Parameter estimation for multivariate exponential sums via iterative rational approximation

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.