Core principle: random sampling
Monte Carlo methods represent integrals and expectations as averages of random samples drawn from a known distribution. Accuracy improves as 1/βN, independent of problem dimensionality.
- In d dimensions, grid-based methods need N^d points; Monte Carlo always needs N β huge advantage for d > 4.
- Random number quality matters: use cryptographically secure or high-period PRNGs (Mersenne Twister).
- Each independent sample contributes equally β embarrassingly parallelizable on modern hardware.
Variance reduction techniques
Crude Monte Carlo converges at 1/βN. Variance reduction methods achieve the same accuracy with far fewer samples.
- Stratified sampling: divide domain into strata, sample each β reduces variance significantly.
- Importance sampling: weight samples toward high-contribution regions.
- Antithetic variates: use paired samples (u, 1βu) to exploit negative correlation.
Engineering reliability applications
Monte Carlo is the engineering standard for computing probability of failure when analytical methods are insufficient β for example, in complex limit state functions.
- First-order reliability method (FORM) is an analytical approximation; MC validates it.
- Failure probability Pf = fraction of simulations where limit state function g(X) < 0.
- Importance-sampled MC estimates rare failure probabilities (Pf < 10β»βΆ) with feasible sample counts.