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l1 regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the split Bregman and the majorization-minimization iterative methods that turn this nonsmooth minimization problem into a sequence of steps that include solving an -regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these iterative methods. The generalized cross validation method and chi2 degrees of freedom test are extended to these inner problems. In particular, for the chi2 test this includes extending the result for problems in which the regularization operator has more rows than columns and showing how to use the -weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and fixed to convergence.
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Dynamically accelerating the power iteration with momentum
In this article, we propose, analyze and demonstrate a dynamic momentum method to accelerate power and inverse power iterations with minimal computational overhead. The method can be applied to real diagonalizable matrices, is provably convergent with acceleration in the symmetric case, and does not require a priori spectral knowledge. We review and extend background results on previously developed static momentum accelerations for the power iteration through the connection between the momentum accelerated iteration and the standard power iteration applied to an augmented matrix. We show that the augmented matrix is defective for the optimal parameter choice. We then present our dynamic method which updates the momentum parameter at each iteration based on the Rayleigh quotient and two previous residuals. We present convergence and stability theory for the method by considering a power‐like method consisting of multiplying an initial vector by a sequence of augmented matrices. We demonstrate the developed method on a number of benchmark problems, and see that it outperforms both the power iteration and often the static momentum acceleration with optimal parameter choice. Finally, we present and demonstrate an explicit extension of the algorithm to inverse power iterations.
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- Award ID(s):
- 2045059
- PAR ID:
- 10626931
- Publisher / Repository:
- Wiley
- Date Published:
- Journal Name:
- Numerical Linear Algebra with Applications
- Volume:
- 31
- Issue:
- 6
- ISSN:
- 1070-5325
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1002/nla.2584
