ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - July 2011 - Volume 17, Issue 03  

Research Article

Convergence and regularization results for optimal control problems with sparsity functional

Wachsmuth, Gerda1 and Wachsmuth, Daniela2

a1 Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany.

a2 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrae 69, 4040 Linz, Austria. daniel.wachsmuth@ricam.oeaw.ac.at

Abstract

Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from a L1-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approximations are studied. A-priori as well as a-posteriori error estimates are developed and confirmed by numerical experiments.

(Received August 27 2009)

(Revised February 9 2010)

(Revised May 18 2010)

(Online publication August 06 2010)

Key Words:

  • Non-smooth optimization;
  • sparsity;
  • regularization error estimates;
  • finite elements;
  • discretization error estimates

Mathematics Subject Classification:

  • 49M05;
  • 65N15;
  • 65N30;
  • 49N45