3.1 Overview of the Method

The Stochastic Response Surface Method (SRSM), as the name suggests, can be viewed as an extension to the classical deterministic Response Surface Method (RSM), on which extensive literature is available [22,21,70,123]. The main difference between the SRSM and the RSM is that in the former the inputs are random variables, where as in the latter, the inputs are deterministic variables.

The SRSM was developed as a part of this work with the objective of reducing the number of model simulations required for adequate estimation of uncertainty, as compared to conventional methods. This is accomplished by approximating both inputs and outputs of the uncertain system through series expansions of standard random variables; the series expansions of the outputs contain unknown coefficients which can be calculated from the results of a limited number of model simulations.

Conceptually, the propagation of input uncertainty through a model using the SRSM consists of the following steps: (1) input uncertainties are expressed in terms of a set of standard random variables, (2) a functional form is assumed for selected outputs or output metrics, and (3) the parameters of the functional approximation are determined.

The SRSM is founded on the principle that random variables with well-behaved (square-integrable) pdfs can be represented as functions of independent random variables [25,49,83,198]. The above integrability requirement is usually satisfied by uncertain quantities of interest, so this method is applicable to a wide variety of of transport-transformation models. Random variables with normal distributions, N(0,1), are often selected to represent input uncertainties due to the mathematical tractability of functions of these random variables [49,160]. In the present work these random variables are referred to as ``Standard Random Variables'' (srvs).

Once the inputs are expressed as functions of the selected srvs, the output metrics can also be represented as functions of the same set of srvs. The minimum number of srvs needed to represent the inputs is defined as the ``number of degrees of freedom'' in input uncertainty. Since model outputs are deterministic functions of model inputs, they have at most the same number of degrees of freedom in uncertainty.

Consider a model $\ensuremath{\mathbf{y}} =\ensuremath{\underline{\underline{\mathcal{F}}}}\,(\ensuremath{\mathbf{x}} )$, where the set of random inputs is represented by the vector , and a set of selected random outputs or output metrics is represented by the vector . First, the vector of input random variables is expressed as a function of the form $\ensuremath{\mathbf{x}} = h({\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} })$, where denotes the vector of the selected srvs. Then, a functional representation for outputs, of the form $\ensuremath{\mathbf{y}} =f({\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} },{\ensuremath{\textstyle\mbox{\boldmath${a}$ }} })$, where denotes a parameter vector, is assumed. The specific method employed in the estimation of parameters of approximation depends on the complexity of the model, which can be broadly categorized into the following:

• Models consisting of simple linear algebraic operators: the outputs can be directly obtained as analytical functions of srvs, as $\ensuremath{\mathbf{y}} = \ensuremath{\underline{\underline{\mathcal{F}}}} ^{-1... ...nsuremath{\mathbf{x}} ) = g({\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} })$. For many cases, even a specific functional form for need not be assumed a priori.
• Models consisting of nonlinear operators with mathematically manipulable equations: a functional form for outputs, such as $\ensuremath{\mathbf{y}} =f({\ensuremath{\textstyle\mbox{\boldmath${\xi}$ }} },{\ensuremath{\textstyle\mbox{\boldmath${a}$ }} })$, where denotes the parameters of approximation, is assumed. The parameters are then calculated by substituting the transformed inputs and outputs into the model equations, and minimizing an appropriate norm of the resultant error in approximation. Galerkin's method used in the Deterministic Equivalent Modeling Method [198,208] is an example of one such case.
• Black-box type models (possibly reflecting the numerical implementation of a complex mechanistic model that one may decide to treat as a ``black-box'' input-output system): a functional representation for the output is assumed, as described above, and model outputs for several realizations of the srvs are used as ``data'' for the estimation of parameters of the functional approximation. This approach can be used for cases where it is not possible to directly substitute the input representations into the model. Sometimes this approach may be preferable in the case of complex nonlinear models, where the direct solution requires a significant amount of algebraic manipulation.

Figure 3.1 further offers a schematic illustration of the application of the SRSM for various scenarios.

The following sections describe the SRSM in more detail, and elaborate on the following steps that are involved in the application of the SRSM:

I

representation of uncertain inputs,

II

representation of model outputs,

III

estimation of approximation parameters,

IV

calculation of statistical properties of the outputs, and

V

evaluation of the approximation of model outputs.

  

Figure 3.1: Schematic depiction of the Stochastic Response Surface Method