Applying recursive numerical integration techniques for solving high dimensional integrals

TL;DR

This paper explores recursive numerical integration (RNI) as an alternative to MCMC for high-dimensional integrals, demonstrating its superior error scaling in quantum physics models like the topological rotor and anharmonic oscillator.

Contribution

It introduces and applies RNI to high-dimensional integrals, showing exponential error scaling and potential advantages over traditional MCMC methods.

Findings

RNI exhibits at least exponential error scaling with the number of integration points.

Application to quantum models demonstrates RNI's effectiveness.

RNI outperforms MCMC in error reduction efficiency.

Abstract

The error scaling for Markov-Chain Monte Carlo techniques (MCMC) with $N$ samples behaves like $1/\sqrt{N}$. This scaling makes it often very time intensive to reduce the error of computed observables, in particular for applications in lattice QCD. It is therefore highly desirable to have alternative methods at hand which show an improved error scaling. One candidate for such an alternative integration technique is the method of recursive numerical integration (RNI). The basic idea of this method is to use an efficient low-dimensional quadrature rule (usually of Gaussian type) and apply it iteratively to integrate over high-dimensional observables and Boltzmann weights. We present the application of such an algorithm to the topological rotor and the anharmonic oscillator and compare the error scaling to MCMC results. In particular, we demonstrate that the RNI technique shows an error scaling in the number of integration points $m$ that is at least exponential.