Polynomial Chaos Expansion
Represent the output as a polynomial series in the uncertain inputs, capturing the distribution with far fewer runs than sampling.
A Structured Alternative to Sampling
Polynomial chaos expansion (PCE) represents a code's output as a series of orthogonal polynomials in the uncertain inputs. Once the coefficients are found, the series is a cheap surrogate that reproduces the output's mean, variance, and full distribution without further code runs. For smooth responses and modest numbers of inputs, PCE reaches a given accuracy with far fewer runs than Monte Carlo.
How It Works
- Choose polynomials orthogonal with respect to the input distributions; Hermite for Gaussian inputs, Legendre for uniform, and so on.
- Determine the expansion coefficients, either by projection using quadrature runs or by regression from a set of sample runs.
- Read statistics directly from the coefficients: the mean is the first coefficient, the variance is the sum of squares of the rest.
Sensitivity for Free
A useful bonus is that Sobol sensitivity indices come almost directly from the expansion coefficients. Grouping the coefficients by which inputs they involve gives the variance attributable to each input and each interaction, so a single PCE yields both the output distribution and its sensitivity decomposition without a separate study.
Where It Breaks Down
PCE assumes the response is smooth in the inputs. Discontinuities, sharp thresholds, and bifurcations make the polynomial series converge slowly or oscillate, degrading accuracy. The number of terms also grows quickly with the number of uncertain inputs, the curse of dimensionality, so PCE is best for problems with a handful of important inputs, often identified first by sensitivity screening. For rough or high-dimensional responses, Monte Carlo or adaptive surrogates are more robust.
Used within its range of validity, PCE turns an expensive propagation into an inexpensive one, and its accuracy should be confirmed against a modest set of independent Monte Carlo runs before the surrogate is trusted for decisions.