Monte Carlo Integration
Estimating integrals by random sampling, with error that shrinks independently of dimension.
Integration by averaging random samples
Monte Carlo integration estimates an integral by averaging the integrand's value at randomly chosen points and multiplying by the domain size. The law of large numbers guarantees the estimate converges to the true value as more samples are drawn. Its defining property is the convergence rate: the error shrinks in proportion to one over the square root of the number of samples, regardless of the dimension of the integral.
Beating the curse of dimensionality
Deterministic quadrature rules suffer the curse of dimensionality: to maintain accuracy, the number of points grows exponentially with dimension. Monte Carlo does not; its square-root convergence is the same in one dimension or a thousand. This makes it the only practical choice for very high-dimensional integrals, such as those over particle phase space or over many uncertain parameters.
Variance reduction
The catch is that square-root convergence is slow, and halving the error requires quadrupling the samples. Variance reduction techniques improve the constant in front of the rate without changing the rate itself. Importance sampling draws more points where the integrand is large; stratified sampling divides the domain into regions sampled separately; control variates subtract a correlated quantity with known integral. These can cut the required sample count by orders of magnitude.
- Importance sampling: sample from a distribution shaped like the integrand
- Stratified sampling: partition the domain and sample each part
- Quasi-Monte Carlo: use low-discrepancy sequences for faster convergence on smooth problems
- Control variates: exploit a correlated function with a known integral
Use in fusion computation
Monte Carlo methods are central to neutron and photon transport, where particles are tracked through complex geometry and scattering, and to plasma kinetic simulation. Neutron transport for blanket and shielding design, including tritium-breeding calculations, is dominated by Monte Carlo particle tracking because the geometry and energy dependence are too complex for deterministic methods. The Hyperion breeder's tritium-breeding-ratio design target of 1.8 is the kind of quantity evaluated by such transport calculations.