Restoring a smooth function from its noisy integrals

TL;DR

This paper introduces the bin hierarchy method, an efficient technique for reconstructing smooth functions from sampled integral data, applicable to various data qualities including Monte Carlo and physical Green's function data.

Contribution

The paper presents a novel bin hierarchy method that maximally utilizes data to restore smooth functions from sampled integrals, improving accuracy and efficiency.

Findings

Effective reconstruction of smooth functions from sampled integrals.

Robust performance across different data qualities, including Monte Carlo data.

Successful application to physical Green's function data.

Abstract

Numerical (and experimental) data analysis often requires the restoration of a smooth function from a set of sampled integrals over finite bins. We present the bin hierarchy method that efficiently computes the maximally smooth function from the sampled integrals using essentially all the information contained in the data. We perform extensive tests with different classes of functions and levels of data quality, including Monte Carlo data suffering from a severe sign problem and physical data for the Green's function of the Fr\"ohlich polaron.