SIAM Journal on Scientific Computing, Volume 43, Issue 1, Page A109-A138, January 2021.
Local Fourier analysis is a useful tool for predicting and analyzing the performance of many efficient algorithms for the solution of discretized PDEs, such as multigrid and domain decomposition methods. The crucial aspect of local Fourier analysis is that it can be used to minimize an estimate of the spectral radius of a stationary iteration, or the condition number of a preconditioned system, in terms of a symbol representation of the algorithm. In practice, this is a “minimax” problem, minimizing with respect to solver parameters the appropriate measure of solver work, which involves maximizing over the Fourier frequency. Often, several algorithmic parameters may be determined by local Fourier analysis in order to obtain efficient algorithms. Analytical solutions to minimax problems are rarely possible beyond simple problems; the status quo in local Fourier analysis involves grid sampling, which is prohibitively expensive in high dimensions. In this paper, we propose and explore optimization algorithms to solve these problems efficiently. Several examples, with known and unknown analytical solutions, are presented to show the effectiveness of these approaches.

中文翻译：

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通过鲁棒优化和局部傅里叶分析调整多网格方法

SIAM科学计算杂志，第43卷，第1期，第A109-A138页，2021年1月。
局部傅里叶分析是一种有用的工具，可用于预测和分析用于离散化PDE的许多有效算法的性能，例如多网格和域分解方法。局部傅里叶分析的关键方面是，就算法的符号表示而言，它可以用于最小化静态迭代的频谱半径或预处理系统的条件数的估计。实际上，这是一个“极小值”问题，相对于求解器参数而言，使求解器工作的适当度量最小化，这涉及使傅立叶频率最大化。通常，可以通过局部傅立叶分析来确定几个算法参数，以便获得有效的算法。除简单问题外，极小值问题的解析解决方案几乎不可能。局部傅里叶分析的现状涉及网格采样，这在高维度上过于昂贵。在本文中，我们提出并探索了可以有效解决这些问题的优化算法。给出了带有已知和未知分析解决方案的几个示例，以显示这些方法的有效性。