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 ${\xi }_2\sim \mathcal{N}(0,1)$

Using the generalized Polynomial Chaos (gPC) Expansion

$$X(\theta )=\sum _{i=0}^{\infty }{q}_i{\Phi }_i(\xi (\theta ))$$

Step 1: Select Basis Functions

• For Uniform Distribution choose Legendre polynomials $L_i$ over ${\xi }_1$
  • ${\mathrm{L}}_0\left({\xi }_1\right)=1$
  • ${\mathrm{L}}_1\left({\xi }_1\right)={\xi }_1$
  • ${\mathrm{L}}_2\left({\xi }_1\right)=(3/2){\xi }_1{}^2-1/2$
• For Gaussian Distribution choose Hermite polynomials $H_i$ over ${\xi }_2$
  • ${\mathrm{H}}_0\left({\xi }_2\right)=1$
  • ${\mathrm{H}}_1\left({\xi }_2\right)={\xi }_2$
  • ${\mathrm{H}}_2\left({\xi }_2\right)={\xi }_2{}^2-1$

Step 2: Calculate the number of Polynomials

$$n=\frac{\left(n_{\mathrm{in}}+p\right)!}{n_{\mathrm{in}}!p!}=\frac{(2+2)!}{2!\cdot 2!}=6$$

where $n_{\text{in}}\to$ no. of input variables (we have 2 germs), $p\to$ degree of expansion (here both are $2$) Step 3: Take all possible combinations

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

Step 3: Represent $X$ as a Series Expansion The gPC expansion of the model function $M(\xi )$ up to total polynomial degree 2 is:

$$X=M(\xi )\approx q_0{\Phi }_0(\xi )+q_1{\Phi }_1(\xi )+q_2{\Phi }_2(\xi )+q_3{\Phi }_3(\xi )+q_4{\Phi }_4(\xi )+q_5{\Phi }_5(\xi )$$

Substituting the polynomial expressions:

$$X=M(\xi )\approx q_0+q_1{\xi }_1+q_2{\xi }_2+q_3\left(\frac{3}{2}{\xi }_1^2-\frac{1}{2}\right)+q_4{\xi }_1{\xi }_2+q_5\left({\xi }_2^2-1\right)$$

The coefficients ${\mathrm{q}}_0$ through ${\mathrm{q}}_5$ would be determined using methods like non-intrusive projection, stochastic collocation, or regression with data from model evaluations.

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