Polynomial Chaos Expansion

Resources

UQ Class Notes at CU Boulder
Smith, Ralph C. 2013a. “Chapter 9: Uncertainty Propagation in Models.” In Uncertainty Quantification: Theory, Implementation, and Applications, 187–206. Computational Science & Engineering. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973228.ch9
Smith, Ralph C. 2013b. “Chapter 10: Stochastic Spectral Methods.” In Uncertainty Quantification: Theory, Implementation, and Applications, 207–37. Computational Science & Engineering. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973228.ch10.
Wentz, Jacqueline, and Alireza Doostan. 2023. “GenMod: A Generative Modeling Approach for Spectral Representation of PDEs with Random Inputs.” Journal of Computational Physics 472 (January):111691. https://doi.org/10.1016/j.jcp.2022.111691.

Uncertainty Quantification

Main Idea

For random variables $y : Ω \to R^{d}$ and a quantity of interest $u = f (y)$ , polynomial chaos expansions approximate $f$ . Originally, this was with normally distributed $y$ , where the optimal polynomials are Hermite, but this has been generalized to other distributions (Wiener-Askey schemes).

Intrusive

For solving Partial Differential Equations, this idea can be introduced intrusively as a Stochastic Galerkin method, where $x$ are just more variables making up the domain:

\begin{matrix} \hat{u} (x, t, y) = \sum_{j = 0}^{P} \underset{unknown}{\underset{⏟}{u_{j} (x, t)}} \underset{known}{\underset{⏟}{ψ_{j} (y)}} \\ ⟨ {\hat{u}}_{t} - N (x, t, y, \hat{u}, {\hat{u}}_{x}, \dots), ψ_{i} ⟩ = 0, i = 0, 1, \dots, P \end{matrix}

This condition of Galerkin Projection (bottom) is a system of $P + 1$ coupled, deterministic PDEs for the unknowns $u_{j} (x, t)$ (for the original scalar PDE).

Non-Intrusive

Alternatively, PCE can be done non-intrusively, using a block box model for $f$ (rather than specifying it implicitly as the solution of a PDE above). With $u = f (y)$ , and $y$ following some distribution, we solve for $c_{j}$ such that

u (y) = \sum_{j = 0}^{P} c_{j} ψ_{j} (y),

where $ψ_{j}$ are the associated Wiener-Askey polynomials. For instance, if $y \sim U ([- 1, 1])$ , $ψ_{j}$ are the Legendre Polynomials. Note that these polynomials are orthogonal with respect to the measure associated with the density of $y$ ( $p$ ). If also normalized, this means

\int_{- \infty}^{\infty} ψ_{i} (y) ψ_{j} (y) d p = δ_{i j} .

Non-intrusively, we sample $N$ instances of $y$ and compute the associated $u$ . We want to choose the coefficients of our expansion so that it matches this data ${(y_{i}, u_{i})}_{i = 1}^{N}$ . For instance, we may wish to minimize the square of the error (Least Squares):

min_{c} \sum_{i = 1}^{N} (u_{i} - u (y_{i}; c))^{2} .

This is solved through the normal equations when $N > P + 1$ , creating the standard $N \times (P + 1)$ measurement matrix $Ψ$ and $N \times 1$ solution vector $u$ ,

\Psi {#T} \Psi \, \mathbf{c} = \Psi ^ T \mathbf{u} .

This system should be solved with factorization that makes use of the positive definite structure (Singular Value Decomposition or QR Decomposition). Note that in the limit of $N \to \infty$ , $\Psi {#T} \Psi \to \mathbf{I}$ , due to the aforementioned orthonormality.

Compressive Sensing

The above least squares requires $N > P + 1$ , but for a $d$ -dimensional $y$ , and polynomials of maximum total order $p$ ,

P + 1 = (\binom{p + d}{d}) = \frac{(p + d)!}{p! d!} .

The total number of terms grows rapidly in both $p$ and $d$ . Thus, we may need very many samples in order to even be determined (let alone sufficiently overdetermined).

Resources

Related

Main Idea

Intrusive

Non-Intrusive

Compressive Sensing