Table of Contents
ISRN Applied Mathematics
Volume 2014, Article ID 974305, 11 pages
Research Article

Numerical Optimal Control for Problems with Random Forced SPDE Constraints

Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

Received 22 September 2013; Accepted 10 December 2013; Published 20 February 2014

Academic Editors: H. Homeier and F. Zirilli

Copyright © 2014 R. Naseri and A. Malek. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. D. Gunzburger and L. S. Hou, “Finite-dimensional approximation of a class of constrained nonlinear optimal control problems,” SIAM Journal on Control and Optimization, vol. 34, no. 3, pp. 1001–1043, 1996. View at Google Scholar · View at Scopus
  2. M. Gunzburger, Perspectives in Flow Control and Optimization, SIAM, 1987.
  3. I. Babuška, R. Tempone, and G. E. Zouraris, “Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation,” Computer Methods in Applied Mechanics and Engineering, vol. 194, no. 12–16, pp. 1251–1294, 2005. View at Publisher · View at Google Scholar · View at Scopus
  4. K. Karhunen, “Uberlineare Methoden in der Wahrscheinlichkeitsrechnung,” Annales Academiæ Scientiarum Fennicæ, vol. 37, p. 79, 1947. View at Google Scholar
  5. M. Loève, “Fonctions aléatoire de second ordre,” La Revue Scientifique, vol. 84, pp. 195–206, 1946. View at Google Scholar
  6. R. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Springer, Berlin, Germany, 1991.
  7. E. Haber and L. Hanson, “Model problems in PDE-constrained optimization,” Tech. Rep. TR-2007-009, Emory, Atlanta, Ga, USA, 2007. View at Google Scholar
  8. M. Hinze, R. Pinnau, M. Ulbrich, and S. Ulbrich, Optimization with Partial Differential Equations, vol. 23 of Mathematical Modelling, Theory and Applications, Springer, Heidelberg, Germany, 2009.
  9. N. Wiener, “The homogeneous chaos,” The American Journal of Mathematics, vol. 60, pp. 897–938, 1938. View at Google Scholar
  10. C. F. van Loan and N. Pitsianis, “Approximation with Kronecker products,” in Linear Algebra for Large Scale and Real-Time Applications, M. S. Moonen, G. H. Golub, and B. L. R. de Moor, Eds., pp. 293–314, Kluwer Academic, Dordrecht, Germany, 1993. View at Google Scholar
  11. M. Heinkenschloss, “Numerical solution of implicitly constrained optimization problems,” Tech. Rep. TR08-05, Department of Computational and Applied Mathematics, Rice University, Houston, Tex, USA, 2008. View at Google Scholar
  12. C. T. Kelley, Iterative Methods for Optimization, SIAM, Philadelphia, Pa, USA, 1999.
  13. J. Nocedal and S. J. Wright, Numerical Optimization, Springer, Berlin, Germany, 2nd edition, 2006.
  14. E. Ullmann, “A kronecker product preconditioner for stochastic galerkin finite eleme nt discretizations,” SIAM Journal on Scientific Computing, vol. 32, no. 2, pp. 923–946, 2010. View at Publisher · View at Google Scholar · View at Scopus
  15. C. E. Powell and E. Ullmann, “Preconditioning stochastic Galerkin saddle point systems,” SIAM Journal on Matrix Analysis and Applications, vol. 31, no. 5, pp. 2813–2840, 2010. View at Google Scholar
  16. J. Schöberl and W. Zulehner, “Symmetric indefinite preconditioners for saddle point problems with applications to PDE-constrained optimization problems,” SIAM Journal on Matrix Analysis and Applications, vol. 29, no. 3, pp. 752–773, 2007. View at Publisher · View at Google Scholar · View at Scopus