Multilevel coarse graining and nano--pattern discovery in many particle stochastic systems

TL;DR

This paper introduces a hierarchical Monte Carlo approach for efficiently sampling equilibrium states in complex stochastic lattice systems, enabling multi-resolution analysis and reducing computational costs.

Contribution

It presents a novel multilevel Monte Carlo framework that couples coarse and microscopic states, improving efficiency over traditional methods in systems with competing interactions.

Findings

Significant reduction in computational cost.

Effective sampling of phase transitions and pattern formations.

Flexible control over approximation errors.

Abstract

In this work we propose a hierarchy of Monte Carlo methods for sampling equilibrium properties of stochastic lattice systems with competing short and long range interactions. Each Monte Carlo step is composed by two or more sub - steps efficiently coupling coarse and microscopic state spaces. The method can be designed to sample the exact or controlled-error approximations of the target distribution, providing information on levels of different resolutions, as well as at the microscopic level. In both strategies the method achieves significant reduction of the computational cost compared to conventional Markov Chain Monte Carlo methods. Applications in phase transition and pattern formation problems confirm the efficiency of the proposed methods.