在球面和极坐标几何中进行计算 I. 球面
Computing with functions in spherical and polar geometries I. The sphere
AI总结：
本文提出了一种基于双傅里叶球方法的结构保持迭代高斯消去算法，用于高效计算球面上的光滑函数，实现了Poisson方程的最优复杂度直接求解。
摘要：
本文描述了一组用于在单位球面上对光滑函数进行数值计算的算法。通过使用一种保持结构的迭代高斯消去变体与双傅里叶球方法，函数可被近似到几乎机器精度。该方法允许稳定求导，减少极点附近函数的过采样，并对某些解析函数收敛。函数评估、求导和积分等操作特别高效，可通过几乎一维算法计算。亮点是使用谱方法的Poisson方程在球面上的最优复杂度直接求解器。未并行化的情况下，一台标准笔记本电脑可在一分钟内解决具有1亿自由度的Poisson方程。文中展示了数值结果。在 companion paper（第二部分）中，我们将此处提出的方法扩展到盘面上的函数计算。
英文摘要：
A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
