Uncertainty Quantification — Monte Carlo, PCE & Sobol Indices

What is the difference between aleatory and epistemic uncertainty?

Aleatory uncertainty is inherent randomness — material scatter between samples, load variability, manufacturing tolerances. It cannot be eliminated no matter how much data you collect. Epistemic uncertainty is lack of knowledge — unknown material constants, unvalidated model assumptions, insufficient data. Epistemic uncertainty can in principle be reduced by more testing or better models.

How does Polynomial Chaos Expansion work and when is it better than Monte Carlo?

PCE represents the output as a polynomial in the uncertain inputs (Hermite polynomials for Gaussian inputs). You run the simulation at a small set of collocation points, fit polynomial coefficients, then evaluate the fitted PCE thousands of times at negligible cost. PCE is most efficient when output depends smoothly on inputs and the number of uncertain parameters is moderate (< ~20). Monte Carlo is simpler and scales better with very high dimensions.