\chapter{Summary and future work}
\label{Ch7}

The goal of this thesis was to develop numerical methods for optimal design problems. In particular, we have proposed new
criteria of optimal design approaches for applications in areas such as medical imaging and geophysics.

First, in Chapter \ref{Ch2}, we have introduced two new design criteria for the $A$-optimal design that minimize the Bayes risk based 
on sparsity. Also, we have developed numerical approaches based on Krylov subspace methods for large scale ill-posed problems. 
For numerical experiments, we have tested our algorithms on a 1D problem as well as a large scale super resolution problem 
and demonstrated the efficiency of our approach.

In Chapter \ref{Ch3}, we have developed numerical methods for the $E$-optimal design. Inverse iteration has been applied to 
approximate the smallest eigenvalue of the information matrix and the corresponding derivative techniques have been discussed.
Numerical experiments have been done on the same 1D problem as was used in Chapter \ref{Ch2} for the purpose of
comparison and also a borehole ray tomography example.

In Chapters \ref{Ch4} and \ref{Ch5}, an optimal $\ell^{2}$ regularization approach has been proposed and two designs based 
on availability of different types of prior information have been developed. Also, several sparsity patterns have been imposed 
on the optimal regularization matrix in order to reduce the computational cost with very little loss of reconstruction quality.
Matrix-based derivative techniques have been discussed in detail for all different sparsity patterns and several numerical 
experiments have shown that our optimal regularization approach gives very promising results. Especially when the model to 
reconstruct is similar to the training set, the training design provides much better results than the traditional differential 
operator.

In Chapter \ref{Ch6}, we have applied our optimal design idea to a geophysical problem: $CO_{2}$ injection monitoring.
Special spatial constraints on placement of sensors have been introduced and different treatments have 
been discussed for sources and receivers, respectively. We have done experiments on a synthetic problem based on a
target zone and the results are quite promising.

Future work to be done is to continue developing numerical methods for solving optimal experimental design problems. 
There are many other optimality criteria that have proved to be very useful for various applications. One of the popular examples 
is the $D$-optimality. As far as we know, there have not been any numerical methods developed for solving ill-posed 
large-scale problems. My goal is to explore efficient methods for large-scale determinant optimization.

Also, we intend to improve the optimal regularization techniques for solving nonlinear inverse problems. The difficulty 
arises because, in this case, there is no simple decomposition of the bias and variance; hence, there is no closed formula 
for the MSE. Also, the formulation will involve a bi-level optimization problem, which is very difficult to solve.  

Furthermore, since the $CO_{2}$ problem we mentioned is in a large class of real-world discrete optimization problems, 
we intend to further develop algorithms to solve discrete optimal experimental design problems.