Compression Approaches for the Regularized Solutions of Linear Systems from Large-Scale Inverse Problems

TL;DR

This paper presents novel compression techniques for large linear systems in inverse problems, utilizing wavelet-inspired sparsification and low-rank SVD to achieve significant size reduction while preserving solution features.

Contribution

It introduces combined wavelet-based sparsification and low-rank SVD methods for efficient regularized solutions of large linear systems in inverse problems.

Findings

Significant compression gains achieved on synthetic and seismic data.

Methods preserve main features of solutions despite size reduction.

Analytical results support the effectiveness of the approaches.

Abstract

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.