FWF/DFG Project within priority program 1253

Numerical analysis and discretization strategies for optimal control problems with singularities

• Project Leaders:
  • Prof. Dr. Thomas Apel  (UniBw Munich)
  • Prof. Dr. Arnd Rösch  (Uni Duisburg-Essen)
  • Prof. Dr. Boris Vexler (Technische Universität München)

• Project Staff:
  • Olaf Benedix (Technische Universität München)
  • Thomas Flaig  (UniBw Munich)
  • Martin Naß  (Uni Duisburg-Essen)
  • Johannes Pfefferer  (UniBw Munich)
  • Dieter Sirch  (UniBw Munich)
  • Gunter Winkler (UniBw Munich, until 30.06.2007)

Funding Periods

• second period: 9/2009 - ongoing
  • funding by the DFG priority program 1253  "Optimierung mit partiellen Differentialgleichungen"
• first period: 9/2006 - 9/2009
  • joint project of FWF  and DFG 
  • FWF Project No. P18971-N18, Project Abstract at FWF 
  • DFG funding by the priority program 1253  "Optimierung mit partiellen Differentialgleichungen"

Project Abstract

Optimization of technological processes plays an increasing role in science and engineering. This project deals with different types of optimal control problems governed by elliptic or parabolic partial differential equations and characterized by additional pointwise inequality constraints for control and state. Of particular interest are problems with all kinds of singularities including those due to reentrant corners and edges, nonsmooth coefficients, small parameters, and inequality constraints. The project targets two goals: First, starting from a priori error estimates, families of meshes are generated that ensure optimal approximation rates. Second, reliable posteriori error estimators are developed and used for adaptive mesh refinement. A challenge is the incorporation of pointwise inequality constraints for control and state. Both techniques lead to efficient and reliable numerical results and allow to calculate numerical solutions of the optimal control problems with given accuracy at a small multiple of the cost of the pure numerical simulation. While the first period of the project was mainly devoted to problems with a linear elliptic state equation, we will consider semilinear and parabolic state equations in the second period. As a non-standard application with a parabolic semilinear state equation and inequality constraints, we will continue to model, analyze and simulate the optimization of the thermo-mechanic properties of concrete during hydration.

Keywords and AMS Classification

• optimal control
• singularities
• inequality constraints
• finite element discretization
• a priori error analysis
• a posteriori error estimation
• mesh refinement

AMS Subject Classification: 49K20, 49M25, 49N10, 49N60

Related Publications:

• Rösch, A., Vexler, B., Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing , SIAM Journal Numerical Analysis, 44(5):1903-1920, 2006.
• Apel, T., Rösch, A., Winkler, G., Discretization error estimates for an optimal control problem in a nonconvex domain . In: A. Bermudez 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
• Apel, T., Rösch, A., Winkler, G., Optimal control in non-convex domains: a priori discretization error estimates, Calcolo 44: 137-158, 2007.
• Vexler, B., Wollner, W., Adaptive finite elements for elliptic optimization problems with control constraints , SIAM Journal on Control and Optimization, 47(1):509-534, 2008.
• Winkler, G., Control constrained optimal control problems in non-convex three dimensional polyhedral domains, PhD thesis, TU Chemnitz, 2008.
• Apel, T., Winkler, G., Optimal Control Under Reduced Regularity , Appl. Numer. Math. 59(9): 2050-2064, 2009.
• Meidner, D., Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part I: Problems without Control Constraints, SIAM Journal on Control and Optimization, 47(3): 1150-1177, 2008.
• Meidner, D., Vexler, B., A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part II: Problems with Control Constraints, SIAM Journal on Control and Optimization, 47(3): 1301-1329, 2008.
• Benedix, O., Vexler, B., A posteriori error estimation and adaptivity for elliptic optimal control problems with state constraints, Computational Optimization and Applications 44(1): 3-25, 2009.
• Apel, T., Sirch, D., L²-error estimates for the Dirichlet and Neumann problem on anisotropic finite element meshes, accepted by Appl. Math., 2008.
• Apel, T., Sirch, D., Winkler, G., Error estimates for control constrained optimal control problems: Discretization with anisotropic finite element meshes, Preprint SPP1253-02-06, DFG Priority Program 1253, Erlangen, 2008. Submitted to Math. Program.
• Merino, P., Tröltzsch, F., Vexler, B., Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space, ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 44(1), 167 - 188, 2010.
• Apel, T., Rösch, A., Sirch, D., L^∞-Error Estimates on Graded Meshes with Application to Optimal Control , SIAM Journal on Control and Optimization, 48: 1771-1796, 2009.
• de los Reyes, J. C., Meyer, C., Vexler, B., Finite element error analysis for state-constrained optimal control of the Stokes equations, Control and Cybernetics, 37(2): 251-284, 2008.
• Kröner, A., Vexler, B., A priori error estimates for elliptic optimal control problems with bilinear state equation, Journal of Computational and Applied Mathematics, 230(2): 781-802, 2009.
• May, S., Rannacher, R., Vexler, B., Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems, submitted, 2008.
• Apel, T., Benedix, O., Sirch, D., Vexler, B., A priori mesh grading for an elliptic problem with Dirac right-hand side, submitted, 2009.
• Nicaise, S., Sirch, D., Optimal control of the Stokes equations: Conforming and non-conforming finite element methods under reduced regularity, Computational Optimization and Applications, published online, 2009. (doi )
• Apel, T., Pfefferer, J., Rösch, A., Finite element error estimates for Neumann boundary control problems on graded meshes , submitted, 2010.
• Sirch, D., Finite Element Error Analysis for PDE-constrained Optimal Control Problems: The Control Constrained Case Under Reduced Regularity, PhD thesis, TU München, 2010.
• Apel, T., Sirch, D., A priori mesh grading for distributed optimal control problems, Accepted for: G. Leugering et al. (eds.): Themenband SPP 1253 (Arbeitstitel), 2010.
• Apel, T., Flaig, T. G., Crank-Nicolson Schemes for Optimal Control Problems with Evolution Equations, submitted, 2010.

Software:

• Gascoigne 
• RoDoBo