ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Preconditioning for PDEs, Optimisation and Data Assimilation

Mon 11:30–12:00, Auditorium Max Weber
Preconditioning Newton methods for optimal control problems with sparsity constraints
Margherita Porcelli (University of Florence)
Joint work with Valeria Simoncini (University of Bologna)

We address the solution of PDE-constrained optimal control problems via semismooth Newton methods. Specifically, we consider problems with control constraints and with nonsmooth costs that are known to promote sparse optimal controls, i.e. controls which are identically zero on large parts of the control domain [1,2]. A typical example is the L$^1$ cost that has been used, e.g., for the optimal placement of control devices [2].

Following a discretize-then-optimize approach, we analyze the convergence properties of the Newton method applied to the discretization of optimal control problems with nonsmooth regularization terms. Moreover, we present the study of the impact of the control sparsity on the structure of the arising linear systems and propose preconditioners which exploit this information. Numerical experiments on 3D problems are presented.

  1. Roland Herzog, Georg Stadler, and Gerd Wachsmuth. Directional sparsity in optimal control of partial differential equations. SIAM Journal on Control and Optimization, 50(2):943–963, 2012.
  2. G. Stadler. Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Computational Optimization and Applications, 44(2):159–181, 2009.