A Fast Stochastic Galerkin Method for a Constrained Optimal Control Problem Governed by a Random Fractional Diffusion Equation
A Fast Stochastic Galerkin Method for a Constrained Optimal Control Problem Governed by a Random Fractional Diffusion Equation
复制标题
随机分数扩散方程约束最优控制问题的快速随机伽辽金法
DOI：
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发表时间：
2018-03
影响因子：
0.9
通讯作者：
Wanfang Shen
中科院分区：
文献类型：
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作者：
Ning Du;Wanfang Shen
对于一类具有确定性约束控制的随机空间分数阶扩散方程的最优控制问题，提出了一种快速随机Galerkin方法。由分数阶扩散方程控制的最优控制问题能更好地描述非均匀介质中的输运或传导过程。然而，分数阶控制问题引入了显着的计算复杂性，由于分数阶微分算子的非局部性质，这是进一步恶化的大量随机空间维数离散的概率空间。通过对空间和概率空间的离散化，我们用梯度算法和随机Galerkin方法相结合来逼近最优性系统。所得到的线性系统可以对随机变量和空间变量进行解耦，从而分别求解。提出了一种快速预处理双共轭梯度稳定化方法，有效地求解空间中由分数阶扩散算子导出的解耦系统。数值实验表明了该方法的有效性。
We develop a fast stochastic Galerkin method for an optimal control problem governed by a random space-fractional diffusion equation with deterministic constrained control. Optimal control problems governed by a fractional diffusion equation tends to provide a.better description for transport or conduction processes in heterogeneous media. However, the fractional control problem introduces significant computation complexity due to the nonlocal nature of fractional differential operators, and this is further worsen by the large number of random space dimensions to discretize the probability space. We approximate the optimality system by a gradient algorithm combined with the stochastic.Galerkin method through the discretization with respect to both the spatial space and the probability space. The resulting linear system can be decoupled for the random and spatial variable, and thus solved separately. A fast preconditioned Bi-Conjugate Gradient.Stabilized method is developed to efficiently solve the decoupled systems derived from the fractional diffusion operators in the spatial space. Numerical experiments show the utility of the method.