A Fast Gradient Projection Method for a Constrained Fractional Optimal Control

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• 作者：Ning Du ; Hong Wang ; Wenbin Liu
• 刊名：Journal of Scientific Computing
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：68
• 期：1
• 页码：1-20
• 全文大小：499 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algorithms
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Mathematical and Computational Physics
  
• 出版者：Springer Netherlands
• ISSN：1573-7691
• 卷排序：68

文摘

Optimal control problems governed by a fractional diffusion equation tends to provide a better description than one by a classical second-order Fickian diffusion equation in the context of transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significantly increased computational complexity and storage requirement than the corresponding classical control problem, due to the nonlocal nature of fractional differential operators. We develop a fast gradient projection method for a pointwise constrained optimal control problem governed by a time-dependent space-fractional diffusion equation, which requires the computational cost from \(O(M N^3)\) of a conventional solver to \(O(M N\log N)\) and memory requirement from \(O(N^2)\) to O(N) for a problem of size N and of M time steps. Numerical experiments show the utility of the method.KeywordsConstrained optimal controlFractional diffusion equationFast Fourier transformPreconditioned conjugate gradient method