4.3 Coupling of SRSM and ADIFOR

The coupling of SRSM and ADIFOR follows the same steps as the SRSM with respect to input and output transformations. The coupled method, SRSM-ADIFOR, SRSM approximates uncertain model outputs in terms of a set of ``standard random variables'' (srvs), denoted by the set $$\ensuremath{\{\xi_{i}\}_{i=1}^{n}}$$. The following steps are involved in the application of the SRSM-ADIFOR:

ithe model inputs are expressed as functions of selected srvs, as described in Section 3.2, ian approximation for the model outputs is assumed in the same form as presented in Section 3.3:

 

$$y_r = a_{r,0} + \sum_{i_{1}=1}^{n}a_{r,i_{1}}\Gamma_{1}(\xi_{i_{1}}) + ... ...^{n}\ \sum_{i_{2}=1}^{i_{1}}a_{r,i_{1}i_{2}}\Gamma_{2}(\xi_{i_{1}},\xi_{i_{2}})$$

$$+ \sum_{i_{1}=1}^{n}\ \sum_{i_{2}=1}^{i_{1}}\ \sum_{i_{3}=1}^{i_{2}}a_{r,i_{1}i_{2}i_{3}} \Gamma_{3}(\xi_{i_{1}},\xi_{i_{2}},\xi_{i_{3}}) + \ldots$$ (4.7)



where, $y_r$ is the $r$th output metric (or random output) of the model, the $a_{r,i_1\ldots}$'s are deterministic constants to be estimated, and the $\Gamma_{p}(\xi_{i_{1}},\ldots,\xi_{i_{p}})$ are multi-dimensional Hermite polynomials of degree $p$, given by


$$\Gamma_{p}(\xi_{i_{1}},\ldots,\xi_{i_{p}}) = (-1)^{p}\ e^{\frac{1}{2}{\ensuremath{\textstyle\mbox{\boldmath${\... ...th${\xi}$ }} }^{\!T}\!\!{\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} }} \ ,$$ (4.8)



icorrespondingly, the first order partial derivatives of the $r$th model output with respect to $j$th srv$\xi_j$, given by $$\frac{\partial y_r}{\partial \xi_j}$$are expressed as

 

$$\displaystyle\frac{\partial y_r}{\partial \xi_j} = \sum_{i_{1}=1}^{n}a_{r,i_{1}}\displaystyle\frac{\partial \Gamma_{... ...displaystyle\frac{\partial \Gamma_{2}(\xi_{i_{1}},\xi_{i_{2}})}{\partial \xi_j}$$

$$+ \sum_{i_{1}=1}^{n}\ \sum_{i_{2}=1}^{i_{1}}\ \sum_{i_{3}=1}^{i_{2}... ...rtial \Gamma_{3}(\xi_{i_{1}},\xi_{i_{2}},\xi_{i_{3}})}{\partial \xi_j} + \ldots$$ (4.9)



ithe model derivative code is generated using the original model code and ADIFOR and modified so that it can calculate the first order partial derivatives of model outputs with respect to the srvs, ia set of sample points is selected using the sampling algorithm presented in Section 3.4.2; the number of sample points selected is smaller than the number of coefficients to be estimated, as shown in Equation 4.11, ithe model outputs and the partial derivatives with respect to the srvs are calculated at these sample points, ioutputs at these points are equated with the polynomial chaos approximation (Equation 4.8, resulting in a set of linear equations in $a_i$'s, and partial derivatives at these points are equated with the corresponding approximation given by Equation 4.10, iThe resultant equations are solved using singular value decomposition, to obtain estimates of $a_i$'s, as illustrated in Section 4.4.

4.3.1 Selection of Sample Points

In the application of SRSM-ADIFOR to the uncertainty analysis of a model with $M$ inputs, $P$ outputs, for a given order of expansion, Equation 4.8 is used to calculate the number of coefficients to be estimated (say $K$). Thus, $K$ coefficients need to be estimated for each output. The execution of the model derivative code at one sample point gives the model calculations for the outputs and $M$ first order partial derivatives for each output. Thus, equations 4.8 and 4.10 in conjunction with these calculations result in $M+1$ linear equations at each sample point.

Here, the number of recommended sample points is based on the rationale behind the regression based SRSM: the number of resulting equations should be higher than the number of coefficients estimated in order to obtain robust estimates of the coefficients. Here, the recommended number of equations is about twice the number of coefficients to be estimated.

Since for each sample point, the number of resultant equations is $M+1$, the number of sample points required for SRSM-ADIFOR, $N$, is approximately given by^4.1:

$$N \approx \frac{2K}{M+1}$$ (4.10)

Footnotes

... by^4.1

the number is approximate because $2K$ may not always be exactly divisible by $M+1$