  We introduce and compare new compression approaches to obtain regularized
solutions of large linear systems which are commonly encountered in large scale
inverse problems. We first describe how to approximate matrix vector operations
with a large matrix through a sparser matrix with fewer nonzero elements, by
borrowing from ideas used in wavelet image compression. Next, we describe and
compare approaches based on the use of the low rank SVD, which can result in
further size reductions. We describe how to obtain the approximate low rank SVD
of the original matrix using the sparser wavelet compressed matrix. Some
analytical results concerning the various methods are presented and the results
of the proposed techniques are illustrated using both synthetic data and a very
large linear system from a seismic tomography application, where we obtain
significant compression gains with our methods, while still resolving the main
features of the solutions.
