Stochastic Collocation
Stochastic collocation propagates uncertainty by running the model at a structured set of sample points and interpolating the response.
Sample and interpolate
Stochastic collocation evaluates a deterministic simulation at a chosen set of points in the uncertain-input space, then builds an interpolant (usually a polynomial) that reproduces the response at those points. Statistics such as mean, variance, and quantiles are computed cheaply from the interpolant. Because each run is independent, the method is non-intrusive: it treats the simulation as a black box.
Choosing collocation points
Points are the nodes of a quadrature rule matched to the input distribution: Gauss-Hermite for Gaussian inputs, Gauss-Legendre for uniform, and so on. Tensor products of one-dimensional nodes work in low dimension but explode combinatorially, motivating sparse grids (see Sparse-Grid Methods) for moderate dimension.
Relation to polynomial chaos
Stochastic collocation and polynomial chaos expansion are closely linked: collocation can compute polynomial-chaos coefficients by quadrature. Collocation is often preferred when only black-box evaluations are available, since it requires no modification of the simulation code.
Accuracy and smoothness
- Converges rapidly (spectrally) for smooth responses
- Degrades for discontinuous or sharply varying responses
- Gibbs-like oscillations appear near discontinuities in output
Cautions
The method assumes the response is smooth in the uncertain inputs; bifurcations, thresholds, or regime changes break the polynomial interpolant and can produce misleading statistics. Detecting non-smoothness (for example by checking interpolation residuals at extra points) is important before trusting collocation results. For high dimension, combine with dimension reduction or adaptive sparse grids.