论文标题
奇点的指数节点聚类，用于理性近似，正交和PDES
Exponential node clustering at singularities for rational approximation, quadrature, and PDEs
论文作者
论文摘要
如果两极呈指数聚类，则具有奇异性函数的有理近似值可以以根指数的速率收敛。我们首先通过“ Lightning”方法在Minimax，最小二乘和AAA近似中审查这种效果。涉及广泛而广泛的数值实验。然后，我们提出了进一步的实验，表明在所有这些应用中，使用指数群集的有利，其在对数尺度上的密度并不统一，而是在奇异性附近线性地挖出到零。我们基于HERMITE轮廓积分和潜在理论给出了理论上的锥形效果的理论解释，这表明渐变的收敛速度翻了一番。最后，我们表明相关数学适用于指数（未锥形）与双重指数（锥形）正交公式之间的关系。在这里，是高斯 - 塔卡哈斯 - 莫里轮廓的积分。
Rational approximations of functions with singularities can converge at a root-exponential rate if the poles are exponentially clustered. We begin by reviewing this effect in minimax, least-squares, and AAA approximations on intervals and complex domains, conformal mapping, and the numerical solution of Laplace, Helmholtz, and biharmonic equations by the "lightning" method. Extensive and wide-ranging numerical experiments are involved. We then present further experiments showing that in all of these applications, it is advantageous to use exponential clustering whose density on a logarithmic scale is not uniform but tapers off linearly to zero near the singularity. We give a theoretical explanation of the tapering effect based on the Hermite contour integral and potential theory, showing that tapering doubles the rate of convergence. Finally we show that related mathematics applies to the relationship between exponential (not tapered) and doubly exponential (tapered) quadrature formulas. Here it is the Gauss--Takahasi--Mori contour integral that comes into play.
