Speaker
Description
We systematically investigated the inverse discrete Fourier transform of quasi-distributions from the perspective of inverse problem theory. Mathematically, we have demonstrated that this transformation satisfies two of Hadamard’s well-posedness criteria, existence and uniqueness of the solution, but critically violates the stability requirement. To address this instability, we employed and compared four classical and widely-used inversion methods: Tikhonov regularization, the Backus–Gilbert method, Bayesian inference, and neural networks. The efficacy of these approaches is validated through controlled toy model studies and real lattice QCD results for quasi-distribution amplitudes. The reconstructed solutions are consistent with those from the physics-driven \lambda-extrapolation method. Our analysis shows that the inverse Fourier problem within the large-momentum effective theory (LaMET) framework constitutes a moderately tractable ill-posed problem. Except for the Backus–Gilbert method which has been shown mathematically to be flawed, all other approaches successfully reconstruct the quasi-distributions in momentum space. Depending on the specific behavior of the quasi-distribution data, it is essential to adopt tailored strategies for data processing and to systematically estimate the associated systematic uncertainties.
