Approaches to the Inverse Fourier Transformation with Limited and Discrete Data

TL;DR

This paper compares multiple methods, including regularization, Bayesian, neural networks, and physics-driven approaches, to solve the inverse Fourier transform problem with limited and discrete data, demonstrating their relative effectiveness and stability.

Contribution

It systematically evaluates and compares various computational approaches for the limited inverse Fourier transform, highlighting the advantages of Bayesian and neural network methods.

Findings

Most methods successfully reconstruct quasi-distributions, except Backus-Gilbert.

Bayesian and neural network approaches provide more stable and accurate results.

Careful method selection and uncertainty assessment are crucial for reliable reconstructions.

Abstract

We investigate several approaches to address the inverse problem that arises in the limited inverse Fourier transform (L-IDFT) of quasi-distributions. The methods explored include Tikhonov regularization, the Backus-Gilbert method, the Bayesian approach with Gaussian Random Walk (GRW) prior, and the feedforward artificial neural networks (ANNs). We evaluate the performance of these methods using both mock data generated from toy models and real lattice data from quasi distribution, and further compare them with the physics-driven $\lambda$-extrapolation approach. Our results demonstrate that the L-IDFT constitutes a moderately tractable inverse problem Except for the Backus-Gilbert method, all the other approaches are capable of correctly reconstructing the quasi-distributions in momentum space. In particular, the Bayesian approach with GRW and the feedforward ANNs yield more stable and accurate reconstructions. Based on these investigations, we conclude that, for a given L-IDFT problem, selecting an appropriate reconstruction method according to the input data and carefully assessing the potential systematic uncertainties are essential for obtaining reliable results.