Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.
Citation: Xiaowei Pang, Haiming Song, Xiaoshen Wang, Jiachuan Zhang. Efficient numerical methods for elliptic optimal control problems with random coefficient[J]. Electronic Research Archive, 2020, 28(2): 1001-1022. doi: 10.3934/era.2020053
Efficient numerical methods for solving Poisson equation constraint optimal control problems with random coefficient are discussed in this paper. By applying the finite element method and the Monte Carlo approximation, the original optimal control problem is discretized and transformed into an optimization problem. Taking advantage of the separable structures, Algorithm 1 is proposed for solving the problem, where an alternating direction method of multiplier is used. Both computational and storage costs of this algorithm are very high. In order to reduce the computational cost, Algorithm 2 is proposed, where the multi-modes expansion is introduced and applied. Further, for reducing the storage cost, we propose Algorithm 3 based on Algorithm 2. The main idea is that the random term is shifted to the objective functional, which could be computed in advance. Therefore, we only need to solve a deterministic optimization problem, which could reduce all the costs significantly. Moreover, the convergence analyses of the proposed algorithms are established, and numerical simulations are carried out to test the performances of them.
[1] | A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev (2010) 52: 317-355. |
[2] | A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. (2007) 45: 1005-1034. |
[3] | Solving elliptic boundary value problems with uncertain coefficients by the finite element method: The stochastic formulation. Comput. Methods. Appl. Mech. Engrg. (2005) 194: 1251-1294. |
[4] | Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. (2011) 119: 123-161. |
[5] | R. E. Caflisch, Monte-Carlo and quasi-Monte Carlo methods, in Acta Numerica, Acta Numer., 7, Cambridge University Press, Cambridge, 1998, 1–49. doi: 10.1017/S0962492900002804 |
[6] | $O(1/t)$ complexity analysis of the alternating direction method of multipliers. Sci. China Math. (2019) 62: 795-808. |
[7] | An efficient Monte Carlo method for optimal control problems with uncertainty. Comput. Optim. Appl. (2003) 26: 219-230. |
[8] | Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty. J. Comput. Phys. (2019) 385: 163-186. |
[9] | P. G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and its Applications, 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. |
[10] | An efficient numerical method for acoustic wave scattering in random media. SIAM/ASA J. Uncertain. Quantif. (2015) 3: 790-822. |
[11] | A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients. Int. J. Uncertain. Quantif. (2016) 6: 429-443. |
[12] | D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001. doi: 10.1007/978-3-642-61798-0 |
[13] | Error estimates of stochastic optimal Neumann boundary control problems. SIAM J. Numer. Anal. (2011) 49: 1532-1552. |
[14] | An alternating direction method of multipliers for the optimization problem constrained with a stationary Maxwell system. Commun. Comput. Phys. (2018) 24: 1435-1454. |
[15] | M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, Mathematical Modelling: Theory and Applications, 23, Springer, New York, 2009. doi: 10.1007/978-1-4020-8839-1 |
[16] | Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs. J. Math. Anal. Appl. (2011) 384: 87-103. |
[17] | D. P. Kouri, M. Heinkenschloss, D. Ridzal and B. G. van Bloemen Waanders, A trust-region algorithm with adaptive stochastic collocation for PDE optimization under uncertainty, SIAM J. Sci. Comput., 35 (2013), A1847–A1879. doi: 10.1137/120892362 |
[18] | Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs. SIAM J. Control Optim. (2013) 51: 2442-2471. |
[19] | An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations. Numer. Algorithms (2018) 78: 161-191. |
[20] | On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys. (1983) 88: 309-318. |
[21] | J. C. De los Reyes, Numerical PDE-Constrained Optimization, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13395-9 |
[22] | Stochastic Galerkin method for elliptic SPDEs: A white noise approach. Discrete Contin. Dyn. Syst. Ser. B (2006) 6: 941-955. |
[23] | Stochastic Galerkin method for constrained optimal control problem governed by an elliptic integro-differential PDE with stochastic coefficients. Int. J. Numer. Anal. Model. (2015) 12: 593-616. |
[24] | Lower bounds for higher eigenvalues by finite difference methods. Pacific J. Math. (1958) 8: 339-368. |
[25] | Fast numerical methods for robust optimal design. Eng. Optim. (2008) 40: 489-504. |
[26] | An alternating direction method of multipliers for elliptic equation constrained optimization problem. Sci. China Math. (2017) 60: 361-378. |