Monte Carlo Method
A family of methods that estimate quantities by averaging over many random samples.
Definition
The Monte Carlo method estimates a quantity by drawing many random samples and averaging the results. It is especially valuable for high-dimensional integrals and for simulating stochastic processes where deterministic methods are infeasible.
Quasi-Monte Carlo methods replace pure randomness with carefully spread low-discrepancy sequences, improving convergence for suitable problems. The choice between true random and quasi-random sampling depends on the smoothness and dimension of the integrand.
Its defining advantage is that convergence does not depend on dimension, which is why it dominates high-dimensional integration where grid methods fail. The price is slow error reduction, proportional to the inverse square root of the sample count, so halving the error requires quadrupling the samples. Variance-reduction techniques, importance sampling, control variates, and quasi-random sequences, are the practical tools for extracting more accuracy from a fixed sampling budget.
Properties
- Error shrinks as one over the square root of the sample count.
- Convergence is independent of dimension, unlike grid methods.
- Variance-reduction techniques improve efficiency.
- Results come with a statistical uncertainty.
Why it matters
Monte Carlo methods make otherwise intractable high-dimensional problems solvable, from finance to physics to Bayesian inference. The slow square-root convergence is the price for dimension-independence, addressed with variance reduction and more samples.
Fusion connection
Monte Carlo particle transport is a standard tool for modeling neutron and particle behavior in fusion systems, tracking many simulated particles to estimate quantities such as heating and material loading.