of systems that arise from the first order conditions of the corresponding Lagrangian. For time-dependent problems we introduce a space-time formulation combined with an all-at-Once approach. This means that we solve for all time-steps simultaneously. For this process no accurate solution of the PDE-constraints is required as we typically only evaluate approximations of the PDE-discretization often using multigrid techniques. We focus here on time-dependent PDEs and in particular the interesting class of time-periodic problems.### Duration and Funding

- since 2011: MPI Magdeburg

### Related Publications

169 | Preconditioning for Allen-Cahn variational inequalities with non-local constraints; Luise Blank; Lavinia Sarbu; Martin Stoll; Journal of Computational Physics, 231, 5406-5420 : 2012. http://dx.doi.org/10.1016/j.jcp.2012.04.035. |

166 | Preconditioning for partial differential equation constrained optimization with control constraints; Martin Stoll; Andy Wathen; Numerical Linear Algebra with Applications : 19:53–71;2012. |

168 | Preconditioning for Allen-Cahn variational inequalities with non-local constraints; Luise Blank; Lavinia Sarbu; Martin Stoll; Oberwolfach report - not refereed : 2010. |

165 | Block triangular preconditioners for PDE constrained optimization; Tyrone Rees; Martin Stoll; Numerical Linear Algebra with Applications Volume 17, Issue 6, pages 977–996, December 2010 : 2010. |

164 | All-at-once preconditioning in PDE-constrained optimization; Tyrone Rees; Martin Stoll; Andy Wathen; Kybernetika : Vol. 46 (2);2010. |