3.3 Step II: Functional Approximation of Outputs

An output of a model may be influenced by any number of model inputs. Hence, any general functional representation of uncertainty in model outputs, should take into account uncertainties in all inputs. For a deterministic model with random inputs, if the inputs are represented in terms of the set , the output metrics can also be represented in terms of the same set, as the uncertainty in the outputs is solely due to the uncertainty of the inputs [198]. This work addresses one specific form of representation, the series expansion of normal random variables, in terms of Hermite polynomials; the expansion is called ``polynomial chaos expansion'' [83].

When normal random variables are used as srvs, an output can be approximated by a polynomial chaos expansion on the set , given by:

 

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

$$+ \sum_{i_{1}=1}^{n}\ \sum_{i_{2}=1}^{i_{1}}\ \sum_{i_{3}=1}^{i_{2}}a_{i_{1}i_{2}i_{3}} \Gamma_{3}(\xi_{i_{1}},\xi_{i_{2}},\xi_{i_{3}}) + \ldots,$$ (3.7)

where, $y$ is any output metric (or random output) of the model, the $a_{i_1\ldots}$'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

 

$$\Gamma_{p}(\xi_{i_{1}},\ldots,\xi_{i_{p}}) = (-1)^{p}\ e^{\frac{1}{2}{\ensuremath{\textstyle\mbox{\boldmath${\... ...th${\xi}$ }} }^{\!T}\!\!{\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} }} \ ,$$ (3.8)

where is the vector of $p$ iid normal random variables $\{\xi_{i_{k}}\}_{k=1}^{p}$, that are used to represent input uncertainty. Hermite polynomials on are random variables, since they are functions of the random variables . Furthermore, the Hermite polynomials defined on are orthogonal with respect to an inner product defined as the expectation of the product of two random variables [83]. Thus,

$$\mbox{E}[\Gamma_p \Gamma_q] = 0 \ \ {\rm iff}\ \ \Gamma_p \neq \Gamma_q\ ,$$

It is known that the set of multi-dimensional Hermite polynomials forms an orthogonal basis for the space of square-integrable pdfs, and that the polynomial chaos expansion is convergent in the mean-square sense [83]. In general, the accuracy of the approximation increases as the order of the polynomial chaos expansion increases. The order of the expansion can be selected to reflect accuracy needs and computational constraints.

For example, an uncertain model output $U$, can be expressed as second and third order Hermite polynomial approximations, $U_2$ and $U_3$ as follows:

  

$${U_2 = a_{0,2} + \sum_{i=1}^{n}a_{i,2}\xi_{i} + \sum_{i=1}^{n}a_{ii,2}(\xi_{i}^{2}-1) + \sum_{i=1}^{n-1}\ \sum_{j>i}^{n}a_{ij,2}\xi_{i}\xi_{j}}$$

$${U_3 = a_{0,3} + \sum_{i=1}^{n}a_{i,3}\xi_{i} + \sum_{i=1}^{n}a_{ii,3}(\xi_{i}^{2}-1) + \sum_{i=1}^{n}a_{iii,3}(\xi_{i}^{3}-3\xi_{i})}$$

$$+ \sum_{i=1}^{n-1}\ \sum_{j>i}^{n}a_{ij,3}\xi_{i}\xi_{j} + \sum_{i=... ...sum_{i=1}^{n-2}\ \sum_{j>i}^{n-1}\ \sum_{k>j}^{n}a_{ijk,3}\xi_{i}\xi_{j}\xi_{k}$$ (3.10)

where $n$ is the number of srvs used to represent the uncertainty in the model inputs, and the coefficients $a_{i,m2}, a_{i,m3}, a_{ij,m2}, a_{ij,m3}, \mbox{~and}~ a_{ijk,m3}$ are the coefficients to be estimated.

From the above equation, it can be seen that the number of unknowns to be determined for second and third order polynomial chaos expansions of dimension $n$, denoted by $N_2$ and $N_3$, respectively, are:

$$N_{2} = 1+2n+\frac{n(n-1)}{2}$$ (3.11)

$$N_{3} = 1+3n+\frac{3n(n-1)}{2}+\frac{n(n-1)(n-2)}{6}.$$ (3.12)