Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-24T02:01:16.455Z Has data issue: false hasContentIssue false
  • English
  • Français

Multigrid Methods for Elliptic Optimal Control Problems with Pointwise State Constraints

Published online by Cambridge University Press:  28 May 2015

Michelle Vallejos
Michelle Vallejos*
Affiliation:
Institute of Mathematics, University of the Philippines, Diliman, Quezon City, Philippines
Get access

Abstract

An elliptic optimal control problem with constraints on the state variable is considered. The Lavrentiev-type regularization is used to treat the constraints on the state variable. To solve the problem numerically, the multigrid for optimization (MGOPT) technique and the collective smoothing multigrid (CSMG) are implemented. Numerical results are reported to illustrate and compare the efficiency of both multigrid strategies.

Keywords

49K2065N0665N55Elliptic optimal control problemsLavrentiev regularizationmultigrid methodspoint-wise state constraints
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 5 , Issue 1 , February 2012 , pp. 99 - 109
Copyright
Copyright © Global Science Press Limited 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bergounioux, M., Haddou, M., Hintermüller, M., and Kunisch, K., A comparison of a Moreau-Yosida-based active set strategy and interior point methods for constrained optimal control problems, SIAM Journal on Optimization, 11 (2000), pp. 495–521.Google Scholar
[2] Bergounioux, M. and Kunisch, K., Primal-Dual Strategy for State-Constrained Optimal Control Problems, Computational Optimization and Applications, 22 (2002), pp. 193–224.Google Scholar
[3] Bergounioux, M. and Kunisch, K., On the structure of Lagrange multipliers for state-constrained optimal control problems, Systems and Control Letters, 48 (2003), pp. 169–176.Google Scholar
[4] Borzì, A., Smoothers for control- and state-constrained optimal control problems, Computing and Visualization in Science, 11 (2008), pp. 59–66.Google Scholar
[5] Borzì, A. and Kunisch, K., A multigrid scheme for elliptic constrained optimal control problems, Computational Optimization and Applications, 31 (2005), pp. 309–333.Google Scholar
[6] Borzì, A., Kunisch, K., and Kwak, D.Y., Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system, SIAM Journal on Control and Optimization, 41 (2002), pp. 1477–1497.Google Scholar
[7] Borzì, A. and Schulz, V., Multigrid methods for PDE optimization, SIAM Review, 51 (2009), pp. 361–395.Google Scholar
[8] Casas, E., Control of an elliptic problem with pointwise state constraints, SIAM Journal on Control and Optimization, 24 (1986), pp. 1309–1318.Google Scholar
[9] Casas, E. and L.Fernández, A., Optimal control of semilinear elliptic equations with pointwise constraints on the gradient of the state, Applied Mathematics and Optimization, 27 (1993), pp. 35–56.Google Scholar
[10] Hintermüller, M. and Kunisch, K., Stationary optimal control problems with pointwise state constraints, Numerical PDE constrained optimization, Lecture Notes in Computational Science and Engineering, 72 (2009).Google Scholar
[11] Kelley, C.T., Iterative methods for optimization, Kluwer, New York (1987).Google Scholar
[12] Lass, O., Vallejos, M., Borzì, A., and Douglas, C.C., Implementation and analysis of multigrid schemes with finite elements for elliptic optimal control problems, Computing, 84 (2009), pp. 27–48.Google Scholar
[13] Lewis, R.M. and Nash, S., Model problems for the multigrid optimization of systems governed by differential equations, SIAM Journal on Scientific Computing, 26 (2005), pp. 1811–1837.Google Scholar
[14] Meyer, C., Rösch, A., and Tröltzsch, F., Optimal control of PDEs with regularized pointwise state constraints, Computational Optimization and Applications, 33 (2006), pp. 209–228.Google Scholar
[15] Meyer, C. and Tröltzsch, F., On an elliptic optimal control problem with pointwise mixed control-state constraints, Recent Advances in Optimization, Lecture Notes in Economics and Mathematical System, Springer, 563 (2006), pp. 187–204.Google Scholar
[16] Meyer, C., PrÜfert, U., and Tröltzsch, F., On two numerical methods for state-constrained elliptic control problems, Optimization Methods and Software, 22 (2007), pp. 871–899.Google Scholar
[17] Nash, S., A multigrid approach to discretized optimization problems, Optimization Methods and Software, 14 (2000), pp. 99–116.Google Scholar
[18] Oh, S., Bouman, C., and Webb, K.J., Multigrid tomographic inversion with variable resolution data and image spaces, IEEE Transactions on Image Processing, 15 (2006), pp. 2805–2819.Google Scholar
[19] Oh, S., Milstein, A., Bouman, C. and Webb, K.J., A general framework for nonlinear multigrid inversion, IEEE Transactions on Image Processing, 14 (2005), pp. 125–140.Google Scholar
[20] Tröltzsch, F., Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints, SIAM Journal on Optimization, 15 (2005), pp. 616–634.Google Scholar
[21] Tröltzsch, F. and Yousept, I., A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints, Computational Optimization and Applications, 42 (2009), pp. 43–66.Google Scholar
[22] Vallejos, M. and Borzì, A., Multigrid methods for control-constrained elliptic optimal control problems, Numerical Mathematics and Advanced Applications, (2010), pp. 883–891.Google Scholar
[23] Vallejos, M. and Borzì, A., Multigrid optimization methods for linear and bilinear elliptic optimal control problems, Computing, 82 (2008), pp. 31–52.Google Scholar