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