When is Monte Carlo integration preferable to ordinary quadrature?

For low-dimensional smooth integrals, deterministic quadrature is more accurate. Monte Carlo wins once the dimension is high, typically beyond four or five, because its error does not depend on dimension whereas grid methods need an exponentially growing number of points.

How is Monte Carlo integration different from Metropolis sampling?

Monte Carlo integration draws independent points to estimate a fixed integral, often using importance sampling from a known distribution. Metropolis sampling generates a correlated Markov chain to sample a complicated distribution, such as a Boltzmann ensemble, that cannot be drawn from directly.

Monte Carlo Integration in Physics

When an integral runs over many dimensions, grid-based quadrature becomes hopeless, and Monte Carlo integration wins by estimating the integral as an average over random points with an error that ignores the dimension.

Definition

Monte Carlo integration estimates a definite integral as the average of the integrand evaluated at randomly chosen points in the domain, times the domain volume, with statistical error decreasing as the inverse square root of the number of points.

Scope

This topic covers Monte Carlo evaluation of high-dimensional physical integrals: plain sampling, variance-reduction by importance and stratified sampling, and adaptive schemes such as VEGAS, with applications to partition functions, scattering cross sections and phase-space integrals. It treats integration specifically, distinct from configuration sampling.

Core questions

Why does Monte Carlo integration beat grid quadrature in high dimensions?
How does importance sampling reduce the variance of an integral estimate?
How does stratified sampling distribute points to lower error?
How do adaptive algorithms like VEGAS learn the shape of a peaked integrand?

Key theories

Dimension-independent error: The statistical error of a Monte Carlo integral scales as the inverse square root of the sample count regardless of dimension, whereas grid quadrature error grows exponentially worse as dimension increases.
Variance reduction: Importance sampling concentrates points where the integrand is large by drawing from a tailored distribution, and stratified sampling partitions the domain, both reducing the variance of the estimate for a fixed number of evaluations.
Adaptive integration: The VEGAS algorithm iteratively refines a separable importance-sampling grid to match the integrand, making it effective for the sharply peaked, high-dimensional integrals that arise in particle physics.

Clinical relevance

Monte Carlo integration evaluates phase-space integrals and scattering cross sections in particle physics, partition-function and free-energy integrals in statistical mechanics, and any multidimensional integral where deterministic quadrature is infeasible.

History

Monte Carlo integration grew out of the same 1940s Los Alamos work that founded Monte Carlo methods; adaptive importance-sampling schemes such as VEGAS, introduced by Lepage in 1978, made high-dimensional integrals in particle physics routinely computable.

Key figures

G. Peter Lepage
Stanislaw Ulam
John von Neumann

Seminal works

lepage1978
press2007

Frequently asked questions

When is Monte Carlo integration preferable to ordinary quadrature?: For low-dimensional smooth integrals, deterministic quadrature is more accurate. Monte Carlo wins once the dimension is high, typically beyond four or five, because its error does not depend on dimension whereas grid methods need an exponentially growing number of points.
How is Monte Carlo integration different from Metropolis sampling?: Monte Carlo integration draws independent points to estimate a fixed integral, often using importance sampling from a known distribution. Metropolis sampling generates a correlated Markov chain to sample a complicated distribution, such as a Boltzmann ensemble, that cannot be drawn from directly.