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A COMBINED WAVELET-SPECTRAL TWO-STEP REGULARIZATIONMETHOD FOR LINEAR ILL-POSED PROBLEMS María F. Acosta, Gisela L. Mazzieriy and Ruben D. Spies Instituto de Matemática Aplicada del Litoral, IMAL, CONICET-UNL, Santa Fe, Argentina.In this work we consider regularization methods for linear inverse problems of the form Tf = g where T is a bounded linear operator with non-closed range between two infinite dimensional Hilbert spaces X and Y. Here f is the unknown and g represents the data of the problem, which is supposed to be known or approximately known, with a certain degree of error. Under these hypotheses the problem is ill-posed, which translates into the unboundedness of Ty, the pseudo-inverse of T. Solving this problem necessarily requiresof some type of regularization at some step of the process. There is a wide variety of regularization methodsin the literature such as Tikhonov-Phillips, Lavrentiev, Truncated Singular Value Decomposition (TSVD), Landweber iterations, etc., to name just a few (Engl et al 1996, Regularization of inverse problems, vol. 375 of Mathematics and its Applications). Although all of them perform well under appropriate conditions,good performance in certain type of problems requires of more ad-hoc, versatile and adaptive methods.Thus, for example, combined (Mazzieri et al 2012, DOI 10.1016/j.jmaa.2012.06.039) as well as adaptive penalization methods which allow for the inclusion of a-priori and data-driven information have recently appeared (Ibarrola et al 2017, DOI 10.1016/j.jmaa.2017.01.00) giving rise to many new tools for tacklingparticular types of heterogeneous and severely ill-posed problems. In a similar manner, the use of wavelet methods has provided new insights and more versatility into some regularization tools (Dicken et al 1996, DOI 10.1515/jiip.1996.4.3.203 and Donoho 1995, DOI 10.1006/acha.1995.1008). In 2006 Klann et al (Klann et al 2006, DOI 10.1515/156939406778474523) introduced a hybrid approach combining the classical Tikhonov-Phillips method with wavelet decomposition. A similar approach is followed in this work, in which adhering to the philosophy of capturing the advantages of both the classical and modern ideas, a two-step wavelet-spectral method is presented. A wavelet expansion followed by thresholding constitutes the first step while the second consists on the application of a spectral regularization method having appropriate classical qualification. Orders of convergence, in terms of the noise level, are obtained and it is proved that under certain general hypotheses these two-step methods yield optimal orders.Finally some numerical experiments are presented to show the performance of the methods.