A Note on Multigrid Preconditioning for Fractional PDE-Constrained Optimization Problems
A Note on Multigrid Preconditioning for Fractional PDE-Constrained Optimization Problems
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DOI：
10.1016/j.rinam.2020.100133
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发表时间：
2020-10
期刊：
影响因子：
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通讯作者：
Harbir Antil;Andrei Draganescu;K. Green
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文献类型：
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作者：
Harbir Antil;Andrei Draganescu;K. Green
在这篇文章中，我们提出了一种多重网格预条件方法来求解受分数次扩散方程约束的二次优化问题。求解一阶最优Karush-Kuhn-Tucker(KKT)系统的一次性方法中的多重网格法得到了广泛的应用，但其发展依赖于底层系统的稀疏性。另一方面，对于大多数离散化，分数次算子的矩阵表示被期望是稠密的。我们为我们的问题开发了一种基于简化方法的预条件策略，即我们使用控制到状态映射来消除状态约束。我们的多重网格预处理方法显示了CG迭代次数的显著减少。我们用光谱距离来评估预处理器的质量。最后，我们对这个预条件算子进行了部分理论分析，并提出了一个猜想，该猜想得到了我们的数值实验的明确支持。
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 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.