Example Problem for Experimental Design

Problem Description

Problem description: http://files.charlesmartinreid.com/VUQ_Toy_Problem.pdf

Inputs and Outputs

There are several (polynomial) response surfaces being fit for the Monte Carlo simulations. This is because there is one response surface for each output or observable. (Technically, these are all part of one large multivariate response surface, but it is easier to think about them as independent response surfaces).

The variables that are included in the response surface analysis are:

• $ z_i $ (for $ i=1,2,3 $) - the location of measurement of axial concentrations
• $ \dot{m} = \dot{m}_1 = \dot{m}_2 $ - the mass flowrate of the inlet streams of A and B
• $ L_{mix} $ - mixing length (parameter for the mixing model)
• $ k(T) $ - reaction rate for the reaction $ A + B \rightarrow^{k} P $

Variables: I/U Map

Variable Name	Input value (I)	Uncertainty (U)	Logarithmic Scale?
$ z_1 $	$ 0.5 m $	$ \pm 0.02 m $	no
$ z_2 $	$ 1.5 m $	$ \pm 0.02 m $	no
$ z_3 $	$ 2.5 m $	$ \pm 0.02 m $	no
$ \dot{m}_1 $	$ 1.0 $	$ \pm 0.05 $	no
	$ 2.0 $	$ \pm 0.10 $
$ \dot{m}_2 $	$ 1.0 $	$ \pm 0.05 $	no
	$ 2.0 $	$ \pm 0.10 $
$ k(T) $	$ 1 $	$ 10^{0 \pm 2} $	yes
$ L_{mix} $		$ 0.3-3.0 $	yes

Response Surfaces

Product at Exit Response Surface

This response surface maps the response of the mass fraction of product at the exit, $ y_{P,exit} $.

This response surface is a function of several variables:

$ y_{P,exit} = y_{P,exit} \left( \dot{m}, k, L_{mix} \right) $

However, the axial location of measurement of product $ z_i $ is not considered, because it does not affect the measurement of P at the exit.

Product at Axial Location Response Surfaces

This response surface maps the response of the mass fraction of product at several axial locations, $ y_{P,z1}, y_{P,z2}, y_{P,z3} $.

This response surface is a function of all variables:

$ y_{P,zi} = y_{P,exit} \left( \dot{m}, k, L_{mix}, z_i \right) $

(Note, however, that only the corresponding $ z_i $ will be a response surface independent variable, since other $ z_i $ values have no affect).

Dealing with Multimodal Variables

Sometimes, when constructing response surfaces, modal variables appear. Modal variables are variables that have multiple modes, or distinct sets of values. There are two variations of modal variables:

1 uncertainty range (sampled with N parameter values)

These types of modal variables have a single range of uncertainty assigned to them, but the values within that range of uncertainty are discrete. In order to sample the parameter within the range of uncertainty, the parameter must be sampled at distinct, discrete values.

For example, if I am using the discrete ordinates model (DOM) for radiation calculations, the DOM requires a number of ordinate directions. This is a discrete value with distinct sets of values - e.g. 3, 6, 8, 24, etc.

Each discrete value in this case composes a single range of uncertainty. Using the DOM example, that range of uncertainty would be $ [3, 24] $.

N uncertainty ranges

The other type of modal variables have several ranges of uncertainty assigned to them, with no restriction on values within that range of uncertainty being discrete or distinct. Essentially this can be thought of as a bimodal uncertainty distribution, where the two modes are distinct. Each mode can be sampled as usual, the only sticking point is that there is more than 1, and that they are distinct.

This case provides an excellent example. The variable $ \dot{m} $ is a modal variable - the two modes are 1.0 and 2.0 - but each mode also has a range of uncertainty, namely $ 5% $ each.

How to Deal

Multimodal variables can be dealt with by creating a response surface for each distinct mode. For the case of single distinct values, such as the number of oordinates in DOM, the modal variable is eliminated as a response surface input parameter/variable. If $ N $ distinct values for the number of ordinates are used to cover a range of possible numbers of ordinates,

An example of a "true" response, which is unknown to the modeler.	The modeler is only interested in distinct regions of the input parameter $ x $ (shown in gray). The remaining regions are left out of the response surface.	The response surfaces actually obtained by the user (blue dotted line). There is a separate response surface obtained by the user (2 distinct blue lines) for each mode (gray region).