Convergence and regularization results for optimal control problems with sparsity functional
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 858-886.

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.

DOI: 10.1051/cocv/2010027
Classification: 49M05, 65N15, 65N30, 49N45
Mots-clés : non-smooth optimization, sparsity, regularization error estimates, finite elements, discretization error estimates
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     title = {Convergence and regularization results for optimal control problems with sparsity functional},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
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     publisher = {EDP-Sciences},
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Wachsmuth, Gerd; Wachsmuth, Daniel. Convergence and regularization results for optimal control problems with sparsity functional. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, pp. 858-886. doi : 10.1051/cocv/2010027. http://archive.numdam.org/articles/10.1051/cocv/2010027/

[1] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976) 147-190. | MR | Zbl

[2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Numer. 10 (2001) 1-102. | MR | Zbl

[3] C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187-1202. | Numdam | MR | Zbl

[4] 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. | MR | Zbl

[5] 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. | Numdam | MR | Zbl

[6] I. Daubechies, M. Defrise and C. De Mol, An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Comm. Pure Appl. Math. 57 (2004) 1413-1457. | MR | Zbl

[7] D. Donoho, For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution. Comm. Pure Appl. Math. 59 (2006) 797-829. | MR | Zbl

[8] A.L. Dontchev, W.W. Hager, A.B. Poore and B. Yang, Optimality, stability and convergence in optimal control. Appl. Math. Optim. 31 (1995) 297-326. | MR | Zbl

[9] R.S. Falk, Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973) 28-47. | MR | Zbl

[10] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term. Inv. Prob. 24 (2008) 055020. | MR | Zbl

[11] R. Griesse and D.A. Lorenz, A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inv. Prob. 24 (2008) 035007. | MR | Zbl

[12] R. Griesse, T. Grund and D. Wachsmuth, Update strategies for perturbed nonsmooth equations. Optim. Methods Softw. 23 (2008) 321-343. | MR | Zbl

[13] M. Hintermüller, R.H.W. Hoppe, Y. Iliash and M. Kieweg, An a posteriori error analysis of adaptive finite element methods for distributed elliptic control problems with control constraints. ESAIM: COCV 14 (2008) 540-560. | Numdam | MR | Zbl

[14] M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comp. Optim. Appl. 30 (2005) 45-63. | MR | Zbl

[15] A.D. Ioffe and V.M. Tichomirov, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979). | MR | Zbl

[16] B. Jin, D.A. Lorenz and S. Schiffler, Elastic-net regularization: error estimates and active set methods. Inv. Prob. 25 (2009) 115022. | MR | Zbl

[17] K. Krumbiegel and A. Rösch, A new stopping criterion for iterative solvers for control constrained optimal control problems. Archives of Control Sciences 18 (2008) 17-42. | Zbl

[18] R. Li, W. Liu, H. Ma and T. Tang, Adaptive finite element approximation for distributed elliptic optimal control problems. SIAM J. Control Optim. 41 (2002) 1321-1349. | MR | Zbl

[19] R. Li, W. Liu and N. Yan, A posteriori error estimates of recovery type for distributed convex optimal control problems. J. Sci. Comput. 33 (2007) 155-182. | MR | Zbl

[20] W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39 (2001) 73-99. | MR | Zbl

[21] D.A. Lorenz, Convergence rates and source conditions for Tikhonov regularization with sparsity constraints. J. Inverse Ill-Posed Probl. 16 (2008) 463-478. | MR | Zbl

[22] D.A. Lorenz and A. Rösch, Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems. Appl. Anal. (to appear). | MR | Zbl

[23] C. Meyer and A. Rösch, Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43 (2004) 970-985. | MR | Zbl

[24] C. Meyer, J.C. De Los Reyes and B. Vexler, Finite element error analysis for state-constrained optimal control of the Stokes equations. Control Cybern. 37 (2008) 251-284. | MR | Zbl

[25] R. Ramlau and G. Teschke, A Tikhonov-based projection iteration for nonlinear ill-posed problems with sparsity constraints. Numer. Math. 104 (2006) 177-203. | MR | Zbl

[26] A. Schiela, Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20 (2009) 1002-1031. | MR | Zbl

[27] G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices. Comp. Optim. Appl. 44 (2009) 159-181. | MR | Zbl

[28] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl

[29] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg, Wiesbaden (2005). | Zbl

[30] G. Wachsmuth, Elliptische Optimalsteuerungsprobleme unter Sparsity-Constraints. Diploma Thesis, Technische Universität Chemnitz (2008) http://www.tu-chemnitz.de/mathematik/part_dgl/publications.php+.

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