参数化最优控制问题的初值校正法
An Initial Value Correction Method for the Parameterized Optimal Control Problem
DOI: 10.12677/ORF.2022.124154, PDF,    国家自然科学基金支持
作者: 叶昌伦：贵州大学数学与统计学院，贵州 贵阳
关键词: 参数化最优控制；共轭梯度法；蒙特卡洛法；Parameterized Optimal Control； Conjugate Gradient Method； Monte Carlo Method
摘要: 针对带随机系数椭圆方程约束的最优控制问题，我们提出一种结合共轭梯度法的初值校正新方法。该方法通过少量的样本求解最优控制，然后得到的解作为抽取其它样本的初始迭代值，从而减少整个系统求解最优控制平均值的成本。我们理论上说明该方法的合理性。两个数值例子表明我们的方法是充分优于未初值校正方法。
Abstract: For optimal control problems with random elliptic equation constraints, we propose an initial value correction new method combined with the conjugate gradient method. This method solves the optimal control through a small number of samples, and then the solution is used as the initial iteration value of extracting other samples, thus reducing the cost of solving the optimal control average for the whole system. We theoretically illustrate the rationality of the method. Two numerical examples show that our method is sufficiently superior to the no initial value correction method.
文章引用：叶昌伦. 参数化最优控制问题的初值校正法[J]. 运筹与模糊学, 2022, 12(4): 1465-1474. https://doi.org/10.12677/ORF.2022.124154

参考文献

[1] Hinze, M., Pinnau, R., Ulbrich, M. and Ulbrich, S. (2009) Optimization with PDE Constraints. Mathematical Mod-elling: Theory and Applications (MMTA, Volume 23), Springer, New York.
[2] Liu, W., Ma, H., Tang, T. and Yan, N. (2004) A Posteriori Error Estimates for Discontinuous Galerkin Time-Stepping Method for Optimal Control Problems Governed by Parabolic Equations. SIAM Journal on Numerical Analysis, 42, 1032-1061. [Google Scholar] [CrossRef] 
[3] Tröltzsch, F. (2010) Optimal Control of Partial Differential Equations: Theory, Procedures, and Applications. American Mathematical Society, Providence. [Google Scholar] [CrossRef] 
[4] Meidner, D. and Vexler, B. (2008) A Priori Error Estimates for Space-Time Finite Element Discretization of Parabolic Optimal Control Problems. Part I: Problems without Control Constraints. SIAM Journal on Control and Optimization, 47, 1150-1177. [Google Scholar] [CrossRef] 
[5] Gong, W. and Yan, N. (2011) Mixed Finite Element Method for Di-richlet Boundary Control Problem Governed by Elliptic PDEs. SIAM Journal on Control and Optimization, 49, 984-1014. [Google Scholar] [CrossRef] 
[6] Kunoth, A. and Schwab, C. (2013) Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs. SIAM Journal on Control and Optimization, 51, 2442-2471. [Google Scholar] [CrossRef] 
[7] Borzi, A. (2010) Multigrid and Sparse-Grid Aschemes for Elliptic Control Problems with Random Coefficients. Computing and Visualization in Science, 13, 153-160. [Google Scholar] [CrossRef] 
[8] Gong, B., Sun, T., Shen, W. and Liu, W. (2016) A Priori Error Estimate of Stochastic Galerkin Method for Optimal Control Problem Governed by Random Parabolic PDE with Constrained Control. International Journal of Computational Methods, 13, Article ID: 1650028. [Google Scholar] [CrossRef] 
[9] Ahmad Ali, A., Ullmann, E. and Hinze, M. (2017) Multi-level Monte Carlo Analysis for Optimal Control of Elliptic PDEs with Random Coefficients. SIAM/ASA Journal on Uncertainty Quantification, 5, 466-492. [Google Scholar] [CrossRef] 
[10] Babuška, I., Nobile, F. and Tempone, R. (2010) A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data. SIAM Review, 52, 317-355. [Google Scholar] [CrossRef] 
[11] Gunzburger, D., Webster, G. and Zhang, G. (2014) Stochastic Finite Element Methods for Partial Differential Equations with Random Input Data. Acta Numerica, 23, 521-650. [Google Scholar] [CrossRef] 
[12] Hou, L., Lee, J. and Manouzi, H. (2011) Finite Element Approximations of Stochastic Optimal Control Problems Constrained by Stochastic Elliptic PDEs. Journal of Mathematical Analysis and Applications, 384, 87-103. [Google Scholar] [CrossRef] 
[13] Borzi, A. and Winckel, G. (2009) Multigrid Methods and Sparse-Grid Collocation Techniques for Parabolic Optimal Control Problems with Random Coefficients. SIAM Journal on Scientific Computing, 31, 2172-2192. [Google Scholar] [CrossRef] 
[14] Barel, A.V. and Vandewalle, S. (2017) Robust Optimization of PDEs with Random Coefficients Using a Multilevel Monte Carlo Method. SIAM/ASA Journal on Uncertainty Quantifica-tion, 7, 174-202. [Google Scholar] [CrossRef] 
[15] Guth, P.A., Kaarnioja, V., Kuo, F., Schillings, C. and Sloan, I.H. (2021) A Quasi-Monte Carlo Method for an Optimal Control Problem under Uncertainty. SIAM/ASA Journal on Uncertainty Quantification, 9, 354-383. [Google Scholar] [CrossRef] 
[16] Alexanderian, A., Petra, N., Stadler, G. and Ghattas, O. (2017) Mean-Variance Risk-Averse Optimal Control of Systems Governed by PDEs with Random Parameter Fields Using Quadratic Approximations. SIAM/ASA Journal on Uncertainty Quantification, 5, 1166-1192. [Google Scholar] [CrossRef] 
[17] Cliffe, K.A., Giles, M.B., Scheichl, R. and Teckentrup, A.L. (2011) Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients. Computing and Visualization in Science, 14, Article No. 3. [Google Scholar] [CrossRef] 
[18] Lord, G.J., Powell, C.E. and Shardlow, T. (2014) An Intro-duction to Computational Stochastic PDEs. Cambridge University Press, Cambridge. [Google Scholar] [CrossRef] 
[19] Dai, Y.-H. and Yuan, Y. (1999) A Nonlinear Conjugate Gradient Method with a Strong Global Convergence Property. SIAM Journal on Optimization, 10, 177-182. [Google Scholar] [CrossRef] 
[20] Evans, L.C. (2010) Partial Differential Equations. Second Edition, American Mathematical Society, Providence. [Google Scholar] [CrossRef] 

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