To solve for this set of coefficients, we need to evaluate the model at special "points" in the space. First, we can define the residual of the problem that we would like to minimize. For the deterministic set of model equations (that is, the "black-box" model), the residual is given by , assuming fixed values for and an invariant set of values for . For the stochastic problem, where and are distributions, the probabilistic residual becomes:

(20–12)

Applying PCEs, Equation 20–12 is transformed to

(20–13)

where is a set of approximating functions (basis functions) that are orthogonal, such that the inner product of the residual and each member of the set of approximating functions is equal to zero [162]. The residual is now cast in terms of , which are the set of coefficients of the polynomial chaos expansions of the vector of outputs.

In order to solve Equation 20–13 using an arbitrary "black-box" model, we need to find a way to perform the integration without knowing the exact form of the residual. This can be achieved using a special quadrature method, collocation, described by Tatang and Wang[162], [163], where the model is evaluated at a finite number of points. The key to this method is that it selects points corresponding to the roots of the "key" polynomials used in the response PCEs in place of the basis functions in Equation 20–13 . As a result, the residual minimization problem becomes:

(20–14)

where are the collocation "points" or sets of values where the model residual will be minimized. Equation 20–14 is equivalent to:

(20–15)

where is the number of collocation points required to determine the coefficients. In this way, the collocation method transforms the stochastic model into a deterministically equivalent model, because it solves the deterministic model at several chosen values of uncertain parameters.

The collocation method uses the roots of the key polynomial as the sampling points, because these points provide good coverage of the distribution function, capturing high-probability and transition regions. This is illustrated in Figure 20.1: Comparison of collocation points for normal PDF and sampling points for Monte Carlo method , where the location of the collocation points for a normal distribution (roots of the Hermite polynomial) are compared with the Monte Carlo approach that requires many more sampling points. The roots of Hermite polynomials are given in Table 20.3: Roots of Hermite Polynomials below. In the collocation method, we take the roots of the polynomial that represents one order higher than the one used in the PCEs of the model outputs. This choice is made to allow more of the sampling points to be high-probability points than would be possible if the roots were of the same order as the PCEs. Because of this choice, however, we have more roots than we need to define the sampling points. To select the collocation points, we start by defining an anchor point. The anchor point is the one consisting of the highest-probability root for all values. We then fill in the remaining points by substituting the highest-probability root for the next-highest-probability root successively into each of the values. This process is repeated until we have defined the collocation points necessary to determine the response coefficients.

Figure 20.1: Comparison of collocation points for normal PDF and sampling points for Monte Carlo method^[3]