Convergence and regularization results for optimal control problems with sparsity functional
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 3, p. 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 : https://doi.org/10.1051/cocv/2010027
Classification:  49M05,  65N15,  65N30,  49N45
Keywords: non-smooth optimization, sparsity, regularization error estimates, finite elements, discretization error estimates
@article{COCV_2011__17_3_858_0,
author = {Wachsmuth, Gerd and Wachsmuth, Daniel},
title = {Convergence and regularization results for optimal control problems with sparsity functional},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {17},
number = {3},
year = {2011},
pages = {858-886},
doi = {10.1051/cocv/2010027},
zbl = {1228.49032},
mrnumber = {2826983},
language = {en},
url = {http://www.numdam.org/item/COCV_2011__17_3_858_0}
}

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://www.numdam.org/item/COCV_2011__17_3_858_0/

[1] N. Aronszajn, Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 (1976) 147-190. | MR 425608 | Zbl 0342.46034

[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 2009692 | Zbl 1105.65349

[3] C. Carstensen, Quasi-interpolation and a posteriori error analysis in finite element methods. ESAIM: M2AN 33 (1999) 1187-1202. | Numdam | MR 1736895 | Zbl 0948.65113

[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 2241699 | Zbl 1214.49019

[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 2775195 | Zbl 1213.49041

[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 2077704 | Zbl 1077.65055

[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 2217606 | Zbl 1113.15004

[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 1316261 | Zbl 0821.49022

[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 686788 | Zbl 0268.49036

[10] M. Grasmair, M. Haltmeier and O. Scherzer, Sparse regularization with lq penalty term. Inv. Prob. 24 (2008) 055020. | MR 2438955 | Zbl 1157.65033

[11] R. Griesse and D.A. Lorenz, A semismooth Newton method for Tikhonov functionals with sparsity constraints. Inv. Prob. 24 (2008) 035007. | MR 2421961 | Zbl 1152.49030

[12] R. Griesse, T. Grund and D. Wachsmuth, Update strategies for perturbed nonsmooth equations. Optim. Methods Softw. 23 (2008) 321-343. | MR 2424163 | Zbl 1195.46044

[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 2434065 | Zbl 1157.65039

[14] M. Hinze, A variational discretization concept in control constrained optimization: the linear-quadratic case. Comp. Optim. Appl. 30 (2005) 45-63. | MR 2122182 | Zbl 1074.65069

[15] A.D. Ioffe and V.M. Tichomirov, Theorie der Extremalaufgaben. VEB Deutscher Verlag der Wissenschaften, Berlin (1979). | MR 527119 | Zbl 0403.49001

[16] B. Jin, D.A. Lorenz and S. Schiffler, Elastic-net regularization: error estimates and active set methods. Inv. Prob. 25 (2009) 115022. | MR 2558682 | Zbl 1188.49026

[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 1187.49025

[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 1971952 | Zbl 1034.49031

[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 2342593 | Zbl 1128.65048

[20] W. Liu and N. Yan, A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal. 39 (2001) 73-99. | MR 1860717 | Zbl 0988.49018

[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 2442066 | Zbl 1161.65041

[22] D.A. Lorenz and A. Rösch, Error estimates for joint Tikhonov- and Lavrentiev-regularization of constrained control problems. Appl. Anal. (to appear). | MR 2683675 | Zbl 1203.49027

[23] C. Meyer and A. Rösch, Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43 (2004) 970-985. | MR 2114385 | Zbl 1071.49023

[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 2472877 | Zbl 1235.49068

[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 2242613 | Zbl 1101.65056

[26] A. Schiela, Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20 (2009) 1002-1031. | MR 2534773 | Zbl 1201.90201

[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 2556849 | Zbl 1185.49031

[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 192177 | Zbl 0151.15401

[29] F. Tröltzsch, Optimale Steuerung partieller Differentialgleichungen. Vieweg, Wiesbaden (2005). | Zbl 1142.49001

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