基于偏微分方程模型降阶方法的最优控制
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摘要
为了快速准确地求解含有偏微分方程约束(PDE)的优化问题,提出了一种基于偏微分方程模型降阶的最优控制问题求解方法.含有偏微分方程约束会使得优化问题的求解耗费大量的时间,难以满足现有控制与优化的需求.在研究了偏微分方程性质的基础上,得出了一种新的模型降阶方法.通过使用奇异值分解法来提取原模型的主要特性,得到低维空间的基函数,再使用伽辽金投影法,将原模型投影到现有基函数构成的低维空间中,从而达到降低模型阶次来快速计算PDE优化问题的目的.实验结果表明在降阶模型阶次较低的情况下,依然能对原模型有较好的逼近效果.该方法用于快速准确地求解含有偏微分方程约束的优化问题是可行的、有效的.
In order to solve constrained optimization problems containing partial differential equations(PDE)quickly and accurately,a model has been proposed on the basis of the optimal control problem solved in partial differential order reduction method.Because the issue of partial differential equations containing constraint solving optimization,problems will be time-consuming,which is difficult to meet the needs of existing control and optimization.For such complex issues in the study of the nature of partial differential equations on the basis,we have proposed a new model reduction method.By extracting the main characteristics of the original model in singular value decomposition method to obtain a low-dimensional space of basic functions,and in the Galerkin projection method,low-dimensional space of the original model has been projected onto the existing base function constituted,which can reduce model the purpose of the order to quickly calculate PDE optimization problems.The simulation experiment results show that under the reduced-order model order times lower case,the original model is still able to have a better approximation results.It follows that the method for quickly and accurately solving constrained optimization problems containing partial differential equations is feasible and effective.
In order to solve constrained optimization problems containing partial differential equations(PDE)quickly and accurately,a model has been proposed on the basis of the optimal control problem solved in partial differential order reduction method.Because the issue of partial differential equations containing constraint solving optimization,problems will be time-consuming,which is difficult to meet the needs of existing control and optimization.For such complex issues in the study of the nature of partial differential equations on the basis,we have proposed a new model reduction method.By extracting the main characteristics of the original model in singular value decomposition method to obtain a low-dimensional space of basic functions,and in the Galerkin projection method,low-dimensional space of the original model has been projected onto the existing base function constituted,which can reduce model the purpose of the order to quickly calculate PDE optimization problems.The simulation experiment results show that under the reduced-order model order times lower case,the original model is still able to have a better approximation results.It follows that the method for quickly and accurately solving constrained optimization problems containing partial differential equations is feasible and effective.
引文
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[2]周宏宇,王爱民.非线性偏微分方程数值求解的自适应方法研究[J].计算机应用与软件,2012,47(20):38-40,84.
[3]杨健,赖晓霞.辅助函数法求解非线性偏微分方程精确解[J].计算机技术与发展,2017,27(10):196-200.
[4] FU H,RUI H.Characteristic-Mixed Methods on Dynamic Finite Element Spaces for Two-Phase Miscible flow in Porous Media[J].Chinese Journal of Engineering Mathematics,2013,27(2):369-374.
[5]李志华,喻军,杨红光.偏微分方程与微分代数方程的一致求解方法[J].中国机械工程,2015,26(4):441-445.
[6] FENG X,NEILAN M.Finite Element Approximations of General Fully Nonlinear Second Order Elliptic Partial Differential Equations Based on the Vanishing Moment Method[J].Computers&Mathematics with Applications,2014,68(12):2182-2204.
[7]李明,崔向照,赵金娥.求解高次有限元方程的外推瀑布型多重网格法[J].西南师范大学学报(自然科学版),2016,41(1):20-23.
[8] JIN B,LAZAROV R,LIU Y K,et al.The Galerkin Finite Element Method for a Multi-Term Time-Fractional Diffusion Equation[J].Journal of Computational Physics,2015,281:825-843.
[9] TAN B T,GHATTAS O.A PDE-Constrained Optimization Approach to the Discontinuous Petrov-Galerkin Method with a Trust Region Inexact Newton-CG Solver[J].Computer Methods in Applied Mechanics&Engineering,2014,278(6):20-40.
[10]李洋洋.基于自由度缩减的局部无网格伽辽金方法研究[D].长沙:湖南大学,2013.
[11]陈军胜.基于四元数矩阵分解的线性方程组解的存在性判断研究[J].西南师范大学学报(自然科学版),2015,40(5):34-38.
[12]谭伟杰,冯西安,张杨梅.基于Hankel矩阵分解的互素阵列高分辨目标定向[J].西南大学学报(自然科学版),2016,38(7):191-198.
[13]李葵,范玉刚,吴建德.基于二次奇异值分解和最小二乘支持向量机的轴承故障诊断方法[J].计算机应用,2014,34(8):2438-2441.