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Journal of Scientific Computing
Published in: Journal of Scientific Computing 1/2018

26-04-2018

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

Authors: Liang Ge, Lianhai Wang, Yanzhen Chang

Published in: Journal of Scientific Computing | Issue 1/2018

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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.
previous article A Hybrid Staggered Discontinuous Galerkin Method for KdV Equations
next article Hybrid Optimized Low-Dissipation and Adaptive MUSCL Reconstruction Technique for Hyperbolic Conservation Laws

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Literature
1.
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)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
5.
Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grids. Adv. Comput. Math. 12, 273–288 (2000)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
7.
8.
Doltsinis, I.: Inelastic deformation processes with random parametersmethods of analysis and design. Comput. Methods Appl. Mech. Eng. 192, 2405–2423 (2003)MATHCrossRef
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)MathSciNetMATHCrossRef
10.
Ghanem, R., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, Berlin (1991)MATHCrossRef
11.
Glowinski, R., Lions, J.L.: Exact and Approximate Controllability for Distributed Parameter Systems. Cambridge University Press, Cambridge (1996)MATH
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)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
14.
Kleiber, M., Hien, T.D.: The Stochastic Finite Element Method. Wiley, Chichester (1992)MATH
15.
Lee, H.C., Lee, J.: A stochastic Galerkin method for stochastic control problems. Commum. Comput. Phys. 14(1), 77–106 (2013)MathSciNetCrossRef
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)CrossRef
17.
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Springer, Berlin (1971)MATHCrossRef
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)MATHCrossRef
19.
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)
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)MathSciNetMATHCrossRef
22.
Loëve, M.: Probability Theory I. Grad. Texts in Math. 45, 4th edn. Springer, New York (1977)MATH
23.
24.
Mosco, U.: Approximation of solution of some variational inequalities. Ann. Sculoa Normale Sup. 21, 373–394 (1967)MathSciNetMATH
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)MathSciNetMATHCrossRef
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)MathSciNetMATHCrossRef
27.
Øksendal, B.: Stochastic Differential Equations, An Introducation with Application, 5th edn. Spring, Berlin (1998)
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)MathSciNetMATHCrossRef
29.
Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)MATH
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)MATHCrossRef
32.
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)MathSciNetMATH
33.
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
34.
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
35.
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)MathSciNetMATH
36.
37.
Tröltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods, and Applications, vol. 112. American Mathematical Society, Providence (2010)MATH
38.
Wasilkowski, G.W., Wozniakowski, H.: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 11, 1–56 (1995)MathSciNetMATHCrossRef
39.
40.
Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)MathSciNetMATHCrossRef
41.
Yang, Q.: The upwind finite volume element method for two-dimensional burgers equation. In: Abstract and Applied Analysis, Vol. 2013, Article ID 351619
Metadata
Title
A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations
Authors
Liang Ge
Lianhai Wang
Yanzhen Chang
Publication date
26-04-2018
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-018-0713-y

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