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[J]. Electronic Research Archive, 2020, 28(2): 977-1000. doi: 10.3934/era.2020052
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.
[1] | Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations. SIAM/ASA J. Uncertain. Quantif. (2017) 5: 1166-1192. |
[2] | Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. (2011) 119: 123-161. |
[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. |
[4] | Multigrid methods and sparse-grid collocation techniques for parabolic optimal control problems with random coefficients. SIAM J. Sci. Comput. (2009) 31: 2172-2192. |
[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 |
[6] | Nonnegative tensor factorizations using an alternating direction method. Front. Math. China (2013) 8: 3-18. |
[7] | On the $O(1/t)$ convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput. Optim. Appl. (2014) 57: 339-363. |
[8] | $O(1/t)$ complexity analysis of the generalized alternating direction method of multipliers. Sci. China Math. (2019) 62: 795-808. |
[9] | An efficient Monte Carlo method for optimal control problems with uncertainty. Comput. Optim. Appl. (2003) 26: 219-230. |
[10] | Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. (2006) 45: 1586-1611. |
[11] | Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations. Numer. Math. (2016) 133: 67-102. |
[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 |
[13] | An efficient numerical method for acoustic wave scattering in random media. SIAM/ASA J. Uncertain. Quantif. (2015) 3: 790-822. |
[14] | An efficient Monte Carlo-transformed field expansion method for electromagnetic wave scattering by random rough surfaces. Commun. Comput. Phys. (2018) 23: 685-705. |
[15] | A multimodes Monte Carlo finite element method for elliptic partial differential equations with random coefficients. Int. J. Uncertain. Quantif. (2016) 6: 429-443. |
[16] | An efficient Monte Carlo interior penalty discontinuous Galerkin method for elastic wave scattering in random media. Comput. Methods Appl. Mech. Engrg. (2017) 315: 141-168. |
[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 |
[18] | On the $O(1/n)$ convergence rate of the Douglas-Rachford alternating direction method. SIAM J. Numer. Anal. (2012) 50: 700-709. |
[19] | On non-ergodic convergence rate of Douglas-Rachford alternating direction method of multipliers. Numer. Math. (2015) 130: 567-577. |
[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 |
[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 |
[22] | Convexification for the inversion of a time dependent wave front in a heterogeneous medium. SIAM J. Appl. Math. (2019) 79: 1722-1747. |
[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 |
[24] | Existence and optimality conditions for risk-averse PDE-constrained optimization. SIAM/ASA J. Uncertain. Quantif. (2018) 6: 787-815. |
[25] | 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. (2014) 2: 82-105. |
[26] | An efficient alternating direction method of multipliers for optimal control problems constrained by random Helmholtz equations. Numer. Algorithms (2018) 78: 161-191. |
[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 |
[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 |
[29] | Solving constrained total-variation image restoration and reconstruction problems via alternating direction methods. SIAM J. Sci. Comput. (2010) 32: 2710-2736. |
[30] | Stochastic collocation for optimal control problems with stochastic PDE constraints. SIAM J. Control Optim. (2012) 50: 2659-2682. |
[31] | Alternating direction algorithms for $\ell_1$-problems in compressive sensing. SIAM J. Sci. Comput. (2011) 33: 250-278. |
[32] | A scalable framework for the solution of stochastic inverse problems using a sparse grid collocation approach. J. Comput. Phys. (2008) 227: 4697-4735. |
[33] | An alternating direction method of multipliers for elliptic equation constrained optimization problem. Sci. China Math. (2017) 60: 361-378. |