New Compression and Regularization Techniques for Large Scale Tomographic Inversion
Abstract
Many tomographic inverse problems can be expressed in the form of a linear system Ax = b. Since A is often ill-conditioned and b is noisy, regularization techniques are used to obtain an acceptable solution. The main challenges are the very large size of the matrix A and the use of an effective regularization scheme which gives the most detailed solution out of the available data. For the first problem, we show our newly devised method of using thresholded wavelet approximations for matrix vector operations and low rank decompositions of the data matrix to accurately approximate the solution in terms of smaller matrices, using the ill-conditioning of A to our advantage. We are able to program our schemes without having to load the whole A into memory, which in our case is several terabytes in size. Using the compression techniques we implement, only matrices which are a small fraction of the size of A are necessary to load for accurate computation. For the second problem, we present a new fast iterative algorithm which attempts to combine the contributions of wavelet regularized sparsity constrained solutions with a set of different sharp and smooth wavelets and of classical ell_2 penalized solutions. The aim of the new approach is to recover sharp and smooth solution features at different scales through the use of multiple different basis functions. We illustrate the results with examples from our inverse problem.
- Publication:
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AGU Fall Meeting Abstracts
- Pub Date:
- December 2013
- Bibcode:
- 2013AGUFM.S33A2385V
- Keywords:
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- 7290 SEISMOLOGY Computational seismology;
- 7270 SEISMOLOGY Tomography