Spectral Expansion
Uncertainty propagation techniques map the variation of model inputs to the statistics of the corresponding outputs. Through this mathematical framework, generalized polynomial chaos represents stochastic processes by using orthogonal polynomials of random variables. This expansion provides a fast surrogate model that replaces slow differential equations during statistical simulations.
Numerical Approximation
Evaluating the coefficients of the expansion requires either intrusive or non-intrusive mathematical projection methods. With generalized polynomial chaos, these coefficients are computed at specific quadrature points to reconstruct the output distribution. This calculation avoids the heavy computational burden of traditional sampling routines.
Stochastic Projection
The choice of the polynomial basis corresponds directly to the probability distribution of the input uncertainties. Classical Hermite polynomials model normal distributions, whereas Legendre polynomials are selected for uniform distributions. Analysts must match these bases carefully to achieve exponential convergence rates in their calculations.
If the input distribution has an unusual shape, custom orthogonal polynomials must be constructed to maintain the efficiency of generalized polynomial chaos.
Sensitivity Extraction
Variance-based sensitivity indicators are obtained directly from the algebraic combination of the calculated expansion coefficients. This direct extraction allows engineers to identify which input parameters dominate the system output variance without running additional simulations. The resulting sensitivity profile guides the optimization of sensor placement and calibration focus.
