4.4 An Illustration of the Application of the SRSM-ADIFOR

This section presents an illustration of the SRSM-ADIFOR method by revisiting the example presented in Section 3.7 involving the application of the SRSM. The probability distributions of the three model inputs, $X_1, X_2,$ and $X_3$, are given by Equation 3.16, and the two model outputs, $Y_1$ and $Y_2$, are approximated by polynomial chaos expansions

 

$$Y_1 = a_0 + a_1\xi_1 + a_2\xi_2 + a_3\xi_3 + a_4(\xi_1^2-1) + a_5(\xi_2^2-1) + a_6(\xi_3^2-1)$$

$$+ a_7\xi_1\xi_2 + a_8\xi_2\xi_3 + a_9\xi_1\xi_3$$ (4.11)

$$Y_2 = b_0 + b_1\xi_1 + b_2\xi_2 + b_3\xi_3 + b_4(\xi_1^2-1) + b_5(\xi_2^2-1) + b_6(\xi_3^2-1)$$

$$+ b_7\xi_1\xi_2 + b_8\xi_2\xi_3 + b_9\xi_1\xi_3$$

Clearly, the partial derivatives of $Y_1$ and $Y_2$ with respect to the srvs are given by:

$$\begin{array}{ccc} \displaystyle\frac{\partial Y_1}{\partial ... ...\xi_3} & = & b_3 + 2b_6\xi_3 + b_8\xi_2 + b_9\xi_1 \end{array}$$ (4.12)

Based on Equation 4.11, the number of sample points, $N$, required to estimate the 10 unknown coefficients equals 5. Thus, using SRSM-ADIFOR, the uncertainty in the model outputs can be estimated by obtaining the model outputs at $N=5$ sample points, compared to 20 sample points required in the application of the SRSM.

The model equations corresponding to the estimation of unknown coefficients, are given by the following equations:

$$\hspace*{-0.5in}\begin{array}{cccc} {\ensuremath{\textstyle\m... ... \xi_{3,N}}\\ \end{array}\right] \mbox {, \ where} \end{array}$$ (4.13)

$${\ensuremath{\textstyle\mbox{\boldmath${Z}$ }} } = \left[\beg... ...,N}\xi_{3,N} & 0 & \xi_{2,N} & \xi_{1,N}\\ \end{array}\right]$$ (4.14)

  

Figure 4.4: Schematic depiction of the steps involved in the application of the SRSM-ADIFOR