Directional sparsity in optimal control of partial differential equations

SIAM Journal on Control and Optimization

Volumn 50, Issue 2, 2012, Pages 943-963

  Herzog, Roland ^a   Stadler, Georg ^b   Wachsmuth, Gerd ^a  

^a CHEMNITZ UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

Control device placement; L1 norm minimization; Optimal control; Semismooth; Sparsity

Indexed keywords

CONTROL DEVICE; L1 NORM; OPTIMAL CONTROLS; SEMISMOOTH; SPARSITY;

OPTIMAL CONTROL SYSTEMS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS;

CONTROL;

EID: 84861581924     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/100815037     Document Type: Article

References (34)

1. S. S. Adavani and G. Biros, Multigrid algorithms for inverse problems with linear parabolic PDE constraints, SIAM J. Sci. Comput., 31 (2008), pp. 369-397.
2. M. Bergounioux, K. Ito, and K. Kunisch, Primal-dual strategy for constrained optimal control problems, SIAM J. Control Optim., 37 (1999), pp. 1176-1194.
3. A. Borzì and V. Schulz, Multigrid methods for PDE optimization, SIAM Rev., 51 (2009), p. 361.
4. E. Candès and Y. Plan, Near-ideal model selection by l1 minimization, Ann. Statist., 37 (2009), pp. 2145-2177.
5. E. Casas, R. Herzog, and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with L1 cost functional, SIAM J. Optim., to appear.
6. T. F. Chan and X.-C. Tai, Identification of discontinuous coefficients in elliptic problems using total variation regularization, SIAM J. Sci. Comput., 25 (2003), pp. 881-904.
7. X. Chen, Z. Nashed, and L. Qi, Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J. Numer. Anal., 38 (2000), pp. 1200-1216.
8. C. Clason and K. Kunisch, A duality-based approach to elliptic control problems in nonreflexive Banach spaces, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 243-266.
9. A. Defant and K. Floret, Tensor Norms and Operator Ideals, North-Holland Math. Stud. 176, North-Holland, Amsterdam, 1993.
10. N. Dinculeanu, Vector Measures, Deutscher Verlag der Wissenschaften, Berlin, 1966.
11. D. Donoho, Compressed sensing, IEEE Trans. Inform. Theory, 52 (2006), pp. 1289-1306.
12. D. Donoho, For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution, Comm. Pure Appl. Math., 59 (2006), pp. 797-829.
13. M. Fornasier, R. Ramlau, and G. Teschke, The application of joint sparsity and total variation minimization algorithms in a real-life art restoration problem, Adv. Comput. Math., 31 (2009), pp. 301-329.
14. M. Fornasier and H. Rauhut, Recovery algorithms for vector-valued data with joint sparsity constraints, SIAM J. Numer. Anal., 46 (2008), pp. 577-613.
15. E. Haber, L. Horesh, and L. Tenorio, Numerical methods for experimental design of largescale linear ill-posed inverse problems, Inverse Problems, 24 (2008), p. 055012.
16. M. Hintermüller, K. Ito, and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method, SIAM J. Optim., 13 (2002), pp. 865-888.
17. M. Hintermüller, S. Volkwein, and F. Diwoky, Fast solution techniques in constrained optimal boundary control of the semilinear heat equation, Internat. Ser. Numer. Math., 155 (2007), pp. 119-147.
18. M. Hladnik and M. Omladič, Spectrum of the product of operators, Proc. Amer. Math. Soc., 102 (1988), pp. 300-302.
19. S. Hüeber, G. Stadler, and B. I. Wohlmuth, A primal-dual active set algorithm for threedimensional contact problems with Coulomb friction, SIAM J. Sci. Comput., 30 (2008), pp. 572-596.
20. A. D. Ioffe and V. M. Tichomirov, Theorie der Extremalaufgaben, VEB Deutscher Verlag der Wissenschaften, Berlin, 1979.
21. K. Ito and K. Kunisch, The primal-dual active set method for nonlinear optimal control problems with bilateral constraints, SIAM J. Control Optim., 43 (2004), pp. 357-376.
22. J. Liu, S. Ji, and J. Ye, SLEP: Sparse Learning with Efficient Projections, Arizona State University, Tempe, AZ, 2009.
23. M. Nikolova, Local strong homogeneity of a regularized estimator, SIAM J. Appl. Math., 61 (2000), pp. 633-658.
24. M. Nikolova, Analysis of the recovery of edges in images and signals by minimizing nonconvex regularized least-squares, Multiscale Model. Simul., 4 (2005), pp. 960-991.
25. Z. Qin, K. Scheinberg, and D. Goldfarb, Efficient Block-Coordinate Descent Algorithms for the Group Lasso, Technical report 2806, Optimization Online, 2010.
26. G. Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices, Comput. Optim. Appl., 44 (2009), pp. 159-181.
27. D. Sun and J. Han, Newton and quasi-Newton methods for a class of nonsmooth equations and related problems, SIAM J. Optim., 7 (1997), pp. 463-480.
28. F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Grad. Stud. Math. 112, American Mathematical Society, Providence, RI, 2010.
29. J. A. Tropp and S. J. Wright, Computational methods for sparse solution of linear inverse problems, Proc. IEEE, 98 (2010), pp. 948-958.
30. C. R. Vogel, Computational Methods for Inverse Problems, SIAM, Philadelphia, 2002.
31. G. Vossen and H. Maurer, On L1-minimization in optimal control and applications to robotics, Optimal Control Appl. Methods, 27 (2006), pp. 301-321.
32. G. Wachsmuth and D. Wachsmuth, Convergence and regularization results for optimal control problems with sparsity functional, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 858-886.
33. G. Wachsmuth, Elliptische Optimalsteuerungsprobleme unter Sparsity-Constraints, Diploma thesis, Technische Universität Chemnitz, 2008; also available online from http://www.tuchemnitz. de/mathematik/part dgl/publications.php.
34. S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, Sparse reconstruction by separable approximation, IEEE Trans. Signal Process., 57 (2009), pp. 2479-2493.