A projection‐based algorithm for ℓ2‐ℓp Tikhonov regularization. (11th June 2018)

• Record Type: Journal Article
• Authors: Borges, Leonardo S.; Bazán, Fermín S.V.; Bedin, Luciano
• Other Names: Bakouch Hassan guestEditor.
• Is Part Of: Mathematical methods in the applied sciences. Volume 41:Number 15(2018)
• Page Start: 5919
• Page End: 5938
• Publication Date: 2018-06-11
• Subjects: discrete ill‐posed problems -- Tikhonov regularization -- ℓ2‐ℓp minimization; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.5110
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : 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.