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 -