用电荷模拟法计算具有（无限）多个分量的紧集的对数容量
Computing the logarithmic capacity of compact sets having (infinitely) many components with the Charge Simulation Method
AI总结：
本文提出用电荷模拟法计算由多个小分量组成的紧集的对数容量，每个分量仅用一个电荷点，结合预条件迭代和快速多极方法，高效且高精度，并在广义康托集和康托尘上验证。
摘要：
我们应用电荷模拟法（CSM）来计算由（无限）多个“小”分量组成的紧集的对数容量。该应用允许每个分量仅使用一个电荷点。因此，所得到的方法比基于边界离散化的方法（例如，我们在[Liesen, Sète, Nasser, 2017]中提出的方法）显著更高效，同时保持非常高的精度。我们研究了CSM中出现的线性代数系统的性质，并展示了如何使用预条件迭代方法高效求解这些系统，其中矩阵-向量乘积使用快速多极方法计算。我们在广义康托集和康托尘上说明了该方法的使用。
英文摘要：
We apply the Charge Simulation Method (CSM) in order to compute the logarithmic capacity of compact sets consisting of (infinitely) many "small" components. This application allows to use just a single charge point for each component. The resulting method therefore is significantly more efficient than methods based on discretizations of the boundaries (for example, our own method presented in [Liesen, Sète, Nasser, 2017]), while maintaining a very high level of accuracy. We study properties of the linear algebraic systems that arise in the CSM, and show how these systems can be solved efficiently using preconditioned iterative methods, where the matrix-vector products are computed using the Fast Multipole Method. We illustrate the use of the method on generalized Cantor sets and the Cantor dust.
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