Multi-Dimensional Polynomial Chaos Expansion

Case 1

$\xi ={\left({\xi }_1,{\xi }_2\right)}^{\top };{\xi }_1\sim \mathcal{U}(-\sqrt{3},\sqrt{3})$ and ${\xi }_2\sim \mathcal{N}(0,1)$

• Step 1: Select Basis Functions
  • Choose Legendre polynomials $L_i$ over ${\xi }_1$
  • Choose Hermite polynomials $H_i$ over ${\xi }_2$
• Step 3: Take all possible combinations

$$\begin{array}{rl}& {\Phi }_0\left({\xi }_1,{\xi }_2\right)=L_0\left({\xi }_1\right)H_0\left({\xi }_2\right)=1\\ & {\Phi }_1\left({\xi }_1,{\xi }_2\right)=L_1\left({\xi }_1\right)H_0\left({\xi }_2\right)={\xi }_1\\ & {\Phi }_2\left({\xi }_1,{\xi }_2\right)=L_0\left({\xi }_1\right)H_1\left({\xi }_2\right)={\xi }_2\\ & {\Phi }_3\left({\xi }_1,{\xi }_2\right)=L_1\left({\xi }_1\right)H_1\left({\xi }_2\right)={\xi }_1{\xi }_2\\ & {\Phi }_4\left({\xi }_1,{\xi }_2\right)=L_2\left({\xi }_1\right)H_0\left({\xi }_2\right)=\frac{3}{2}{\xi }_1^2-\frac{1}{2}\\ & {\Phi }_5\left({\xi }_1,{\xi }_2\right)=L_0\left({\xi }_1\right)H_2\left({\xi }_2\right)={\xi }_2^2-1\end{array}$$

The Number of polynomials is give by

$$n=\frac{\left(n_{\mathrm{in}}+p\right)!}{n_{\mathrm{in}}!p!}$$

where $n_\text{in}\rightarrow$ no. of variables, $p\rightarrow$ degree of expansion (here both are $2$)

• Step 3: Represent $X$ as a Series Expansion:

$$X\approx \sum _{i=0}^{\infty }{q}_i{\Phi }_i\left({\zeta }_1,{\zeta }_2\right)$$

Case 2

Consider $Y_i=u(t_i)$ denote a random solution to different points in time $t_i$

• Step 1: gPC Surrogate Model   

$$\gamma \approx \tilde{\mathcal{M}}(\mathbf{\xi })=\sum _{i=0}^n{q}_i{\Phi }_i(\mathbf{\xi })$$

• Step 2: Approximate Moments using gPC coefficient $q_i$

$${\tilde{M}}^k[r]\approx \sum _{i=0}^n{q}_i^k$$

• Step 3: Evaluate gPC Surrogate $\tilde{\mathcal{M}}$ with different input Random Variables.

Note: $q_i$ is calculated using methods like non-Intrusive Surrogate Modeling