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A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

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Abstract

In this paper, we deal with an optimal control problem governed by the convection diffusion equations with random field in its coefficients. Mathematically, we prove the necessary and sufficient optimality conditions for the optimal control problem. Computationally, we establish a scheme to approximate the optimality system through the discretization by the upwind finite volume element method for the physical space, and by the sparse grid stochastic collocation algorithm based on the Smolyak construction for the probability space, which leads to the discrete solution of uncoupled deterministic problems. Moreover, the existence and uniqueness of the discrete solution are given. A priori error estimates are derived for the state, the co-state and the control variables. Numerical examples are presented to illustrate our theoretical results.

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References

  1. Adams, R.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317–355 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Babuska, 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12, 273–288 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, P., Quarteroni, A., Rozza, G.: Stochastic optimal robin boundary control problems of advection-dominated elliptic equations. SIAM J. Numer. Anal. 51(5), 2700–2722 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clenshaw, C.W., Curtis, A.R.: A method for numerical integration on an automatic computer. Numer. Math. 2, 197–205 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  8. Doltsinis, I.: Inelastic deformation processes with random parametersmethods of analysis and design. Comput. Methods Appl. Mech. Eng. 192, 2405–2423 (2003)

    Article  MATH  Google Scholar 

  9. Ewing, R.E., Lin, T., Lin, Y.P.: On the accuracy of the finite volume element method based on piecewise linear polynomials. SIAM J. Numer. Anal. 39, 1865–1888 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991)

    Book  MATH  Google Scholar 

  11. Glowinski, R., Lions, J.L.: Exact and Approximate Controllability for Distributed Parameter Systems. Cambridge University Press, Cambridge (1996)

    MATH  Google Scholar 

  12. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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, 87–103 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. Wiley, Chichester (1992)

    MATH  Google Scholar 

  15. Lee, H.C., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commum. Comput. Phys. 14(1), 77–106 (2013)

    Article  MathSciNet  Google Scholar 

  16. Li, R.H., Chen, Z.Y., Wei, W.: Generalized Difference memthods for Differential Equations: Numerical Analysis of Finite Volume Methods. Marcel Dekker, New York (2000)

    Book  Google Scholar 

  17. Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  18. Liu, W.B., Tiba, D.: Error estimates for the finite element approximation of a class of nonlinear optimal control problems. J. Numer. Func. Optim. 22, 953–972 (2001)

    Article  MATH  Google Scholar 

  19. Liu, W.B., Yan, N.N.: A posteriori error estimates for convex boundary control problems. SIAM Numer. Anal. 39, 73–99 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Liu, W.B., Yan, N.N.: Adaptive Finite Element Methods for Optimal Control Governed by PDEs, Series in Information and Computational Science 41. Science Press, Beijing (2008)

    Google Scholar 

  21. 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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Loëve, M.: Probability Theory I. Grad. Texts in Math. 45, 4th edn. Springer, New York (1977)

    MATH  Google Scholar 

  23. Loëve, M.: Probability Theory II. Grad. Texts in Math. 46. Springer, New York (1978)

    Book  MATH  Google Scholar 

  24. Mosco, U.: Approximation of solution of some variational inequalities. Ann. Sculoa Normale Sup. 21, 373–394 (1967)

    MathSciNet  MATH  Google Scholar 

  25. 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)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nobile, F., Tempone, R., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411–2442 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  27. Øksendal, B.: Stochastic Differential Equations, An Introducation with Application, 5th edn. Spring, Berlin (1998)

    Google Scholar 

  28. 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)

    Article  MathSciNet  MATH  Google Scholar 

  29. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    MATH  Google Scholar 

  30. Rosseel, E., Wells, G.N.: Optimal control with stochastic PDE constrains and uncertain controls. Comput. Methods Appl. Mech. Eng. 213–216, 152–167 (2012)

    Article  MATH  Google Scholar 

  31. 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)

    MathSciNet  MATH  Google Scholar 

  32. 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)

    MathSciNet  Google Scholar 

  33. Sun, T.J.: Discontinuous Galerkin finite element method with interior penalties for convection diffusion optimal control problem. Int. J. Numer. Anal. Mod. 7(1), 87–107 (2010)

    MathSciNet  Google Scholar 

  34. Sun, T.J., Ge, L., Liu, W.B.: Equivalent a posteriori error estimates for a constrained optimal control problem governed by parabolic equations. Int. J. Numer. Anal. Mod. 10(1), 1–23 (2013)

    MathSciNet  MATH  Google Scholar 

  35. 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, accepted by J.S.C. https://doi.org/10.1007/s10915-015-0091-7

  36. Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112. American Mathematical Society, Providence (2010)

    MATH  Google Scholar 

  37. Wasilkowski, G.W., Wozniakowski, H.: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 11, 1–56 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wiener, N.: The homogeneous chaos. Am. J. Math. 60, 897–936 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  39. Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  40. Yang, Q.: The upwind finite volume element method for two-dimensional burgers equation. In: Abstract and Applied Analysis, Vol. 2013, Article ID 351619

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Correspondence to Yanzhen Chang.

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This project was supported by BUCT Fund for Disciplines Construction and Development (Project No. XK1523) and the National Natural Science Foundation of China (No. 11501326, 11701253).

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Ge, L., Wang, L. & Chang, Y. A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations. J Sci Comput 77, 524–551 (2018). https://doi.org/10.1007/s10915-018-0713-y

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  • DOI: https://doi.org/10.1007/s10915-018-0713-y

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