Introduction to Monte Carlo Methods

Abstract

Monte Carlo methods play an important role in scientific computation, especially when problems have a vast phase space. In this lecture an introduction to the Monte Carlo method is given. Concepts such as Markov chains, detailed balance, critical slowing down, and ergodicity, as well as the Metropolis algorithm are explained. The Monte Carlo method is illustrated by numerically studying the critical behavior of the two-dimensional Ising ferromagnet using finite-size scaling methods. In addition, advanced Monte Carlo methods are described (e.g., the Wolff cluster algorithm and parallel tempering Monte Carlo) and illustrated with nontrivial models from the physics of glassy systems. Finally, we outline an approach to study rare events using a Monte Carlo sampling with a guiding function.

lecture at the third international summer school "Modern Computation Science", 15 - 26 August 2011, Oldenburg (Germany), see http://www.mcs.uni-oldenburg.de

References in corpus (11)

• Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram
• Universality in three-dimensional Ising spin glasses: A Monte Carlo study
• Transition Matrix Monte Carlo Method
• Optimizing the ensemble for equilibration in broad-histogram Monte Carlo simulations
• A Cluster Monte Carlo Algorithm for 2-Dimensional Spin Glasses
• Lower Critical Dimension of Ising Spin Glasses
• Absence of an Almeida-Thouless line in Three-Dimensional Spin Glasses
• Make life simple: unleash the full power of the parallel tempering algorithm
• Ultrametricity and clustering of states in spin glasses: A one-dimensional view
• Robust Parameter Selection for Parallel Tempering
• Rejuvenation and Memory in model Spin Glasses in 3 and 4 dimensions

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• Correlations between the dynamics of parallel tempering and the free-energy landscape in spin glasses
• Reconstructing a non-linear interaction in the dark sector with cosmological observations
• borealis - A generalized global update algorithm for Boolean optimization problems
• Collective Monte Carlo updates through tensor network renormalization