# American Institute of Mathematical Sciences

June  2020, 28(2): 1001-1022. doi: 10.3934/era.2020053

## Efficient numerical methods for elliptic optimal control problems with random coefficient

 1 School of Mathematics, Jilin University, Changchun 130012, China 2 Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock 72204, United States 3 Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China 4 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

* Corresponding author: Haiming Song(songhaiming@jlu.edu.cn)

Received  March 2020 Revised  April 2020 Published  June 2020

Fund Project: The work is supported by NSF grant No.11701210, 11871245, and 11971221

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. Electronic Research Archive, 2020, 28 (2) : 1001-1022. doi: 10.3934/era.2020053
##### References:
 [1] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev, 52 (2010), 317-355.  doi: 10.1137/100786356.  Google Scholar [2] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.  Google Scholar [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, Comput. Methods. Appl. Mech. Engrg., 194 (2005), 1251-1294.  doi: 10.1016/j.cma.2004.02.026.  Google Scholar [4] A. Barth, C. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.  Google Scholar [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.  Google Scholar [6] X. Cai and D. Han, $O(1/t)$ complexity analysis of the alternating direction method of multipliers, Sci. China Math., 62 (2019), 795-808.  doi: 10.1007/s11425-016-9184-4.  Google Scholar [7] Y. Cao, M. Y. Hussaini and T. A. Zang, An efficient Monte Carlo method for optimal control problems with uncertainty, Comput. Optim. Appl., 26 (2003), 219-230.  doi: 10.1023/A:1026079021836.  Google Scholar [8] P. Chen, U. Villa and O. Ghattas, Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty, J. Comput. Phys., 385 (2019), 163-186.  doi: 10.1016/j.jcp.2019.01.047.  Google Scholar [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.  Google Scholar [10] X. Feng, J. Lin and C. Lorton, An efficient numerical method for acoustic wave scattering in random media, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 790-822.  doi: 10.1137/140958232.  Google Scholar [11] X. Feng, J. Lin and C. Lorton, A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients, Int. J. Uncertain. Quantif., 6 (2016), 429-443.  doi: 10.1615/Int.J.UncertaintyQuantification.2016016805.  Google Scholar [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.  Google Scholar [13] M. D. Gunzburger, H.-C. Lee and J. Lee, Error estimates of stochastic optimal Neumann boundary control problems, SIAM J. Numer. Anal., 49 (2011), 1532-1552.  doi: 10.1137/100801731.  Google Scholar [14] Y. Hao, H. Song, X. Wang and K. Zhang, An alternating direction method of multipliers for the optimization problem constrained with a stationary Maxwell system, Commun. Comput. Phys., 24 (2018), 1435-1454.  doi: 10.4208/cicp.oa-2017-0117.  Google Scholar [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.  Google Scholar [16] L. S. Hou, J. Lee and H. Manouzi, Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs, J. Math. Anal. Appl., 384 (2011), 87-103.  doi: 10.1016/j.jmaa.2010.07.036.  Google Scholar [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.  Google Scholar [18] A. Kunoth and C. Schwab, Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs, SIAM J. Control Optim., 51 (2013), 2442-2471.  doi: 10.1137/110847597.  Google Scholar [19] J. Li, X. Wang and K. Zhang, An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations, Numer. Algorithms, 78 (2018), 161-191.  doi: 10.1007/s11075-017-0371-4.  Google Scholar [20] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.  doi: 10.1007/BF01213210.  Google Scholar [21] J. C. De los Reyes, Numerical PDE-Constrained Optimization, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13395-9.  Google Scholar [22] L. J. Roman and M. Sarkis, Stochastic Galerkin method for elliptic SPDEs: A white noise approach, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 941-955.  doi: 10.3934/dcdsb.2006.6.941.  Google Scholar [23] W. Shen, T. Sun, B. Gong and W. Liu, Stochastic Galerkin method for constrained optimal control problem governed by an elliptic integro-differential PDE with stochastic coefficients, Int. J. Numer. Anal. Model., 12 (2015), 593-616.   Google Scholar [24] H. F. Weinberger, Lower bounds for higher eigenvalues by finite difference methods, Pacific J. Math., 8 (1958), 339-368.  doi: 10.2140/pjm.1958.8.339.  Google Scholar [25] D. Xiu, Fast numerical methods for robust optimal design, Eng. Optim., 40 (2008), 489-504.  doi: 10.1080/03052150801893631.  Google Scholar [26] K. Zhang, J. Li, Y. Song and X. Wang, An alternating direction method of multipliers for elliptic equation constrained optimization problem, Sci. China Math., 60 (2017), 361-378.  doi: 10.1007/s11425-015-0522-3.  Google Scholar

show all references

##### References:
 [1] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev, 52 (2010), 317-355.  doi: 10.1137/100786356.  Google Scholar [2] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), 1005-1034.  doi: 10.1137/050645142.  Google Scholar [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, Comput. Methods. Appl. Mech. Engrg., 194 (2005), 1251-1294.  doi: 10.1016/j.cma.2004.02.026.  Google Scholar [4] A. Barth, C. Schwab and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), 123-161.  doi: 10.1007/s00211-011-0377-0.  Google Scholar [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.  Google Scholar [6] X. Cai and D. Han, $O(1/t)$ complexity analysis of the alternating direction method of multipliers, Sci. China Math., 62 (2019), 795-808.  doi: 10.1007/s11425-016-9184-4.  Google Scholar [7] Y. Cao, M. Y. Hussaini and T. A. Zang, An efficient Monte Carlo method for optimal control problems with uncertainty, Comput. Optim. Appl., 26 (2003), 219-230.  doi: 10.1023/A:1026079021836.  Google Scholar [8] P. Chen, U. Villa and O. Ghattas, Taylor approximation and variance reduction for PDE-constrained optimal control under uncertainty, J. Comput. Phys., 385 (2019), 163-186.  doi: 10.1016/j.jcp.2019.01.047.  Google Scholar [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.  Google Scholar [10] X. Feng, J. Lin and C. Lorton, An efficient numerical method for acoustic wave scattering in random media, SIAM/ASA J. Uncertain. Quantif., 3 (2015), 790-822.  doi: 10.1137/140958232.  Google Scholar [11] X. Feng, J. Lin and C. Lorton, A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients, Int. J. Uncertain. Quantif., 6 (2016), 429-443.  doi: 10.1615/Int.J.UncertaintyQuantification.2016016805.  Google Scholar [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.  Google Scholar [13] M. D. Gunzburger, H.-C. Lee and J. Lee, Error estimates of stochastic optimal Neumann boundary control problems, SIAM J. Numer. Anal., 49 (2011), 1532-1552.  doi: 10.1137/100801731.  Google Scholar [14] Y. Hao, H. Song, X. Wang and K. Zhang, An alternating direction method of multipliers for the optimization problem constrained with a stationary Maxwell system, Commun. Comput. Phys., 24 (2018), 1435-1454.  doi: 10.4208/cicp.oa-2017-0117.  Google Scholar [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.  Google Scholar [16] L. S. Hou, J. Lee and H. Manouzi, Finite element approximations of stochastic optimal control problems constrained by stochastic elliptic PDEs, J. Math. Anal. Appl., 384 (2011), 87-103.  doi: 10.1016/j.jmaa.2010.07.036.  Google Scholar [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.  Google Scholar [18] A. Kunoth and C. Schwab, Analytic regularity and GPC approximation for control problems constrained by linear parametric elliptic and parabolic PDEs, SIAM J. Control Optim., 51 (2013), 2442-2471.  doi: 10.1137/110847597.  Google Scholar [19] J. Li, X. Wang and K. Zhang, An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations, Numer. Algorithms, 78 (2018), 161-191.  doi: 10.1007/s11075-017-0371-4.  Google Scholar [20] P. Li and S. T. Yau, On the Schrödinger equation and the eigenvalue problem, Comm. Math. Phys., 88 (1983), 309-318.  doi: 10.1007/BF01213210.  Google Scholar [21] J. C. De los Reyes, Numerical PDE-Constrained Optimization, SpringerBriefs in Optimization, Springer, Cham, 2015. doi: 10.1007/978-3-319-13395-9.  Google Scholar [22] L. J. Roman and M. Sarkis, Stochastic Galerkin method for elliptic SPDEs: A white noise approach, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 941-955.  doi: 10.3934/dcdsb.2006.6.941.  Google Scholar [23] W. Shen, T. Sun, B. Gong and W. Liu, Stochastic Galerkin method for constrained optimal control problem governed by an elliptic integro-differential PDE with stochastic coefficients, Int. J. Numer. Anal. Model., 12 (2015), 593-616.   Google Scholar [24] H. F. Weinberger, Lower bounds for higher eigenvalues by finite difference methods, Pacific J. Math., 8 (1958), 339-368.  doi: 10.2140/pjm.1958.8.339.  Google Scholar [25] D. Xiu, Fast numerical methods for robust optimal design, Eng. Optim., 40 (2008), 489-504.  doi: 10.1080/03052150801893631.  Google Scholar [26] K. Zhang, J. Li, Y. Song and X. Wang, An alternating direction method of multipliers for elliptic equation constrained optimization problem, Sci. China Math., 60 (2017), 361-378.  doi: 10.1007/s11425-015-0522-3.  Google Scholar
The FEM convergence order on the state and the control for three algorithms
The theoretical solution and numerical solutions on the control for three algorithms
The ADMM convergence rate for three algorithms
The CPU memory costs for different $h$ and $Q$: unit GB
 $h$ Algorithm 1 Algorithm 2 Algorithm 3 $Q = 1$ $Q = 2$ $Q = 3$ $Q = 1$ $Q = 2$ $Q = 3$ $\frac{1}{4}$ 1.271E-1 1.009E-1 1.195E-1 1.221E-1 1.967E-3 1.007E-3 2.006E-3 $\frac{1}{8}$ 1.655E0 1.277E0 1.676E0 1.685E0 8.136E-3 2.723E-3 5.859E-3 $\frac{1}{16}$ 1.807E1 1.792E1 2.481E1 2.487E1 4.856E-2 8.097E-2 7.375E-2 $\frac{1}{32}$ out of memory 2.677E2 out of memory out of memory 7.546E-1 1.161E0 1.626E0
 $h$ Algorithm 1 Algorithm 2 Algorithm 3 $Q = 1$ $Q = 2$ $Q = 3$ $Q = 1$ $Q = 2$ $Q = 3$ $\frac{1}{4}$ 1.271E-1 1.009E-1 1.195E-1 1.221E-1 1.967E-3 1.007E-3 2.006E-3 $\frac{1}{8}$ 1.655E0 1.277E0 1.676E0 1.685E0 8.136E-3 2.723E-3 5.859E-3 $\frac{1}{16}$ 1.807E1 1.792E1 2.481E1 2.487E1 4.856E-2 8.097E-2 7.375E-2 $\frac{1}{32}$ out of memory 2.677E2 out of memory out of memory 7.546E-1 1.161E0 1.626E0
The CPU memory costs for different $M$ and $Q$: unit GB
 $M$ Algorithm 1 Algorithm 2 Algorithm 3 $Q = 1$ $Q = 2$ $Q = 3$ $Q = 1$ $Q = 2$ $Q = 3$ $1$ 6.143E-2 5.456E-2 8.195E-2 7.839E-2 4.514E-2 6.814E-2 5.881E-2 $10$ 2.498E-1 2.511E-1 2.964E-1 2.976E-1 4.568E-2 6.785E-2 5.901E-2 $10^2$ 1.927E0 1.929E0 2.511E0 2.513E0 4.603E-2 6.887E-2 6.074E-2 $10^3$ 1.807E1 1.792E1 2.481E1 2.487E1 4.856E-2 6.931E-2 7.375E-2
 $M$ Algorithm 1 Algorithm 2 Algorithm 3 $Q = 1$ $Q = 2$ $Q = 3$ $Q = 1$ $Q = 2$ $Q = 3$ $1$ 6.143E-2 5.456E-2 8.195E-2 7.839E-2 4.514E-2 6.814E-2 5.881E-2 $10$ 2.498E-1 2.511E-1 2.964E-1 2.976E-1 4.568E-2 6.785E-2 5.901E-2 $10^2$ 1.927E0 1.929E0 2.511E0 2.513E0 4.603E-2 6.887E-2 6.074E-2 $10^3$ 1.807E1 1.792E1 2.481E1 2.487E1 4.856E-2 6.931E-2 7.375E-2
 [1] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [2] Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 [3] Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057 [4] Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917 [5] Jingshi Li, Jiachuan Zhang, Guoliang Ju, Juntao You. A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations. Electronic Research Archive, 2020, 28 (2) : 977-1000. doi: 10.3934/era.2020052 [6] Heung Wing Joseph Lee, Chi Kin Chan, Karho Yau, Kar Hung Wong, Colin Myburgh. Control parametrization and finite element method for controlling multi-species reactive transport in a circular pool. Journal of Industrial & Management Optimization, 2013, 9 (3) : 505-524. doi: 10.3934/jimo.2013.9.505 [7] Ying Liu, Yanping Chen, Yunqing Huang, Yang Wang. Two-grid method for semiconductor device problem by mixed finite element method and characteristics finite element method. Electronic Research Archive, 2021, 29 (1) : 1859-1880. doi: 10.3934/era.2020095 [8] Ming Yan, Lili Chang, Ningning Yan. Finite element method for constrained optimal control problems governed by nonlinear elliptic PDEs. Mathematical Control & Related Fields, 2012, 2 (2) : 183-194. doi: 10.3934/mcrf.2012.2.183 [9] Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1783-1799. doi: 10.3934/jimo.2019029 [10] Jie-Wen He, Chi-Chon Lei, Chen-Yang Shi, Seak-Weng Vong. An inexact alternating direction method of multipliers for a kind of nonlinear complementarity problems. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 353-362. doi: 10.3934/naco.2020030 [11] Eduardo Casas, Mariano Mateos, Arnd Rösch. Finite element approximation of sparse parabolic control problems. Mathematical Control & Related Fields, 2017, 7 (3) : 393-417. doi: 10.3934/mcrf.2017014 [12] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [13] Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247 [14] Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037 [15] Feng Ma, Jiansheng Shu, Yaxiong Li, Jian Wu. The dual step size of the alternating direction method can be larger than 1.618 when one function is strongly convex. Journal of Industrial & Management Optimization, 2021, 17 (3) : 1173-1185. doi: 10.3934/jimo.2020016 [16] Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317 [17] Yan Gu, Nobuo Yamashita. Alternating direction method of multipliers with variable metric indefinite proximal terms for convex optimization. Numerical Algebra, Control & Optimization, 2020, 10 (4) : 487-510. doi: 10.3934/naco.2020047 [18] Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078 [19] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [20] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339

2020 Impact Factor: 1.833

## Tools

Article outline

Figures and Tables