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## Stability of the IRLS algorithm in deconvolution

The results concerning the conditioning and the eigenvalues of the equations  (6) are unfortunately less accurate. The matrix ATWA is no longer Toeplitz. To compute its eigenvalues would require us to know the singular-value decomposition of A, which increases the cost of the computations.

I could only observe that ATA and ATWA have more or less the same condition number during the IRLS algorithm; consequently, the weighted least-squares problems are also ill-conditioned if the original power spectrum of the filter a is band-limited. However, I have no general results concerning the repartition of the eigenvalues of ATWA.

The choice of is not completely free. The synthetic examples I used showed that the more ill-conditioned the initial matrix ATA is, the larger should be taken, otherwise the IRLS algorithm does not converge. For example, with a condition number equal to 1000, I took , and the convergence (rate 1/10000) was reached in 10 iterations; with a condition number equal to 10000, I took , even with double precision. Increasing makes W closer to the identity matrix, in which case the problem corresponds more to a L2 minimization: we would be solving a mixed L1-L2 problem, as I will explain later.

In conclusion, I may expect the convergence of the CG algorithms to be slow if the power spectrum of the filter a contains some small values. Despite the suggestion to limit the number of iterations in order to cope with the ill-conditioning, I prefer to increase their number, to be closer to the numerical limits. However, the speed of convergence is greatly increased if it is possible to give a first estimate of the unknown x: that is the case in predictive deconvolution, where the first estimate would be the L2-Wiener filter.

Next: SYNTHETIC EXAMPLE Up: IRLS ALGORITHM APPLIED TO L Previous: Ill-conditioning of L deconvolution
Stanford Exploration Project
1/13/1998