Approaches to the Inverse Fourier Transformation with Limited and Discrete Data

TL;DR

The paper tackles the inverse problem of reconstructing momentum-space quasi-distributions from limited lattice data via the limited inverse discrete Fourier transform (L-IDFT). It systematically compares four approaches—Tikhonov regularization, Backus-Gilbert, Bayesian inference with a Gaussian Random Walk prior, and feedforward neural networks—and benchmarks them against physics-driven $\lambda$-extrapolation. The results show that L-IDFT is moderately ill-posed but tractable: BG underperforms for global structure, while Tikhonov, Bayesian GRW, and ANN yield stable, accurate reconstructions, with Bayesian and ANN often providing the best balance of fidelity and regularization. The study provides practical guidance on method selection based on data quality and coverage in $\lambda$, and emphasizes using multiple approaches to assess systematic uncertainties in lattice QCD extractions of parton physics.

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 $λ$-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.

Figures (17)

• Figure 1: A multilayer feedforward neural network architecture.
• Figure 2: Upper panel: The true solution is taken to be the asymptotic form of the distribution amplitude. Lower panel: The error-contaminated distribution $g^{\delta}(\lambda)$ in coordinate space.
• Figure 3: L-curve for determining the regularization parameter $\alpha$ in Tikhonov method for Toy model I
• Figure 4: Reconstruction results of Toy model I obtained by four methods.
• Figure 5: The reconstruction obtained by precondition-BG using different precondition function. Blue band: $a=0.7,b=1.6$. Gray band: $a=0.2,b=0.4$. Blue line: mean value of $f_{\text{pre-rebuild}}^{a=0.7,b=1.6}(x)$. Gray line: mean value of $f_{\text{pre-rebuild}}^{a=0.2,b=0.4}(x)$. Dash line: true solution.