June  2020, 28(2): 977-1000. doi: 10.3934/era.2020052

A multi-mode expansion method for boundary optimal control problems constrained by random Poisson equations

1. 

School of Mathematics and Statistics, Nanjing University of Information Science & Technology, Nanjing 210044, China

2. 

Department of Mathematics, Southern University of Science and Technology, Shenzhen 518055, China

3. 

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

4. 

School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China

5. 

Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China

* Corresponding author: Jiachuan Zhang

Received  March 2020 Revised  April 2020 Published  June 2020

Fund Project: The first author is supported by the Natural Science Foundation of Jiangsu Province (Grant No. BK20190766), the Startup Foundation for Introducing Talent of NUIST (No. 2018r022), and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJB110018). The last three authors are partially supported by the NSF of China No. 11971221 and 11731006, the Shenzhen Sci-Tech Fund No. JCYJ20170818153840322 and JCYJ20190809150413261, and Guangdong Provincial Key Laboratory of Computational Science and Material Design No. 2019B030301001

This paper develops efficient numerical algorithms for the optimal control problem constrained by Poisson equations with uncertain diffusion coefficients. We consider both unconstrained condition and box-constrained condition for the control. The algorithms are based upon a multi-mode expansion (MME) for the random state, the finite element approximation for the physical space and the alternating direction method of multipliers (ADMM) or two-block ADMM for the discrete optimization systems. The compelling aspect of our method is that, target random constrained control problem can be approximated to one deterministic constrained control problem under a state variable substitution equality. Thus, the computing resource, especially the memory consumption, can be reduced sharply. The convergence rates of the proposed algorithms are discussed in the paper. We also present some numerical examples to show the performance of our algorithms.

Citation: 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
References:
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[13]

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[21]

Y. Hwang, J. Lee, J. Lee and M. Yoon, A domain decomposition algorithm for optimal control problems governed by elliptic PDEs with random inputs, Appl. Math. Comput., 364 (2020), 14pp. doi: 10.1016/j.amc.2019.124674.  Google Scholar

[22]

M. V. KlibanovJ. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034.  Google Scholar

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D. P. Kouri, M. Heinkenschloos, 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

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[26]

J. LiX. 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

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R. Naseri and A. Malek, Numerical optimal control for problems with random forced SPDE constraints, ISRN Appl. Math., 2014 (2014), 11pp. doi: 10.1155/2014/974305.  Google Scholar

[29]

M. K. NgP. Weiss and X. Yuang, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.  Google Scholar

[30]

H. TieslerR. M. KirbyD. Xiu and T. Preusser, Stochastic collocation for optimal control problems with stochastic PDE constraints, SIAM J. Control Optim., 50 (2012), 2659-2682.  doi: 10.1137/110835438.  Google Scholar

[31]

J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[32]

N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Comput. Phys., 227 (2008), 4697-4735.  doi: 10.1016/j.jcp.2008.01.019.  Google Scholar

[33]

K. ZhangJ. LiY. 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]

A. AlexanderianN. PetraG. Stadler and O. Ghattas, Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations, SIAM/ASA J. Uncertain. Quantif., 5 (2017), 1166-1192.  doi: 10.1137/16M106306X.  Google Scholar

[2]

A. BarthC. 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

[3]

P. Benner, S. Dolgov, A. Onwunta and M. Stoll, Solving optimal control problems governed by random Navier-Stokes equations using low-rank methods, preprint, arXiv: 1703.06097. Google Scholar

[4]

A. Borzì and G. von Winckel, Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients, SIAM J. Sci. Comput., 31 (2009), 2172-2192.  doi: 10.1137/070711311.  Google Scholar

[5]

A. Bünger, S. Dolgov and M. Stoll, A low-rank tensor method for PDE-constrained optimization with isogeometric analysis, SIAM J. Sci. Comput., 42 (2020), A140–A161. doi: 10.1137/18M1227238.  Google Scholar

[6]

X. CaiY. Chen and D. Han, Nonnegative tensor factorizations using an alternating direction method, Front. Math. China, 8 (2013), 3-18.  doi: 10.1007/s11464-012-0264-8.  Google Scholar

[7]

X. CaiG. Gu and B. He, On the $O(1/t)$ convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators, Comput. Optim. Appl., 57 (2014), 339-363.  doi: 10.1007/s10589-013-9599-7.  Google Scholar

[8]

X. Cai and D. Han, $O(1/t)$ complexity analysis of the generalized alternating direction method of multipliers, Sci. China Math., 62 (2019), 795-808.  doi: 10.1007/s11425-016-9184-4.  Google Scholar

[9]

Y. CaoM. 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

[10]

E. Casas and J.-P. Raymond, Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations, SIAM J. Control Optim., 45 (2006), 1586-1611.  doi: 10.1137/050626600.  Google Scholar

[11]

P. ChenA. Quarteroni and G. Rozza, Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations, Numer. Math., 133 (2016), 67-102.  doi: 10.1007/s00211-015-0743-4.  Google Scholar

[12]

J. Eckstein and M. Fukushima, Some reformulations and applications of the alternating direction method of multipliers, in Large Scale Optimization, Kluwer Acad. Publ., Dordrecht, 1994,115–134. doi: 10.1007/978-1-4613-3632-7_7.  Google Scholar

[13]

X. FengJ. 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

[14]

X. FengJ. Lin and D. P. Nicholls, An efficient Monte Carlo-transformed field expansion method for electromagnetic wave scattering by random rough surfaces, Commun. Comput. Phys., 23 (2018), 685-705.  doi: 10.4208/cicp.oa-2017-0041.  Google Scholar

[15]

X. FengJ. 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

[16]

X. Feng and C. Lorton, An efficient Monte Carlo interior penalty discontinuous Galerkin method for elastic wave scattering in random media, Comput. Methods Appl. Mech. Engrg., 315 (2017), 141-168.  doi: 10.1016/j.cma.2016.10.036.  Google Scholar

[17]

R. Glowinski and A. Marrocco, Sur l'approximation, par éléments finis d'ordre un, et la résolution, par pénalisation-dualité, d'une classe de problèmes de Dirichlet non linéaires, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge Anal. Numér., 9 (1975), 41–76. doi: 10.1051/m2an/197509R200411.  Google Scholar

[18]

B. He and X. Yuan, On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method, SIAM J. Numer. Anal., 50 (2012), 700-709.  doi: 10.1137/110836936.  Google Scholar

[19]

B. He and X. Yuan, On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers, Numer. Math., 130 (2015), 567-577.  doi: 10.1007/s00211-014-0673-6.  Google Scholar

[20]

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

[21]

Y. Hwang, J. Lee, J. Lee and M. Yoon, A domain decomposition algorithm for optimal control problems governed by elliptic PDEs with random inputs, Appl. Math. Comput., 364 (2020), 14pp. doi: 10.1016/j.amc.2019.124674.  Google Scholar

[22]

M. V. KlibanovJ. Li and W. Zhang, Convexification for the inversion of a time dependent wave front in a heterogeneous medium, SIAM J. Appl. Math., 79 (2019), 1722-1747.  doi: 10.1137/18M1236034.  Google Scholar

[23]

D. P. Kouri, M. Heinkenschloos, 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

[24]

D. P. Kouri and T. M. Surowiec, Existence and optimality conditions for risk-averse PDE-constrained optimization, SIAM/ASA J. Uncertain. Quantif., 6 (2018), 787-815.  doi: 10.1137/16M1086613.  Google Scholar

[25]

A. Labovsky and M. Gunzburger, An efficient and accurate method for the identification of the most influential random parameters appearing in the input data for PDEs, SIAM/ASA J. Uncertain. Quantif., 2 (2014), 82-105.  doi: 10.1137/120883785.  Google Scholar

[26]

J. LiX. 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

[27]

J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der mathematischen Wissenschaften, 1, Springer-Verlag, New York-Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8.  Google Scholar

[28]

R. Naseri and A. Malek, Numerical optimal control for problems with random forced SPDE constraints, ISRN Appl. Math., 2014 (2014), 11pp. doi: 10.1155/2014/974305.  Google Scholar

[29]

M. K. NgP. Weiss and X. Yuang, Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods, SIAM J. Sci. Comput., 32 (2010), 2710-2736.  doi: 10.1137/090774823.  Google Scholar

[30]

H. TieslerR. M. KirbyD. Xiu and T. Preusser, Stochastic collocation for optimal control problems with stochastic PDE constraints, SIAM J. Control Optim., 50 (2012), 2659-2682.  doi: 10.1137/110835438.  Google Scholar

[31]

J. Yang and Y. Zhang, Alternating direction algorithms for $\ell_1$-problems in compressive sensing, SIAM J. Sci. Comput., 33 (2011), 250-278.  doi: 10.1137/090777761.  Google Scholar

[32]

N. Zabaras and B. Ganapathysubramanian, A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach, J. Comput. Phys., 227 (2008), 4697-4735.  doi: 10.1016/j.jcp.2008.01.019.  Google Scholar

[33]

K. ZhangJ. LiY. 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

Figure 1.  The FEM disretization errors dominating $ \|\bar{u}-{\bf R}{\bf u}^{Q,k}_{\bf h}\|_{L^2(D)} $ and $ \|y^*-{\bf R}{\bf y}^{Q,k}_{\bf h}\|_{L^2(D)} $ for two algorithms: (Left) algorithm 1; (Right) algorithm 2
Figure 2.  The errors of the ADMM with respect to $ k $-th iteration ($ \|{\bf \bar{u}}_{\bf h}-{\bf u}^{\bf k}_{\bf h}\|_\mathcal{H} $) in algorithm 1 (left) and algorithm 2 (right) with $ k = 100 $ and $ h = 1/32 $
Figure 3.  The convergence rates of $ \varepsilon $ under $ Q = 3 $ in algorithm 1 (left) and algorithm 2 (right)
Table 1.  The error of $ u $ for algorithms 1 and algorithm 2 with different mode number $ Q $
Q Algorithms 1 Algorithms 2
1 1.942e-1 2.276e-1
2 2.315e-2 2.823e-2
3 8.241e-3 7.566e-3
Q Algorithms 1 Algorithms 2
1 1.942e-1 2.276e-1
2 2.315e-2 2.823e-2
3 8.241e-3 7.566e-3
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