Monte Carlo Sampling in Fractal Landscapes

TL;DR

This paper introduces a novel flat-histogram Monte Carlo method for efficiently sampling fractal landscapes, such as escape time functions, by adapting the Wang-Landau algorithm to chaotic systems, enabling polynomial-time computation.

Contribution

It generalizes the Wang-Landau algorithm to fractal landscapes, allowing efficient sampling and accurate estimation of escape time distributions in high-dimensional chaotic systems.

Findings

Method achieves polynomial computational time.

Effective in high-dimensional phase spaces up to 30 dimensions.

Provides accurate density of states and escape time distributions.

Abstract

We propose a flat-histogram Monte Carlo method to efficiently sample fractal landscapes such as escape time functions of open chaotic systems. This is achieved by using a random-walk step which depends on the height of the landscape via the largest Lyapunov exponent of the associated chaotic system. By generalizing the Wang-Landau algorithm, we obtain a method which simultaneously constructs the density of states (escape time distribution) and the correct step-length distribution. As a result, averages are obtained in polynomial computational time, a dramatic improvement over the exponential scaling of traditional uniform sampling. Our results are not limited by the dimensionality of the phase space and are confirmed numerically for dimensions as large as 30.