Directional convergence of spectral regularization method associated to families of closed operators
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Directional convergence of spectral regularization method associated to families of closed operators Gisela L. Mazzieri · Ruben D. Spies · Karina G. Temperini
Received: 7 May 2012 / Revised: 18 October 2012 / Accepted: 18 October 2012 / Published online: 4 April 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013
Abstract We consider regularized solutions of linear inverse ill-posed problems obtained with generalized Tikhonov–Phillips functionals with penalizers given by linear combinations of seminorms induced by closed operators. Convergence of the regularized solutions is proved when the vector regularization rule approaches the origin through appropriate radial and differentiable paths. Characterizations of the limiting solutions are given. Finally, examples of image restoration using generalized Tikhonov–Phillips methods with convex combinations of seminorms are shown.
Communicated by José Eduardo Souza de Cursi. This work was supported in part by Consejo Nacional de Investigaciones Científicas y Técnicas, CONICET, through PIP 2010-2012 Nro. 0219, by Universidad Nacional del Litoral, U.N.L., through project CAI+D 2009-PI-62-315, by Agencia Nacional de Promoción Científica y Tecnológia ANPCyT, through project PICT 2008-1301 and by the Air Force Office of Scientific Research, AFOSR, through Grant FA9550-10-1-0018. G. L. Mazzieri · R. D. Spies · K. G. Temperini Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Güemes 3450, S3000GLN Santa Fe, Argentina G. L. Mazzieri Departamento de Matemática, Facultad de Bioquímica y Ciencias Biológicas, Universidad Nacional del Litoral, Santa Fe, Argentina e-mail: [email protected] R. D. Spies (B) Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Santa Fe, Argentina e-mail: [email protected] K. G. Temperini Departamento de Matemática, Facultad de Humanidades y Ciencias, Universidad Nacional del Litoral, Santa Fe, Argentina e-mail: [email protected]
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Keywords Inverse problem · Ill-posed · Regularization · Tikhonov–Phillips, Closed operators Mathematics Subject Classification (2010)
47A52 · 65J20
1 Introduction Very often an inverse problem can be formulated as the necessity of approximating x in an equation of the form T x = y,
(1)
where T is a linear bounded operator between two infinite dimensional Hilbert spaces X and Y (in general, these will be function spaces), the range of T and R(T ) is non-closed and y is the data, supposed to be known, perhaps with a certain degree of error. It is well known that under these hypotheses, problem (1) is ill-posed in the sense of Hadamard (1902). In this case, the ill-posedness is a result of the unboundedness of T † , the Moore–Penrose pseudo inverse of T . The Moore–Penrose pseudo inverse is a fundamental tool in the treatment of inverse ill-posed problems and their regularized solutions. This is so mainly because the least-squares solution of minimum norm of problem (1
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