Parameter estimation for multivariate exponential sums via iterative rational approximation

TL;DR

The paper addresses the problem of recovering parameters of a $d$-variate exponential sum $f(oldsymbol{t}) = \sum_{j=1}^{M} \gamma_j e^{\langle \boldsymbol{\lambda}_j, \boldsymbol{t} \rangle}$ from a finite set of Fourier coefficients by exploiting the rational structure of these coefficients. It reformulates the multivariate parameter estimation as a multivariate rational interpolation problem and solves it via two complementary strategies: a sparse-grid approach that uses a small subset of line samples and a recursive-dimension-reduction approach that leverages univariate rational methods iteratively across dimensions; both approaches can employ the AAA method or the Loewner pencil to recover the poles (frequencies) and then compute the residues (amplitudes) to obtain $\\boldsymbol{\\lambda}_j$ and $\\gamma_j$. The paper provides detailed extensions to the bivariate and higher-dimensional cases, analyzes computational costs, and demonstrates numerical performance with several examples. The work advances multivariate exponential analysis by providing data-efficient, stable reconstruction pipelines with explicit connections to rational-approximation techniques, and suggests avenues for using function values as input and addressing noisy data in future work.

Abstract

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.

Figures (2)

• Figure 1: The sparse grid of indices of the Fourier coefficients used by Algorithm \ref{['alg1']} for the reconstruction of $f_1$ from Example \ref{['ex61']}. The blue dots visualize the coefficients, which are used for separate computation of the frequencies, while the orange dots indicate the coefficients used for determining the correct pairing. The grey dots represent all coefficients from the 2D cube $[-N,N]^2$.
• Figure 2: The sparse grid of indices of the Fourier coefficients used by Algorithm \ref{['alg3']} for the reconstruction of $f_2$ from Example \ref{['ex61']}. The blue dots visualize the coefficients, which are used for separate computation of the frequencies, while the orange dots indicate the coefficients used for determining the correct pairing. The grey dots represent all coefficients from the 3D cube $[-N,N]^3$.

Theorems & Definitions (7)

• Remark 2.1
• Remark 4.1
• Remark 4.2
• Remark 5.1
• Example 6.1
• Example 6.2
• Example 6.3