GEM - International Journal on Geomathematics

, Volume 6, Issue 2, pp 251–294 | Cite as

Compression approaches for the regularized solutions of linear systems from large-scale inverse problems

Original Paper

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 singular value decomposition (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.

Keywords

Ill-posedness Regularization Singular value decomposition Wavelets Data compression 

Mathematics Subject Classification

65F22 86-08 65T60 15A18 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Applied MathematicsUniversity of ColoradoBoulderUSA
  2. 2.Department of Earth, Atmospheric and Planetary SciencesMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.Géoazur, Université de NiceSophia AntipolisFrance

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