chapter05.tex

\chapter{\uppercase{Extensions and Applications of MUD Points and Skewness} \label{chapter:vector-valued}}
In Chapter~\ref{chapter:geometry}, we introduced the notion of skewness and showed examples of how it impacts the accuracy of approximating SIP solutions with finite sampling.
Here, we demonstrate how an awareness of skewness allows us to a priori define a QoI that will\---on average\---better resolve $\paramref$ by providing information in mutually distinct directions in $\pspace$.
We begin by revisiting the example in Section \ref{subsec:pde-example} involving the Poisson problem and uncertain Neumann boundary condition $g$.
Recall that $\qoi_\text{2D}$ is able to better resolve $\paramref$ (in this case, $\paramref$ is defined by a finite-dimensional representation of an exact $g$), than $\qoi_\text{1D}$.
The map $\qoi_\text{2D}$ is presented as an alternative option for aggregating the same $100$ measurements and using them to construct a more informative 2D map.
In this chapter, we present a more detailed study of the construction of this map, and perform a case-study in designing a problem where the data-constructed QoI map is more informative.

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\section{Conclusions}

We have shown that when posing and solving SIPs, we are motivated to choose data-constructed QoI maps which have (a) as many components as possible (up to the dimension of $\pspace$), and (b) whose components exhibit high-degrees of mutually geometrically distinct information (i.e., low average skewness).
There is a dramatic reduction in the sensitivity to the measurement noise when the same hundred measurements are used to construct a map with lower skewness.
In the next chapter, we discuss some other considerations for constructing SIPs, including measurement precision and placement.
We show how different experimental setups can lead to parameter estimates which exhibit different levels of accuracy and precision.
We also introduce a novel method for leveraging the results shown here in a sequential manner to gain possible computational efficiency.