We propose a minimal generalization of the celebrated Markov-Chain Monte Carlo algorithm which allows for an arbitrary number of configurations to be visited at every Monte Carlo step. This is advantageous when a parallel computing machine is available, or when many biased configurations can be evaluated at little additional computational cost. As an example of the former case, we report a significant reduction of the thermalization time for the paradigmatic Sherrington-Kirkpatrick spin-glass model. For the latter case, we show that, by leveraging on the exponential number of biased configurations automatically computed by Diagrammatic Monte Carlo, we can speed up computations in the Fermi-Hubbard model by two orders of magnitude.
Exchange Monte Carlo for continuous-space Path Integral Monte Carlo simulation
The University of Tokyo（东京大学）；Institute for Physics of Intelligence, The University of Tokyo（东京大学物理智能研究所）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）
We propose a new Markov chain Monte Carlo method in which trial configurations are generated by evolving a state, sampled from a prior distribution, using a Markov transition matrix. We present two prototypical algorithms and derive their corresponding acceptance rules. We first identify the important factors controlling the quality of the sampling. We then apply the method to the problem of sampling polymer configurations with fixed endpoints. Applications of the proposed method range from the design of new generative models to the improvement of the portability of specific Monte Carlo algori…
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this compara…
Markov chain Monte Carlo algorithms are invaluable tools for exploring stationary properties of physical systems, especially in situations where direct sampling is unfeasible. Common implementations of Monte Carlo algorithms employ reversible Markov chains. Reversible chains obey detailed balance and thus ensure that the system will eventually relax to equilibrium. Detailed balance is not necessary for convergence to equilibrium. We review nonreversible Markov chains, which violate detailed balance, and yet still relax to a given target stationary distribution. In particular cases, nonreversib…
We discuss a backward Monte-Carlo technique for muon transport problem, with emphasis on its application in muography. Backward Monte-Carlo allows exclusive sampling of a final state by reversing the simulation flow. In practice it can be made analogous to an adjoint Monte-Carlo, though it is more versatile for muon transport. A backward Monte-Carlo was implemented as a dedicated muon transport library: PUMAS. It is shown for case studies relevant for muography imaging that the implementations of forward and backward Monte-Carlo schemes agree to better than 1%.
Monte Carlo simulations are widely used in many areas including particle accelerators. In this lecture, after a short introduction and reviewing of some statistical backgrounds, we will discuss methods such as direct inversion, rejection method, and Markov chain Monte Carlo to sample a probability distribution function, and methods for variance reduction to evaluate numerical integrals using the Monte Carlo simulation. We will also briefly introduce the quasi-Monte Carlo sampling at the end of this lecture.
We present a simple approach to the fixed phase method in Quantum Monte Carlo. This applies to electrons in molecules and electron gas and is straightforwardly extended to the Schrödinger equation with magnetic field.
We introduce \textit{Policy Guided Monte Carlo} (PGMC), a computational framework using reinforcement learning to improve Markov chain Monte Carlo (MCMC) sampling. The methodology is generally applicable, unbiased and opens up a new path to automated discovery of efficient MCMC samplers. After developing a general theory, we demonstrate some of PGMC's prospects on an Ising model on the kagome lattice, including when the model is in its computationally challenging kagome spin ice regime. Here, we show that PGMC is able to automatically machine learn efficient MCMC updates without a priori knowl…
We introduce and discuss Monte Carlo methods in quantum field theories. Methods of independent Monte Carlo, such as random sampling and importance sampling, and methods of dependent Monte Carlo, such as Metropolis sampling and Hamiltonian Monte Carlo, are introduced. We review the underlying theoretical foundations of Markov chain Monte Carlo. We provide several examples of Monte Carlo simulations, including one-dimensional simple harmonic oscillator, unitary matrix model exhibiting Gross-Witten-Wadia transition and a supersymmetric model exhibiting dynamical supersymmetry breaking.
A preliminary version (v0.2.1) of the DANTON Monte-Carlo package is presented. DANTON allows the exclusive sampling of (decaying) $τ$ generated by $ν_τ$ interactions with the Earth. The particles interactions with matter are simulated in detail, including transverse scattering. Detailed topography data of the Earth can be used as well. Yet, high Monte-Carlo efficiency is achieved by using a Backward Monte-Carlo technique. Some validation results are provided.
