DFG-Projekt: Numerical analysis and discretization strategies for optimal control problems with singularities

im DFG-Schwerpunktprogramm 1253: Optimization with PDE constraints

 

 Leitung: 

  • Thomas Apel
  • Arnd Rösch
  • Boris Vexler

 

 Bearbeiter: 

  • Olaf Benedix
  • Thomas Flaig
  • Martin Naß
  • Johannes Pfefferer
  • Dieter Sirch
  • Gunter Winkler
  • Max Winkler

 

 Kooperationspartner: 

  • Roland Becker
  • Michael Hinze
  • Ronald W. Hoppe
  • Gert Lube
  • Rolf Rannacher
  • Fredi Tröltzsch

 

 Förderungszeiträume: 

  • 2. Förderungszeitraum: Oktober 2009 - Dezember 2013
  • 1. Förderungszeitraum: Oktober 2006 - September 2009

 

 Ziele: 

  • A priori Fehlerabschätzungen für Optimalsteuerprobleme mit Kontrollbeschränkungen
  • A priori Fehlerabschätzungen bei punktweisen Zustandsbeschränkungen
  • A posteriori Fehlerschätzer bei Kontrollbeschränkungen
  • A posteriori Fehlerschätzer bei Zustandsbeschränkungen
  • Verifizierung aller Untersuchungen durch entsprechende numerische Tests

 

 Abschlussarbeiten aus dem Projekt: 

  • Flaig, T. G.: Discretization strategies for optimal control problems with parabolic partial differential equations, PhD thesis, UniBw München, 2013. [im Buchhandel]
  • Sirch, D.: Finite Element Error Analysis for PDE-constrained Optimal Control Problems: The Control Constrained Case Under Reduced Regularity, PhD thesis, TU München, 2010. [Dissertation]
  • Winkler, G.: Control constrained optimal control problems in non-convex three dimensional polyhedral domains, PhD thesis, TU Chemnitz, 2008. [Dissertation]

 

 Veröffentlichungen aus dem Projekt: 

  • Apel, T., Pfefferer, J., Rösch, A.: Locally refined meshes in optimal control for elliptic partial differential equations - an overview. In: G. Leugering, P. Benner, S. Engell, A. Griewank, H. Harbrecht, M. Hinze, R. Rannacher, and S. Ulbrich (eds.): Trends in PDE Constrained Optimization. International Series of Numerical Mathematics 165. Birkhäuser, Basel, 2014, pp. 285-302.
  • Flaig, T. G., Meidner, D.,  Vexler, B.: Petrov-Galerkin Crank-Nicolson Scheme for Parabolic Optimal Control Problems on Nonsmooth Domains, In Günter Leugering, Peter Benner, Sebastian Engell, Andreas Griewank, Helmut Harbrecht, Michael Hinze, Rolf Rannacher, Stefan Ulbrich (ed.): Trends in PDE Constrained Optimization. International Series of Numerical Mathematics 165. Birkhäuser, Basel, 2014, pp. 421-435.
  • Apel, T., Flaig, T. G., Nicaise, S.: A priori error estimates for finite element methods for H^(2,1)-elliptic equations Numerical Functional Analysis and Optimization 35(2): 153-176, 2014. [Paper]
  • Flaig, T. G.: Implicit Runge-Kutta schemes for optimal control problems with evolution equations, submitted, 2013. [Preprint]
  • Apel, T., Flaig, T. G.: Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations, SIAM J. Numer. Anal. 50: 1484-1512, 2012. [Paper] / [Preprint]
  • Apel, T., Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Computational Optimization and Applications 52(1): 3-28, 2012. [Paper] / [Preprint]
  • Apel, T., Sirch, D.: A priori mesh grading for distributed optimal control problems. In: G. Leugering, S. Engell, A. Griewank, M. Hinze, R. Rannacher, V. Schulz, M. Ulbrich, and S. Ulbrich (eds.): Constrained Optimization and Optimal Control for Partial Differential Equations. International Series of Numerical Mathematics 160. Springer, Basel, 2012, pp. 377-389. [Preprint]
  • Apel, T., Benedix, O., Sirch, D., Vexler, B.: A priori mesh grading for an elliptic problem with Dirac right-hand side, SIAM Journal on Numerical Analysis 49(3): 992 - 1005, 2011. [Paper] / [Preprint]
  • Apel, T., Sirch, D.: L2-error estimates for the Dirichlet and Neumann problem on anisotropic finite element meshes. Appl. Math. 56(2): 177–206, 2011. [Paper]
  • Apel, T., Flaig, T. G.: Simulation and Mathematical Optimization of the Hydration of Concrete for avoiding thermal Cracks., in K. Gürlebeck and C. Könke (eds.): 18th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, ISSN 1611-4086, Weimar 2009. [Paper]
  • Nicaise, S., Sirch, D.: Optimal control of the Stokes equations: Conforming and non-conforming finite element methods under reduced regularity, Comput. Optim. Appl. 49(3): 567-600, 2009. [Paper]
  • Apel, T., Sirch, D., Winkler, G.: Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes, submitted, 2008. [Preprint]
  • Apel, T., Rösch, A., Sirch, D.: L∞-error estimates on graded meshes with application to optimal control, SIAM J. Control Optim. 48(2009), 1771-1796. [Paper] / [Preprint]
  • Apel, T., Rösch, A., Winkler, G.: Optimal control in nonconvex domains: a priori discretization error estimates, Calcolo 44(3): 137-158, 2007. [Paper] / [Preprint]
  • Apel, T., Winkler, G.: Optimal Control Under Reduced Regularity. Appl. Numer. Math. 59(9): 2050-2064, 2009. [Paper] / [Preprint]
  • Apel, T., Rösch, A., Winkler, G.: Discretization error estimates for an optimal control problem in a nonconvex domain, in: A. Bermúdez de Castro et al. (eds.): Numerical Mathematics and Advanced Applications, Proceedings of ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications, Santiago de Compostela, Spain, July 2005, 299-307, Springer, Berlin, 2006. [Paper] / [Preprint]