Analysis of the convergence of the 1/t and Wang-Landau algorithms in the calculation of multidimensional integrals

TL;DR

This paper compares the convergence and efficiency of the 1/t and Wang-Landau algorithms for multidimensional integrals, showing that 1/t converges similarly to Monte Carlo, while Wang-Landau does not converge.

Contribution

The study provides a detailed analysis of the convergence behaviors of 1/t and Wang-Landau algorithms in multidimensional integral calculations, highlighting their differences.

Findings

1/t algorithm errors decrease as N^{-1/2}, similar to Monte Carlo.

Wang-Landau errors saturate over time, indicating non-convergence.

The sources of errors in both methods are identified.

Abstract

In this communication, the convergence of the 1/t and Wang - Landau algorithms in the calculation of multidimensional numerical integrals is analyzed. Both simulation methods are applied to a wide variety of integrals without restrictions in one, two and higher dimensions. The errors between the exact and the calculated values of the integral are obtained and the efficiency and accuracy of the methods are determined by their dynamical behavior. The comparison between both methods and the simple sampling Monte Carlo method is also reported. It is observed that the time dependence of the errors calculated with 1/t algorithm goes as N^{-1/2} (with N the MC trials) in quantitative agreement with the simple sampling Monte Carlo method. It is also showed that the error for the Wang - Landau algorithm saturates in time evidencing the non-convergence of the methods. The sources for the error are also determined.