PyLIT: Reformulation and implementation of the analytic continuation problem using kernel representation methods

TL;DR

PyLIT introduces a kernel-based approach with regularization techniques for the ill-posed analytic continuation problem in quantum simulations, providing an open-source tool that improves dynamic structure factor estimation.

Contribution

The paper presents a novel kernel representation method with various regularizers, including Wasserstein distance, implemented in an open-source package for improved analytic continuation.

Findings

Non-uniform grid points reduce unknowns and solution space.

Wasserstein regularizer performs as well as entropic regularizer.

PyLIT's estimates align qualitatively with established methods.

Abstract

Path integral Monte Carlo (PIMC) simulations are a cornerstone for studying quantum many-body systems. The analytic continuation (AC) needed to estimate dynamic quantities from these simulations is an inverse Laplace transform, which is ill-conditioned. If this inversion were surmounted, then dynamical observables (e.g. dynamic structure factor (DSF) $S(q,\omega)$) could be extracted from the imaginary-time correlation functions estimates. Although of important, the AC problem remains challenging due to its ill-posedness. 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. We use least-squares optimization regularized with a Bayesian prior to determine the coefficients of this linear combination. We explore various regularization term, such as the commonly used entropic regularizer, as well as the Wasserstein distance and $L^2$-distance as well as techniques for setting the regularization weight. A key outcome is the open-source package PyLIT (\textbf{Py}thon \textbf{L}aplace \textbf{I}nverse \textbf{T}ransform), which leverages Numba and unifies the presented formulations. PyLIT's core functionality is kernel construction and optimization. In our applications, we find PyLIT's DSF estimates share qualitative features with other more established methods. 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 Wasserstein distance, a previously unexplored regularizer, performs as good as the entropic regularizer while benefiting from its linear gradient. Thirdly, future work can meaningfully combine regularized and stochastic optimization. (text cut for char. limit)