标题： 多元指数和的参数估计通过迭代有理逼近 显示英文标题

标题： Parameter estimation for multivariate exponential sums via iterative rational approximation

摘要： 我们提出了两种新的多元指数分析方法。 在[7]中，我们通过利用指数和傅里叶系数的有理结构，并用自适应Antoulas-Anderson（AAA）方法[15]重建这种有理结构，开发了一种新的用于重构单变量指数和的新算法。在本文中，我们将这些思想扩展到多元设置。与单变量情况类似，多元指数和的傅里叶系数具有有理结构，且多元指数恢复问题可以重新表述为多元有理插值问题。我们通过将其约化为若干个单变量问题来发展两种解决这种特殊多元有理插值问题的方法，然后再次使用单变量AAA方法求解这些单变量问题。 我们的第一种方法基于从稀疏网格中选择傅里叶系数索引，这确保了使用相对少量的输入数据进行高效重构。第二种方法则基于使用傅里叶系数的所有网格索引，并依赖于递归维数约化的思想。我们通过多个数值例子展示了我们方法的性能。 显示英文摘要

摘要： We present two new methods for multivariate exponential analysis. In [7], we developed a new algorithm for reconstruction of univariate exponential sums by exploiting the rational structure of their Fourier coefficients and reconstructing this rational structure with the AAA (adaptive Antoulas-Anderson) method for rational approximation [15]. In this paper, we extend these ideas to the multivariate setting. Similarly as in univariate case, the Fourier coefficients of multivariate exponential sums have a rational structure and the multivariate exponential recovery problem can be reformulated as multivariate rational interpolation problem. We develop two approaches to solve this special multivariate rational interpolation problem by reducing it to the several univariate ones, which are then solved again via the univariate AAA method. Our first approach is based on using indices of the Fourier coefficients chosen from some sparse grid, which ensures efficient reconstruction using a respectively small amount of input data. The second approach is based on using the full grid of indices of the Fourier coefficients and relies on the idea of recursive dimension reduction. We demonstrate performance of our methods with several numerical examples.