Advertisement

Journal of Scientific Computing

, Volume 78, Issue 3, pp 1571–1600 | Cite as

Stochastic Galerkin Method for Optimal Control Problem Governed by Random Elliptic PDE with State Constraints

  • Wanfang ShenEmail author
  • Liang Ge
  • Wenbin Liu
Article
  • 103 Downloads

Abstract

In this paper, we investigate a stochastic Galerkin approximation scheme for an optimal control problem governed by an elliptic PDE with random field in its coefficients. The optimal control minimizes the expectation of a cost functional with mean-state constraints. We first represent the stochastic elliptic PDE in terms of the generalized polynomial chaos expansion and obtain the parameterized optimal control problems. By applying the Slater condition in the subdifferential calculus, we obtain the necessary and sufficient optimality conditions for the state-constrained stochastic optimal control problem for the first time in the literature. We then establish a stochastic Galerkin scheme to approximate the optimality system in the spatial space and the probability space. Then the a priori error estimates are derived for the state, the co-state and the control variables. A projection algorithm is proposed and analyzed. Numerical examples are presented to illustrate our theoretical results.

Keywords

Stochastic optimal control Stochastic Galerkin method Optimal control problem with state constraints 

References

  1. 1.
    Adams, R.: Sobolev Spaces. Academic, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Babuška, I., Chatzipantelidis, P.: On solving elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 191, 4093–4122 (2002)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Babuška, I., Tempone, R., Zouraris, G.E.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800–825 (2004)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Becker, R., Vexler, B.: Optimal control of the convection–diffusion equation using stabilized finite element methods. Numer. Math. 106, 349–367 (2007)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bergounioux, M., Kunisch, K.: Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35, 1524–1543 (1997)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bespalov, A., Powell, C.E., Silvester, D.: Energy norm a posteriori error estimation for parametric operator equations. SIAM J. Sci. Comput. 36, A339–A363 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 2nd edn. Springer, Berlin (2002)zbMATHGoogle Scholar
  8. 8.
    Casas, E.: Boundary control with pointwise state constraints. SIAM J. Control Optim. 31, 993–1006 (1993)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Casas, E.: Errore stimates for the numerical approximation of semilinear elliptic control trol problems with finitely many state constraints. ESIAM Control Optim. Calc. Var. 8, 345–374 (2002)zbMATHGoogle Scholar
  10. 10.
    Casas, E., Mateos, M.: Uniform convergence of the FEM Applications to state constrained control problems. Comput. Appl. Math. 21(1), 67–100 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chen, Y.P., Huang, F.L.: Galerkin spectral approximation of elliptic optimal control problems with H-1-norm state constraint. J. Sci. Comput. 67(1), 65–83 (2016)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ciarlet, P.G.: The finite element method for elliptic problems. In: Classics Appl. Math., vol. 40. SIAM, Philadelphia (2002)Google Scholar
  13. 13.
    Deckelnick, K., Michael Hinze, M.: Convergence of a finite element approximation to a state-constrained elliptic control problem. SIAM J. Numer. Anal. 45, 1937–1953 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Doltsinis, I.: Inelastic deformation processes with random parametersmethods of analysis and design. Comput. Methods Appl. Mech. Eng. 192, 2405–2423 (2003)zbMATHGoogle Scholar
  15. 15.
    Du, N., Shi, J.T., Liu, W.B.: An effective gradient projection method for stochastic optimal control. Int. J. Numer. Anal. Model 10(4), 757–774 (2013)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Evans, L.: Partial differential equations. Grad. Stud. Math. vol. 19. AMS, Providence, RI (1998)Google Scholar
  17. 17.
    Foo, J., Wan, X., Karniadakis, G.E.: The multi-element probabilistic collocation method (ME-PCM): error analysis and applications. J. Comput. Phys. 227(22), 9572–9595 (2008)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991)zbMATHGoogle Scholar
  19. 19.
    Gong, B.X., Sun, T.J., Shen, W.F., Liu, W.B.: A priori error estimate of stochastic Galerkin method for optimal control problem governed by random parabolic PDE. Int. J. Comput. Methods 13(5), 1–26 (2016)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Grisvard, P.: Elliptic Problems in Nonsmooth Doamin. Longman Higher Education, Harlow (1986)Google Scholar
  21. 21.
    Gunzburger, M.D., Lee, H.C., Lee, J.: Error estimates of stochastic optimal Neumann boundary control problems. SIAM J. Numer. Anal. 49(4), 1532–1552 (2011)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differrential equations with random input data. Acta Numer. 23, 521–650 (2014)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Haslinger, J., Neittaanmaki, P.: Finite Element Approximation for Optimal Shape Design. Wiley, Chichester (1989)zbMATHGoogle Scholar
  24. 24.
    Hintermüller, M., Kunisch, K.: Stationary optimal control problems with pointwise state constraints. SIAM J. Optim. 20, 1133–1156 (2009)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Hou, L.S., Lee, J., Manouzi, H.: Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs. J. Math. Anal. Appl. 384(1), 87–103 (2011)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Keese, A.: Numerical solution of systems with stochastic uncertainties: a general purpose framework for stochastic Finite elements. Ph.D. thesis, Technical University Braunschweig, Braunschweig, Germany (2004)Google Scholar
  27. 27.
    Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. Wiley, Chichester (1992)zbMATHGoogle Scholar
  28. 28.
    Knowles, G.: Finite element approximation of parabolic time optimal control problems. SIAM J. Control Optim. 20, 414–427 (1982)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lee, H.C., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commun. Comput. Phys. 14(1), 77–106 (2013)MathSciNetGoogle Scholar
  30. 30.
    Li, R.: On multi-mesh h-adaptive algorithm. J. Sci. Comput. 24, 321–341 (2005)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)zbMATHGoogle Scholar
  32. 32.
    Liu, W.B., Tiba, D.: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. J. Numer. Funct. Optim. 22, 953–972 (2001)zbMATHGoogle Scholar
  33. 33.
    Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Govereved by PDEs. Science Press, Beijing (2008)Google Scholar
  34. 34.
    Liu, W.B., Yang, D.P., Yuan, L., Ma, C.Q.: Finite elemnet approximation of an optimal control problem with integral state constraint. SIAM J. Numer. Anal. 48(3), 1163–1185 (2010)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Malliavin, P.: Stochastic Analysis. Springer, Berlin (1997)zbMATHGoogle Scholar
  36. 36.
    Neittaanmaki, P., Tiba, D.: Optimal control of nonlinear parabolic systems. Theory algorithms and applications. M. Dekker, New York (1994)zbMATHGoogle Scholar
  37. 37.
    Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocationmethod for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309–2345 (2008)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Øksendal, B.: Stochastic Differential Equations, An Introducation with Application, 5th edn. Spring, Berlin (1998)Google Scholar
  39. 39.
    Papadrakakis, M., Papadopoulos, V.: Robust and efficient methods for stochastic finite element analysis using Monte Carlo simulation. Comput. Methods Appl. Mech. Eng. 134, 325–340 (1996)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin (1984)zbMATHGoogle Scholar
  41. 41.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar
  42. 42.
    Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constrains and uncertain controls. Comput. Methods Appl. Mech. Eng. 213–216, 152–167 (2012)zbMATHGoogle Scholar
  43. 43.
    Schwab, C., Todor, R.A.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Shen, W.F., Ge, L., Yang, D.P.: Finite element methods for optimal control problems governed by linear quasi-parabolic integer-differential equations. Int. J. Numer. Anal. Model 10(3), 536–550 (2013). Reference [44] is given in list but not cited in text. Please cite in text or delete from list.MathSciNetzbMATHGoogle Scholar
  45. 45.
    Shen, W.F., Sun, T.J., Gong, B.X., Liu, W.B.: Stochastic Galerkin method for constrained optimal control problem governed by an elliptic integro-differential PDE with stochastic coefficients. Int. J. Numer. Anal. Model. 12(4), 593–616 (2015)MathSciNetGoogle Scholar
  46. 46.
    Sun, T.J., Shen, W.F., Gong, B.X., Liu, W.B.: A priori error estimate of stochastic Galerkin method for optimal control problem governed by stochastic elliptic PDE with constrained control. J. Sci. Comput. 67, 405–431 (2016)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Tiba, D.: Lectures on the Optimal Control of Elliptic Equations. University of Jyvaskyla Press, Jyvaskyla (1995)Google Scholar
  48. 48.
    Tiesler, H., Kirby, R.M., Xiu, D., Preusser, T.: Stochastic collocation for optimal control problems with stochastic PDE constraints. SIAM J. Control Optim. 50(5), 2659–2682 (2012)MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112. American Mathematical Society, Providence (2010)zbMATHGoogle Scholar
  50. 50.
    Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Xiu, D., Karniadakis, G.E.: The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24, 619–644 (2002)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Xiu, D., Lucor, D., Su, C.H., Karniadakis, G.E.: Stochastic modeling of flow-structure interactions using generalized polynomial chaos. ASME J. Fluid Eng. 124, 51–69 (2002)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematic and Quantitative EconomicsShandong University of Finance and EconomicsJinanPeople’s Republic of China
  2. 2.School of Mathematical SciencesUniversity of JinanJinanPeople’s Republic of China
  3. 3.KBSUniversity of KentCanterburyUK

Personalised recommendations