{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,21]],"date-time":"2025-12-21T06:26:18Z","timestamp":1766298378743,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T00:00:00Z","timestamp":1742428800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"INdAM-GNCS 2024 Project \u201cTecniche numeriche per problemi di grandi dimensioni\u201d","award":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"],"award-info":[{"award-number":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"]}]},{"name":"European Union\u2014NextGenerationEU and by the Italian Ministry of University and Research (MUR)","award":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"],"award-info":[{"award-number":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"]}]},{"name":"European Union\u2014NextGenerationEU and by the Italian Ministry of University and Research (MUR)","award":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"],"award-info":[{"award-number":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"]}]},{"name":"Fondazione di Sardegna, Progetto biennale bando 2021, and \u201cComputational Methods and Networks in Civil Engineering (COMANCHE)\u201d","award":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"],"award-info":[{"award-number":["CUP_ E53C24001950001","P2022PMEN2","2022ANC8HL"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider the problem of computed tomography (CT). This ill-posed inverse problem arises when one wishes to investigate the internal structure of an object with a non-invasive and non-destructive technique. This problem is severely ill-conditioned, meaning it has infinite solutions and is extremely sensitive to perturbations in the collected data. This sensitivity produces the well-known semi-convergence phenomenon if iterative methods are used to solve it. In this work, we propose a multigrid approach to mitigate this instability and produce fast, accurate, and stable algorithms starting from unstable ones. We consider, in particular, symmetric Krylov methods, like lsqr, as smoother, and a symmetric projection of the coarse grid operator. However, our approach can be extended to any iterative method. Several numerical examples show the performance of our proposal.<\/jats:p>","DOI":"10.3390\/sym17030470","type":"journal-article","created":{"date-parts":[[2025,3,20]],"date-time":"2025-03-20T09:17:39Z","timestamp":1742462259000},"page":"470","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Multigrid Methods for Computed Tomography"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6456-4150","authenticated-orcid":false,"given":"Alessandro","family":"Buccini","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Cagliari, Via Ospedale 72, 09124 Cagliari, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7958-9126","authenticated-orcid":false,"given":"Marco","family":"Donatelli","sequence":"additional","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0006-6099-3261","authenticated-orcid":false,"given":"Marco","family":"Ratto","sequence":"additional","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,3,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Buzug, T.M. (2011). Computed tomography. Springer Handbook of Medical Technology, Springer.","DOI":"10.1007\/978-3-540-74658-4_16"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Hansen, P.C., J\u00f8rgensen, J., and Lionheart, W.R.B. (2021). Computed Tomography: Algorithms, Insight, and Just Enough Theory, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9781611976670"},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Engl, H.W., Hanke, M., and Neubauer, A. (1996). Regularization of Inverse Problems, Springer Science & Business Media.","DOI":"10.1007\/978-94-009-1740-8"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Hansen, P.C. (1998). Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion, SIAM.","DOI":"10.1137\/1.9780898719697"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hanke, M. (2017). 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