PyLIT: Reformulation and implementation of the analytic continuation problem using kernel representation methods
Alexander Benedix Robles, Phil-Alexander Hofmann, Thomas M. Chuna, Tobias Dornheim, Michael Hecht · Computer Physics Communications · 2025
Path integral Monte Carlo (PIMC) simulations are a cornerstone method for studying quantum many-body systems, such as warm dense matter and ultracold atoms. The analytic continuation needed to estimate dynamic quantities from these simulations amounts to an inverse Laplace transform, which is an ill-conditioned problem. If this challenging problem were surmounted, dynamical observables such as the dynamic structure factor (DSF) S ( q , ω ) —a key property e.g. in x-ray and neutron scattering experiments—could be extracted from the imaginary-time correlation functions F ( q , τ ) estimates. Although of fundamental importance, the analytic continuation problem remains challenging due to its ill-posedness, and state-of-the-art techniques continue to deliver unsatisfactory results. To address this challenge, we express the DSF as a linear combination of kernel functions with known Laplace transforms that have been tailored to satisfy its physical constraints, e.g. , detailed balance. Then we employ least-squares optimization regularized with a Bayesian prior estimate to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the uncommon L 2 -distance and CDF L 2 -distance. We also explore techniques for setting the regularization weight. A key outcome and contribution is the open-source package PyLIT ( Py thon L aplace I nverse T ransform), which leverages Numba for C-level performance, unifying the presented formulations. PyLIT’s core functionality is kernel construction and optimization. In our applications, we find that PyLIT’s DSF estimates share many qualitative features with other more established methods. Drawing from our insights, we identify three key findings. Firstly, independent of the regularization choice, utilizing non-uniform grid point distributions reduced the number of unknowns and thus reduced our space of possible solutions. Secondly, the L 2 -distance and CDF L 2 -distance, previously unexplored regularizers, benefit from their linear gradients and perform about as well as entriopic regularization. Thirdly, future work can meaningfully combine regularized and stochastic optimization.