黄秋月.一类分数阶最优控制问题的高阶快速算法[J].重庆工商大学学报(自然科学版),2020,37(3):52-59
HUANG Qiu-yue.Fast Algorithms for High-order Methods of a Fractional Optimal Control Problem[J].Journal of Chongqing Technology and Business University(Natural Science Edition),2020,37(3):52-59
一类分数阶最优控制问题的高阶快速算法
Fast Algorithms for High-order Methods of a Fractional Optimal Control Problem
  
DOI:
中文关键词:  边值方法  预处理  两点边值问题  FDE-约束优化
英文关键词:boundary value methods  preconditioning  two point boundary value problem  FDE constrained optimization
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作者单位
黄秋月 重庆师范大学 数学科学学院重庆 401331 
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中文摘要:
      分数阶扩散方程约束的分布式最优控制问题广泛地应用于科学和工程领域,包括优化设计、控制和参数识别;针对这类问题,提出了一种高阶的快速算法。对于求解该问题的一阶最优条件所产生的耦合两点边值问题,在空间上利用紧差分,时间上利用边值方法对该问题进行离散,离散后得到一个2×2块线性系统;然后使用带有Kronecker积分裂的迭代算法求解该线性系统,该算法是块状的Kronecker积结构,通过交替的Kronecker积分裂迭代方法得到了这个Kronecker积,并证明了该分裂迭代算法是收敛的;同时使用GMRES方法来加速Kronecker积分裂迭代的收敛;最后数值实验表明了该算法的精确性和计算效率。
英文摘要:
      Distributed optimal control problem with the constraint of fractional order diffusion equation is widely used in the description of scientific and engineering applications including optimal design, control and parameter identification. Aiming at this problem, a high-order fast algorithm is proposed. For the coupled two point boundary value problems arising from the first order optimality conditions for this problem, the problem is discretized in space by compact difference and in time by boundary value method. After discretization, a two-by-two block linear system is obtained. Then we use Kronecker product splitting preconditioning strategy for solving this linear system. The preconditioner is a bloc Kronecker product structure. We obtain this Kronecker product through an alternating Kronecker product splitting iteration method. We prove the convergence of this preconditioner algorithm and use GMRES method to accelerate the convergence of the Kronecker splitting iteration. Finally, numerical experiments show the accuracy and computational efficiency of the algorithm.
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