Schottky-Klein素函数无限乘积表示的快速计算与收敛性分析
Fast computation and convergence analysis of the infinite-product representation of the Schottky--Klein prime function
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中文总结 AI 辅助
该研究针对Schottky-Klein素函数无限乘积表示的数值计算难题，提出基于交叉比势阈值的高效枚举算法，实现了显著的计算加速。
中文摘要 AI 辅助
Schottky-Klein素函数是处理多连通圆形区域边值问题的标准工具。由于该函数表示为Schottky群上的无限乘积，数值计算需截断为有限个因子。标准的词长截断方法计算成本呈指数增长，在边界圆几乎相互接触或接近单位圆时效率低下。为解决这一难题，我们为每个群元素分配一个衡量其贡献大小的交叉比势，仅保留低于预设阈值的项。我们建立了在Schottky群生成元前添加时该势变化的一致闭式界，并据此推导了高效枚举算法。所得相对误差随阈值呈指数衰减，衰减率由Schottky群极限集的Hausdorff维数决定。数值实验表明，在具有挑战性的几何构型下，所提方案相比词长截断实现了显著的计算加速。
英文摘要
The Schottky--Klein prime function is a standard tool for boundary-value problems on multiply connected circular domains. Because this function is represented as an infinite product over a Schottky group, numerical evaluation requires truncation to finitely many factors. The standard word-length truncation grows exponentially in cost and becomes inefficient when the boundary circles nearly touch one another or the unit circle. To address this difficulty, we assign to each group element a cross-ratio potential measuring the size of its contribution, and retain only terms below a prescribed threshold. We establish uniform closed-form bounds on the change in this potential when prepending Schottky-group generators, and from these bounds we derive an efficient enumeration algorithm. The resulting relative error decays exponentially with the threshold at a rate determined by the Hausdorff dimension of the limit set of the Schottky group. Numerical experiments demonstrate that the proposed formulation achieves substantial computational speedups over word-length truncation in challenging geometric configurations.
发表机构
- Department of Mathematical Informatics, Graduate School of Information Science and Technology, The University of Tokyo（东京大学信息理工学研究科数学信息学系）
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