Case 1
and
Using the generalized Polynomial Chaos (gPC) Expansion
Step 1: Select Basis Functions
- For Uniform Distribution choose Legendre polynomials over
- For Gaussian Distribution choose Hermite polynomials over
Step 2: Calculate the number of Polynomials
where no. of input variables (we have 2 germs), degree of expansion (here both are ) Step 3: Take all possible combinations
Step 3: Represent as a Series Expansion The gPC expansion of the model function up to total polynomial degree 2 is:
Substituting the polynomial expressions:
The coefficients through would be determined using methods like non-intrusive projection, stochastic collocation, or regression with data from model evaluations.
Case 2
Consider denote a random solution to different points in time
- Step 1: gPC Surrogate Model
- Step 2: Approximate Moments using gPC coefficient
- Step 3: Evaluate gPC Surrogate with different input Random Variables.
Note: is calculated using methods like non-Intrusive Surrogate Modeling