Polynomial Chaos: A technique for modeling uncertainty
summary
Polynomial chaos represents any finite‑variance random parameter as a deterministic series of orthogonal polynomials, computes coefficients by Galerkin projection with an inverse CDF transform, and evaluates inner products with Gauss‑Hermite quadrature. The uploaded Python uses numpy and scipy, truncates at low orders (5th or 13th) and reconstructs a distribution that matches Monte Carlo means and confidence bands while replacing thousands of sample ODE solves with one P×P system solve; occasionally truncated samples produce nonphysical draws but the moment estimates agree.
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chapters (10)
- 0:00Polynomial Chaos intro
- 2:35Infinite-dimensional PCA
- 8:14Arbitrary distributions with Gaussians
- 11:46Mathematical setup
- 16:48Coefficients via quadrature
- 21:03Example with differential equations
- 27:04Intrusive chaos for ODEs
- 30:39Results and efficiency
- 31:23Q&A: polynomial order
- 33:59Q&A: inverse transform
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