UCLA

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 spec-

troscopy 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 acquisi-

tion time in NMR spectroscopy.

Quantum mechanical signal processing is central to MRI reconstruction. MRI data ac-

quisition 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 com-

puted 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 with l1-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.