| Signals describing the energy levels of quantum mechanical systems are, by definition, sparse in the energy domain. Processing these signals via sparsity promoting methods is thus reasonable and, as this dissertation argues, valuable.;Quantum mechanical energy levels are determined experimentally through NMR spectroscopy where noise, peak blurring, and long experiment times impede progress. We show how l1-penalized optimization can lead to improved signal quality and reduce data acquisition time in NMR spectroscopy.;Quantum mechanical signal processing is central to MRI reconstruction. MRI data acquisition and reconstruction is highly time-consuming and expensive. We provide a fast converging algorithm based on minimizing a combination total variation and framelet norms which produces high-quality images from undersampled MRI data.;In the field of numerical analysis, all the eigenvalues of a Hermitian matrix may be computed by simulating a fictitious quantum dynamical system with Hamiltonian corresponding to the matrix of interest and then determining the energy levels of this fictitious system. By determining the energy levels withl1-penalized optimization we show that the number of simulation steps can be significantly reduced.;Quantum mechanical systems have spatial components as well. When the spatial domain is partitioned according to the location of potential wells, one often finds low-energy wavefunctions tend to localize within the confines of each partition. For a given partition, the energy levels of its corresponding localized wavefunctions often make up only a small fraction of the complete range of energy levels. For situations where only a few eigenpairs are sought we introduce a "projection-correction" method allowing us to efficiently compute only the low-energy eigenpairs which localize within a given spatial partition. In contrast to standard methods for eigenvalue computation which specify only a part of the spectrum, our method also allows one to isolate regions of space where prior information on eigenfunction locality is known. |
