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Convergence and regularization resultsfor optimal control problems with sparsity functional

Published online by Cambridge University Press:  06 August 2010

Gerd Wachsmuth
Affiliation:
Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany.
Daniel Wachsmuth
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrae 69, 4040 Linz, Austria. [email protected]
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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 Newtonmethod. 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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

Aronszajn, N., Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976) 147190. CrossRef
Becker, R. and Rannacher, R., An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1102. CrossRef
Carstensen, C., Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 11871202. CrossRef
E. Casas and M. Mateos, Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Continuous piecewise linear approximations, in Systems, control, modeling and optimization 202, IFIP Int. Fed. Inf. Process., Springer, New York (2006) 91–101.
C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in non-reflexive Banach spaces. ESAIM: COCV (2010) DOI: 10.1051/cocv/2010003. CrossRef
Daubechies, I., Defrise, M. and De Mol, C., An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004) 14131457. CrossRef
Donoho, D., For most large underdetermined systems of linear equations the minimal l 1-norm solution is also the sparsest solution. Comm. Pure Appl. Math. 59 (2006) 797829. CrossRef
Dontchev, A.L., Hager, W.W., Poore, A.B. and Yang, B., Optimality, stability and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297326. CrossRef
Falk, R.S., Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 2847. CrossRef
Grasmair, M., Haltmeier, M. and Scherzer, O., Sparse regularization with lq penalty term. Inv. Prob. 24 (2008) 055020. CrossRef
Griesse, R. and Lorenz, D.A., A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inv. Prob. 24 (2008) 035007. CrossRef
Griesse, R., Grund, T. and Wachsmuth, D., Update strategies for perturbed nonsmooth equations. Optim. Methods Softw. 23 (2008) 321343. CrossRef
Hintermüller, M., Hoppe, R.H.W., Iliash, Y. and Kieweg, M., An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: COCV 14 (2008) 540560. CrossRef
Hinze, M., A variational discretization concept in control constrained optimization: the linear-quadratic case. Comp. Optim. Appl. 30 (2005) 4563. CrossRef
A.D. Ioffe and V.M. Tichomirov, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979).
Jin, B., Lorenz, D.A. and Schiffler, S., Elastic-net regularization: error estimates and active set methods. Inv. Prob. 25 (2009) 115022. CrossRef
Krumbiegel, K. and Rösch, A., A new stopping criterion for iterative solvers for control constrained optimal control problems. Archives of Control Sciences 18 (2008) 1742.
Li, R., Liu, W., Ma, H. and Tang, T., Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41 (2002) 13211349. CrossRef
Li, R., Liu, W. and Yan, N., A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 33 (2007) 155182. CrossRef
Liu, W. and Yan, N., A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39 (2001) 7399. CrossRef
Lorenz, D.A., Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463478. CrossRef
D.A. Lorenz and A. Rösch, Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems. Appl. Anal. (to appear).
Meyer, C. and Rösch, A., Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43 (2004) 970985. CrossRef
Meyer, C., de los Reyes, J.C. and Vexler, B., Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37 (2008) 251284.
Ramlau, R. and Teschke, G., Tikhonov-based, A projection iteration for nonlinear ill-posed problems with sparsity constraints. Numer. Math. 104 (2006) 177203. CrossRef
Schiela, A., Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20 (2009) 10021031. CrossRef
Stadler, G., Elliptic optimal control problems with L 1-control cost and applications for the placement of control devices. Comp. Optim. Appl. 44 (2009) 159181. CrossRef
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189258. CrossRef
F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg, Wiesbaden (2005).
G. Wachsmuth, Elliptische Optimalsteuerungsprobleme unter Sparsity-Constraints. Diploma Thesis, Technische Universität Chemnitz (2008) http://www.tu-chemnitz.de/mathematik/part_dgl/publications.php+.