Efficiency of Monte Carlo Sampling in Chaotic Systems

TL;DR

This study analyzes how the complexity of chaotic phase spaces impacts the efficiency of importance sampling Monte Carlo simulations, revealing polynomial scaling and critical slowing down due to local proposal limitations.

Contribution

It provides an analytical demonstration of the polynomial and sub-optimal scaling of Monte Carlo methods in chaotic systems, highlighting the role of local proposals.

Findings

Computational effort scales polynomially with finite-time.

Scaling is sub-optimal due to critical slowing down.

Results are valid across various Monte Carlo methods.

Abstract

In this paper we investigate how the complexity of chaotic phase spaces affect the efficiency of importance sampling Monte Carlo simulations. We focus on a flat-histogram simulation of the distribution of finite-time Lyapunov exponent in a simple chaotic system and obtain analytically that the computational effort of the simulation: (i) scales polynomially with the finite-time, a tremendous improvement over the exponential scaling obtained in usual uniform sampling simulations; and (ii) the polynomial scaling is sub-optimal, a phenomenon known as critical slowing down. We show that critical slowing down appears because of the limited possibilities to issue a local proposal on the Monte Carlo procedure in chaotic systems. These results remain valid in other methods and show how generic properties of chaotic systems limit the efficiency of Monte Carlo simulations.