chapter06.tex

\chapter{\uppercase{Conclusions, Future Work, and Preliminary Results} \label{chapter:future}}


\section{Concluding Remarks}
In the Chapter~\ref{chapter:mud} we demonstrate that the MUD solution retains the accuracy of least-squares solutions while simultaneously offering the flexibility of specifying initial beliefs.
Normally in order to incorporate such beliefs, practitioners in the machine-learning field would perform Tikhonov regularization, usually with the inclusion of a hyper-parameter which scales the additional parameter-space norm in the objective function.
Mathematically, this scaling factor applied to the norm is equivalent to scaling the matrix representation of an initial (prior) covariance and searching for the MAP point of a Bayesian posterior.
Increasing this scaling factor is interpreted as having less confidence in these initial assumptions.
Conversely, decreasing it is equivalent to putting more emphasis on the prior beliefs than the evidence provided by the data, which causes MAP solutions to drift away from the solution contour (equivalence class) to which $\paramref$ belongs.

The MUD point is not impacted by scaling of the initial covariance, providing \emph{consistent} solutions which demonstrate levels of accuracy that MAP points only exhibit for larger values of scaling factors.
Not only is it robust to the specification of prior assumptions, but it manages to offer the flexibility of such specifications without paying the additional cost of hyper-parameter optimization that would be required for the Tikhonov solution to achieve comparable results; any choice of $\alpha$ would have sufficed.

By contrast, the Tikhonov-regularized solution selects a point that is biased in directions defined by the initial density (covariance).
The data-consistent solution is an update to the initial mean in this same direction but will always exist on the contour $Q^{-1}(\observedMean)$, where $\observedMean$ is the mean of the observed density.


We also show that regardless of how well-informed the initial beliefs are, the convergence rate of the MUD solutions as more data are incorporated\---either by dimension or rank)\---will match those of the Least-Squares solutions.
Moreover, unlike MAP solutions, the MUD point is not sensitive to scaling of the initial covariance (how strongly initial beliefs are held).
This insensitivity provides a strong motivating factor for the consideration of the data-consistent approach within the standard set of solution methods available to scientists and modelers who seek to perform parameter-identification.
We leave the investigation of more connections to the removal of hyper-parameter estimation to future work.

The trouble is, none of these regularization approaches actually guarantee that in under-determined problems, the unique solutions that are selected are close to $\paramref$.
The equivalence-class nature of the solution contour means that by definition there are directions in which uncertainty is unresolved.
Thus, the goal is to aggregate data into components of a vector-valued QoI map to improve estimates of $\paramref$ across more dimensions.
We use skewness as a guide for constructing the QoI and demonstrate its utility in improving estimates to a $\paramref$ related to estimating an uncertain function.


\section{Future Work and Preliminary Results}

\input{extensions/mud_oed.tex}
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%\subsection{Leveraging Data in Different Ways}\label{sec:ch05-data}
%Mention the other map we can use (SSE) here.
%
%\subsection{Addressing Model Assumptions}\label{sec:ch05-variance}
%Both the maps required us to have knowledge of the variance in the measurements.
%What if we got it wrong?
%\emph{What if we don't know the variance? How does mis-estimating it affect our solutions?}
%In this section we pose some questions and provide a brief hint at a research direction but really we do not have adequate time to flesh out the answers to these, just acknowledge that they're similar concerns shared by the Bayesians.
%
%Multiplicative noise - handled in a straightforward way, maybe put an example here and leave it at that? Put it in appendix?


\input{extensions/sequential_inversion.tex}
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