在球面和极几何中进行计算 II. 盘
Computing with functions in spherical and polar geometries II. The disk
AI总结：
本文提出了一种在单位圆上计算光滑函数的算法，通过结合双傅里叶球方法的圆盘类比和一种保持结构的迭代高斯消去变体，实现了低秩近似，为分析函数提供几何收敛的近似解，适用于FFT算法。
摘要：
本文描述了一组用于在单位圆上对光滑函数进行数值计算的算法。在极几何中，低秩近似通过合成圆盘的双傅里叶球方法的类比与一种保持结构的迭代高斯消去变体形成，该变体对某些解析函数表现出几何收敛性。这种自适应过程在其采样策略中接近最优，生成的近似解在微分时稳定，并有助于在两个变量中使用基于FFT的算法。低秩形式的近似在积分和微分等操作中特别有用，将其转化为本质上的一维过程，并被用来制定一种新的快速圆盘泊松求解器，用于计算解的低秩近似。本文补充了关于在单位球面上计算函数的配套论文（第一部分）的工作。
英文摘要：
A collection of algorithms is described for numerically computing with smooth functions defined on the unit disk. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier sphere method with a structure-preserving variant of iterative Gaussian elimination that is shown to converge geometrically for certain analytic functions. This adaptive procedure is near-optimal in its sampling strategy, producing approximants that are stable for differentiation and facilitate the use of FFT-based algorithms in both variables. The low rank form of the approximants is especially useful for operations such as integration and differentiation, reducing them to essentially 1D procedures, and it is also exploited to formulate a new fast disk Poisson solver that computes low rank approximations to solutions. This work complements a companion paper (Part I) on computing with functions on the surface of the unit sphere.
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