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

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.

Figures (8)

• Figure 1: Stability analysis for choosing the reparameterization scale $t_0$ in the $F(s)=1/s^2$ analytic test case. The blue dashed curve (left axis) shows relative differences between consecutive $t_0$ reconstructions (Eq. \ref{['eq:consecutive_solutions_reparameterization']}), and the black curve (right axis) shows the true inversion residual. Both have a clear minimum near $t_0=0.2$. The shaded band marks the stable region, and the black marker shows the residual minimum.
• Figure 2: Numerical inversion and reconstruction test for $F(s)=1/s^{2}$ with analytic inverse $f(t)=t$, performed in the absence of noise.
• Figure 3: Numerical inversion and reconstruction tests for several analytic Laplace-space functions with known inverse transforms, performed in the absence of noise. For each column, the upper panel shows the reconstructed inverse function, while the lower panel shows the corresponding reconstruction of the Laplace-space function. The symbols and curves follow the same conventions as in Fig. \ref{['fig:laplace_test']}.
• Figure 4: Numerical inversion and reconstruction for $F(s)=1/s^{2}$ with analytic inverse $f(t)=t$ and injected Gaussian noise, without additional denoising. The left column corresponds to relative noise level $\delta=10^{-6}$, while the right column shows results for $\delta=10^{-2}$ (Eq. (\ref{['eq:noise_injection']})).
• Figure 5: Illustration of weighted local polynomial smoothing. The black dotted curve shows the noisy signal, the red dashed curve the current local polynomial fit within the sliding window (shaded region), and the purple curve the resulting smoothed signal. The square marker denotes the currently smoothed point.