论文标题
关于分数PDE约束优化问题的多移民预处理的注释
A Note on Multigrid Preconditioning for Fractional PDE-Constrained Optimization Problems
论文作者
论文摘要
在本说明中，我们提出了一种用于求解受分数扩散方程约束的二次优化问题的多机预处理方法。解决第一阶最优性Karush-Kuhn-Tucker（KKT）系统中的全面方法中的多机方法非常受欢迎，但是它们的发展依赖于稀疏的基础系统。另一方面，对于大多数离散化，预计分数操作员的矩阵表示形式将是密集的。我们基于减少的方法为问题制定了一个预处理策略，即我们使用控制对国家地图消除了状态约束。我们的多移民预处理方法显示了CG迭代次数的大幅减少。我们在光谱距离方面评估了预处理的质量。最后，我们为该预处理提供了部分理论分析，并提出了一个猜想，我们的数值实验显然支持。
In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by a fractional diffusion equation. Multigrid methods within the all-at-once approach to solve the first order-order optimality Karush-Kuhn-Tucker (KKT) systems are widely popular, but their development have relied on the underlying systems being sparse. On the other hand, for most discretizations, the matrix representation of fractional operators is expected to be dense. We develop a preconditioning strategy for our problem based on a reduced approach, namely we eliminate the state constraint using the control-to-state map. Our multigrid preconditioning approach shows a dramatic reduction in the number of CG iterations. We assess the quality of preconditioner in terms of the spectral distance. Finally, we provide a partial theoretical analysis for this preconditioner, and we formulate a conjecture which is clearly supported by our numerical experiments.
