This thesis addresses the novel double-bounded positive semidefinite Procrustes problem, which arises from the optimal model regularization problem for wafer alignment.
Despite convexity in the optimization variable A , solving the problem is challenging due to the presence of both an upper and a lower bound, both of which introduce nonlinearity. Several numerical methods have been proposed, including semidefinite programming, alternating projection, and projected gradient methods. Among these, the projected gradient method proves to be the most efficient: by decomposing the upper bound matrix and transforming the variables, the constraints simplify, allowing for straightforward projection onto the feasible region using eigenvalue decomposition.
The optimal regularization approach was tested on various customer datasets, demonstrating overlay improvements of several tens to hundreds of picometers in most cases. However, the method failed for datasets with significant deformation and high variance among wafers, due to incomplete optimization of the regularization matrix. Future research should focus on developing a more sophisticated scheme for combining data-driven optimal regularization with standard bending energy matrices, and extensively validating the method across diverse datasets for accuracy, consistency, and performance.

We introduce a method of designing NMR pulse sequences, with a specific focus toward complete decoupling of carbon-13 spin qubits coupled to a nitrogen-vacancy (NV) center in diamond. Using Average Hamiltonian Theory, we calculate different intermediate Hamiltonians corresponding to different rotations implemented by NMR pulses. Two novel pulse sequences are obtained containing 24 and 12 evolution periods respectively. Assuming ideal and instantaneous pulses, performance for both sequences is better than the WAHUHA + echo sequence for total evolution times of less than 5 ms and comparable for greater evolution times. Analysis of sensitivity points toward non-robustness for angle errors in the applied pulses when the direction of pulses is not taken into account, however this can be corrected for using chirality sums. Suggestions for further research include the study of non-globally applied pulses and application of the design method to non-zero effective Hamiltonians.