AAA-least squares rational approximation and solution of Laplace problems
AAA-least squares rational approximation and solution of Laplace problems
复制标题
拉普拉斯问题的AAA-最小二乘有理逼近与求解
DOI：
--
复制
发表时间：
2021
期刊：
影响因子：
--
通讯作者：
L. Trefethen
中科院分区：
文献类型：
--
作者：
S. Costa;L. Trefethen
介绍了一种通过有理逼近求解平面拉普拉斯问题的两步法。首先，通过AAA近似确定边界数据的复杂有理近似，无论是全局还是局部接近每个角点或其他奇点。然后收集问题域之外的这些近似的极点，并用于对解决方案进行全局最小二乘拟合。典型的问题在一秒钟的笔记本电脑时间内解决了8位数的精度，一直到角落，共轭谐波函数也提供了。AAA最小二乘组合也提供了一种新的方法，以避免虚假极点在其他合理的近似问题，并大大加快他们在许多奇异的情况下。作为一种特殊情况，AAA-LS近似导致计算希尔伯特变换或狄利克雷-诺依曼映射的强大方法。
A two-step method for solving planar Laplace problems via rational approximation is introduced. First complex rational approximations to the boundary data are determined by AAA approximation, either globally or locally near each corner or other singularity. The poles of these approximations outside the problem domain are then collected and used for a global least-squares fit to the solution. Typical problems are solved in a second of laptop time to 8-digit accuracy, all the way up to the corners, and the conjugate harmonic function is also provided. The AAA-least squares combination also offers a new method for avoiding spurious poles in other rational approximation problems, and for greatly speeding them up in cases with many singularities. As a special case, AAA-LS approximation leads to a powerful method for computing the Hilbert transform or Dirichlet-to-Neumann map.