Regularization of ill-posed problems using (symmetric) Cauchy-like preconditioners

Misha E. Kilmer

doi:10.1117/12.325698

2 October 1998 Regularization of ill-posed problems using (symmetric) Cauchy-like preconditioners

Misha E. Kilmer

Proceedings Volume 3461, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII; (1998) https://doi.org/10.1117/12.325698
Event: SPIE's International Symposium on Optical Science, Engineering, and Instrumentation, 1998, San Diego, CA, United States

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.

Citation Download Citation

Misha E. Kilmer "Regularization of ill-posed problems using (symmetric) Cauchy-like preconditioners", Proc. SPIE 3461, Advanced Signal Processing Algorithms, Architectures, and Implementations VIII, (2 October 1998); https://doi.org/10.1117/12.325698

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