TY - JOUR
T1 - Hybrid and iteratively reweighted regularization by unbiased predictive risk and weighted gcv for projected systems
AU - Renaut, Rosemary
AU - Vatankhah, Saeed
AU - Ardestani, Vahid E.
N1 - Funding Information:
This work was supported by NSF grants 1216559 and 1418377, and AFOSR grant 025717.
Publisher Copyright:
© 2017 Societ y for Industrial and Applied Mathematics.
PY - 2017
Y1 - 2017
N2 - Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization, parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation, and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate which can be used for a full problem that is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are overor underdetermined. Numerical simulations for standard one-dimensional test problems and twodimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem [I. Hnětynková, M. Plěsinger, and Z. Strakoš, BIT Numer. Math., 49(2009), pp. 669-696]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.
AB - Tikhonov regularization for projected solutions of large-scale ill-posed problems is considered. The Golub-Kahan iterative bidiagonalization is used to project the problem onto a subspace and regularization then applied to find a subspace approximation to the full problem. Determination of the regularization, parameter for the projected problem by unbiased predictive risk estimation, generalized cross validation, and discrepancy principle techniques is investigated. It is shown that the regularized parameter obtained by the unbiased predictive risk estimator can provide a good estimate which can be used for a full problem that is moderately to severely ill-posed. A similar analysis provides the weight parameter for the weighted generalized cross validation such that the approach is also useful in these cases, and also explains why the generalized cross validation without weighting is not always useful. All results are independent of whether systems are overor underdetermined. Numerical simulations for standard one-dimensional test problems and twodimensional data, for both image restoration and tomographic image reconstruction, support the analysis and validate the techniques. The size of the projected problem is found using an extension of a noise revealing function for the projected problem [I. Hnětynková, M. Plěsinger, and Z. Strakoš, BIT Numer. Math., 49(2009), pp. 669-696]. Furthermore, an iteratively reweighted regularization approach for edge preserving regularization is extended for projected systems, providing stabilization of the solutions of the projected systems and reducing dependence on the determination of the size of the projected subspace.
KW - Discrepancy principle
KW - Generalized cross validation
KW - Golub-Kahan bidiagonalization
KW - Iteratively reweighted schemes
KW - Large-scale inverse problems
KW - Regularization parameter estimation
KW - Unbiased predictive risk estimator
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U2 - 10.1137/15M1037925
DO - 10.1137/15M1037925
M3 - Article
AN - SCOPUS:85018277200
VL - 39
SP - B221-B243
JO - SIAM Journal of Scientific Computing
JF - SIAM Journal of Scientific Computing
SN - 1064-8275
IS - 2
ER -