# Stochastic Approximation Monte Carlo with a Dynamic Update Factor

## TL;DR

This paper introduces SAD, a new Monte Carlo method that adaptively adjusts its update factor during simulations, leading to faster and more robust convergence in calculating the density of states without extensive parameter tuning.

## Contribution

The paper proposes SAD, a stochastic approximation Monte Carlo algorithm with a dynamic update factor, improving convergence and ease of use over existing methods like SAMC and $1/t$-WL.

## Key findings

- SAD converges rapidly to the correct density of states.
- SAD does not require user-specified parameters like $t_0$ or energy ranges.
- SAD outperforms $1/t$-WL when the energy range is unknown.

## Abstract

We present a new Monte Carlo algorithm based on the Stochastic Approximation Monte Carlo (SAMC) algorithm for directly calculating the density of states. The proposed method is Stochastic Approximation with a Dynamic update factor (SAD) which dynamically adjusts the update factor $\gamma_t$ during the course of the simulation. We test this method on the square-well fluid and the 31-atom Lennard-Jones cluster and compare the convergence behavior of several related Monte Carlo methods. We find that both the SAD and $1/t$-Wang-Landau ($1/t$-WL) methods rapidly converge to the correct density of states without the need for the user to specify an arbitrary tunable parameter $t_0$ as in the case of SAMC. SAD requires as input the temperature range of interest, in contrast to $1/t$-WL, which requires that the user identify the interesting range of energies. The convergence of the $1/t$-WL method is very sensitive to the energy range chosen for the low-temperature heat capacity of the Lennard-Jones cluster. Thus, SAD is more powerful in the common case in which the range of energies is not known in advance.