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These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel \u201capproximate Tikhonov regularization method\u201d based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method.<\/jats:p>","DOI":"10.1007\/s11075-022-01407-7","type":"journal-article","created":{"date-parts":[[2022,10,19]],"date-time":"2022-10-19T13:04:20Z","timestamp":1666184660000},"page":"223-245","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["An Arnoldi-based preconditioner for iterated Tikhonov regularization"],"prefix":"10.1007","volume":"92","author":[{"ORCID":"http:\/\/orcid.org\/0000-0002-6456-4150","authenticated-orcid":false,"given":"Alessandro","family":"Buccini","sequence":"first","affiliation":[]},{"given":"Lucas","family":"Onisk","sequence":"additional","affiliation":[]},{"given":"Lothar","family":"Reichel","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,10,19]]},"reference":[{"key":"1407_CR1","unstructured":"Berisha, S., Nagy, J.G.: Iterative methods for image restoration. 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