Efficient Irreversible Monte Carlo Samplers

Area of Science:

• Computational Physics
• Statistical Mechanics
• Monte Carlo Methods

Background:

• Markov chain Monte Carlo (MCMC) methods are crucial for simulating complex systems.
• Traditional MCMC algorithms often face challenges with slow convergence in discrete state spaces.
• The detailed balance condition, while ensuring convergence, can limit algorithmic efficiency.

Purpose of the Study:

• To introduce two novel irreversible Markov chain Monte Carlo algorithms for discrete state systems.
• To improve computational efficiency by relaxing the detailed balance condition using the lifting framework.
• To demonstrate the applicability and performance enhancement of these algorithms on classical spin systems.

Main Methods:

• Development of two irreversible MCMC algorithms based on the random-scan Gibbs sampler and Metropolized-Gibbs sampler.
• Incorporation of the lifting framework with a skewed detailed balance condition to construct irreversible Markov chains.
• Application and testing of the algorithms on the 1D 4-state Potts model and generalization to classical spin systems.

Main Results:

• The proposed algorithms satisfy the balance condition while being irreversible, leading to improved dynamics.
• Application to the 1D 4-state Potts model shows a reduction in the dynamical scaling exponent (z) from ≈1 to ≈1/2 for magnetization and energy density.
• Generalization of an irreversible Metropolis-Hastings algorithm demonstrates a square root reduction in mixing time at high temperatures for classical spin systems.

Conclusions:

• The developed irreversible MCMC algorithms offer a significant speed-up in simulations of discrete systems.
• The lifting framework with skewed detailed balance is an effective strategy for enhancing MCMC efficiency.
• These findings pave the way for more efficient simulations in statistical mechanics and related computational fields.