polynomial chaos
Suppose your simulation's output depends on a couple of uncertain inputs, and you want to know the output's whole distribution — its mean, its spread, the chance it exceeds a danger threshold. Monte Carlo would just sample thousands of times and tally the results, paying the slow 1/sqrt(N) price. Polynomial chaos takes a smarter route when the dependence is smooth: it builds a compact FORMULA — a polynomial in the random inputs — that mimics how the output responds, then reads the statistics off that formula almost for free. The name is historical and a little misleading: there is nothing chaotic about it; 'chaos' here is an old word (from Norbert Wiener's 1938 work) for a random expansion.
Here is the idea in plain steps. Write the uncertain inputs as random variables (say a single one, xi, with a known distribution). Polynomial chaos approximates the output Y as a short series Y is approximately sum_{k=0}^{P} c_k Phi_k(xi), where the Phi_k are special ORTHOGONAL polynomials matched to the input's distribution — Hermite polynomials for a Gaussian input, Legendre for a uniform one, and so on (this matching is the 'generalized' polynomial chaos idea). The clever part is that these polynomials are orthogonal with respect to the input's probability weight, so once you find the coefficients c_k, the output's mean is simply c_0 and its variance is a weighted sum of the other c_k squared — the statistics fall out of the coefficients by hand, no further sampling needed. You find the coefficients either by projection (computing integrals, often with Gaussian quadrature) or by regression (fitting the series to a modest number of simulation runs).
When it works, polynomial chaos can be dramatically cheaper than Monte Carlo: a handful of well-chosen simulation runs can pin down a distribution that would take Monte Carlo thousands of samples to resolve, because for smooth responses the series converges fast (spectrally). It is essentially a spectral / surrogate method transplanted from approximating functions of space to approximating functions of random inputs. The honest limits are sharp. First, it needs the output to depend SMOOTHLY on the inputs — a discontinuity (a sudden phase change, a threshold) wrecks the polynomial fit, just as Runge's phenomenon wrecks high-degree interpolation. Second, it suffers the curse of dimensionality: the number of terms grows fast with the number of uncertain inputs, so it shines for a few uncertain parameters and loses to Monte Carlo for very many. Use it for smooth problems in low-to-moderate dimension; reach for Monte Carlo when the response is rough or the uncertain inputs are legion.
An output Y depends on a uniform random input xi on [-1,1]. Polynomial chaos writes Y is approximately c_0 P_0 + c_1 P_1 + c_2 P_2 using Legendre polynomials P_k. Running the simulation at a few Gauss-Legendre nodes pins the c_k; then mean(Y) = c_0 exactly, and var(Y) = sum_{k>=1} c_k^2 / (2k+1). A handful of runs yields the whole distribution that Monte Carlo would need thousands of samples to match.
A short orthogonal-polynomial series in the random inputs yields the output's mean and variance from its coefficients.
Polynomial chaos needs the output to depend smoothly on the inputs — a discontinuity wrecks the fit just as Runge's phenomenon wrecks high-degree interpolation. And it suffers the curse of dimensionality, so it loses to Monte Carlo when the uncertain inputs are very many.