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Proceedings Paper

Regularization of ill-posed problems using (symmetric) Cauchy-like preconditioners
Author(s): Misha E. Kilmer
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Paper Abstract

In certain applications, the discretization of 2D integral equations can lead to system involving matrices with block Toeplitz-Toeplitz block structure. Iterative Krylov subspace methods are sometimes employed to find regularized solutions to the related 2D discrete ill-posed problems; however, preconditioned which filter noise are needed to speed convergence to regularized solutions. We describe a preconditioning techniques based on the Toeplitz-type structure of the matrix which generalizes the approaches in (1) and (2) to take advantage of symmetry and real arithmetic operations. We use fast sine transforms to transform the original system to a system whose matrix has partially reconstructible Cauchy-like blocks. The preconditioner is a block diagonal, rank m approximation to this matrix, with Cauchy-like blocks each augmented by an identity of appropriate dimension. We note that the initialization cost is in general less than that for the similar 2D preconditioner in (2) which does not take advantage of symmetry. Several examples are given which illustrate the success of our preconditioned methods.

Paper Details

Date Published: 2 October 1998
PDF: 12 pages
Proc. SPIE 3461, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, (2 October 1998); doi: 10.1117/12.325698
Show Author Affiliations
Misha E. Kilmer, Northeastern Univ. (United States)

Published in SPIE Proceedings Vol. 3461:
Advanced Signal Processing Algorithms, Architectures, and Implementations VIII
Franklin T. Luk, Editor(s)

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