A novel framework for spectral density reconstruction via quadrature-based Laplace inversion

TL;DR

This paper introduces a stable, quadrature-based numerical method for inverse Laplace transformation that effectively reconstructs spectral densities from noisy data, with potential applications in lattice QCD.

Contribution

The work presents a new regularized quadrature-based framework for spectral density reconstruction via Laplace inversion, emphasizing stability and robustness under noisy conditions.

Findings

Demonstrates stability and effectiveness on toy models

Accurately reproduces spectral densities from mock data

Maintains reliability in noisy environments

Abstract

In this work, we explore a numerical approach for performing the inverse Laplace transformation, with an emphasis on achieving stability and robustness under noisy conditions. Our quadrature-based method integrates reparameterization, data smoothing, and optimization techniques to regularizing ill-conditioned systems. Together, these elements enable consistency checks that enhance the reliability of the inversion process. Through a series of controlled tests on toy models, we demonstrate the stability and effectiveness of the method in the presence of noise. Using mock data, we approximate spectral densities from Euclidean correlators, generating smoothed and stable results that accurately reproduce the correlator behavior, particularly at large Euclidean times. We conclude by discussing prospects for applications to actual lattice QCD data.