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A projection‐based algorithm for ℓ 2 ‐ ℓ p Tikhonov regularization
Author(s) -
Borges Leonardo S.,
Bazán Fermín S.V.,
Bedin Luciano
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5110
Subject(s) - tikhonov regularization , mathematics , regularization (linguistics) , krylov subspace , algorithm , subspace topology , iterative method , mathematical optimization , inverse problem , mathematical analysis , computer science , artificial intelligence
This paper deals with the minimization problem associated with Tikhonov regularization where the penalty term is a ℓ p ‐norm with p ∈ (1,2). An algorithm based on fixed‐point iterations is proposed, and theoretical convergence results are given. Further, combining projection over the Krylov subspace generated by the Golub‐Kahan bidiagonalization procedure and the proposed iterative algorithm, we derive an iterative procedure that is well suited for large‐scale ℓ 2 ‐ ℓ p Tikhonov regularization. The performance and efficiency of the proposed algorithms on image restoration problems like 2D tomography and super resolution, as well as on classical discrete ill‐posed problems having solutions with flat and smooth parts or nonsmooth solutions are also illustrated.