Monte Carlo for Uncertainty Quantification
Monte Carlo estimates output statistics by random sampling of inputs, a robust dimension-independent workhorse of uncertainty propagation.
The method
Monte Carlo uncertainty propagation is disarmingly simple: draw many samples of the uncertain inputs from their distributions, run the model on each, and estimate output statistics - mean, variance, quantiles, exceedance probabilities - from the resulting ensemble. Its simplicity, generality, and robustness make it the default against which all other UQ methods are compared.
Convergence
The Monte Carlo error of a mean estimate shrinks as the standard deviation divided by the square root of the sample count N. This rate is independent of the number of input dimensions - a decisive advantage over grid methods that suffer the curse of dimensionality. The cost is slow convergence: cutting the error in half requires four times as many samples.
What you get
- Unbiased estimates of means and probabilities
- Confidence intervals from the sample standard error
- Full empirical output distributions, not just moments
- No smoothness assumption - works for discontinuous or noisy responses
Variance reduction
Because plain Monte Carlo is sample-hungry, variance-reduction techniques squeeze more accuracy from each run: stratified and Latin hypercube sampling spread points more evenly, importance sampling focuses on important regions, control variates exploit a correlated cheap model, and antithetic variates use negatively correlated pairs. Each reduces estimator variance without bias.
Multilevel and multifidelity
Multilevel Monte Carlo runs most samples on cheap low-fidelity models and few on expensive high-fidelity ones, combining them to get high-fidelity accuracy at greatly reduced cost. Multifidelity Monte Carlo generalizes this to arbitrary correlated models, including surrogates. These methods make Monte Carlo affordable for expensive simulators.
In fusion UQ
Monte Carlo is the reliable backbone of Kronos uncertainty studies. For expensive models of the machines it is run on surrogates or in multifidelity form, and it serves as the trusted reference to check that faster spectral methods have not introduced bias into the reported uncertainty ranges.