4. COUPLING OF THE SRSM WITH SENSITIVITY ANALYSIS METHODS: DEVELOPMENT AND IMPLEMENTATION OF SRSM-ADIFOR

The objective of this effort is to couple the Stochastic Response Surface Method (SRSM) with sensitivity analysis methods, to obtain savings in the computer resources required for uncertainty propagation. In the construction of an approximate model, sensitivity analysis methods can provide information that can help reduce the number of model simulations required to estimate the approximation parameters. Consequently, the use of sensitivity information in conjunction with uncertainty propagation methods can reduce the model simulations required to estimate uncertainties in model outputs.

The rationale for the coupling of the SRSM with a sensitivity analysis method is illustrated in Figure 4.1 for the simple case of approximation of a curve in a 2-D Euclidean space using a set of sample points. Figure 4.1 (A) represents the model response, which can be approximated accurately using the five sample points, as shown in Figure 4.1 (B). In contrast, using three sample points, only a poor approximation as shown in Figure 4.1 (C) can be obtained. However, using the derivative information at the three sample points (tangents at these points), a good approximation, as shown in Figure 4.1 (D) can be obtained. As an extension of this reasoning to multi-dimensional systems, the partial derivatives of model outputs with respect to model inputs (i.e., tangents in the multi-dimensional space) can be used to reduce the number of sample points required to construct an adequately approximate model.

  

Figure 4.1: Rationale for coupling of the SRSM with a sensitivity analysis method

An overview of various sensitivity metrics is presented in Section 2.4, and the metrics are listed in Table 2.3. Of these sensitivity metrics, the derivative information can be directly utilized in constructing an approximate model. For example, consider a model

$$z = f(x,y)$$

for which the output $z$ is to be approximated as a function of inputs $x$ and $y$ through the expression:

$$z = a_0 + a_1x + a_2y + a_3x^2 + a_4y^2 + a_5xy$$

In order to estimate the coefficients $a_i$'s, the model outputs for $z$ are needed at six sample points, requiring six model runs. This results in a set of six equations with the left hand side consisting of the model outputs and the right hand side consisting of linear equations in the coefficients.

For the same case, if all the first order partial derivatives are available at each sample point, the number of model runs can be reduced to two, and the derivative information at two points, $(x_1,y_1)$ and $(x_2,y_2)$ can be used to construct the following closed system of equations:

$$\left. z \right\vert _{(x_1,y_1)} = a_0 + a_1x_1 + a_2y_1 + a_3{x_1}^2 + a_4{y_1}^2 + a_5x_1y_1$$ (4.1)

$$\left. \frac{\partial z}{\partial x} \right\vert _{(x_1,y_1)} = a_1 + 2a_3{x_1} + a_5y_1$$ (4.2)

$$\left. \frac{\partial z}{\partial y} \right\vert _{(x_1,y_1)} = a_2 + 2a_4{y_1} + a_5x_1$$ (4.3)

$$\left. z \right\vert _{(x_2,y_2)} = a_0 + a_1x_2 + a_2y_2 + a_3{x_2}^2 + a_4{y_2}^2 + a_5x_2y_2$$ (4.4)

$$\left. \frac{\partial z}{\partial x} \right\vert _{(x_2,y_2)} = a_1 + 2a_3{x_2} + a_5y_2$$ (4.5)

$$\left. \frac{\partial z}{\partial y} \right\vert _{(x_2,y_2)} = a_2 + 2a_4{y_2} + a_5x_2$$ (4.6)

Similarly, for a model with $M$ parameters, the number of model runs required to estimate the unknown coefficients decreases by a factor of $M+1$, since the first order partial derivatives with respect to $M$parameters result in $M$ additional equations at each sample point, whereas calculation of the model output at a given point results in only one equation.