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Matrix Anal. Appl."],"published-print":{"date-parts":[[2025,12,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>In recent years, there has been a renewed interest in preconditioning for multilevel Toeplitz systems, a research field that has been extensively explored over the past several decades. This work introduces novel preconditioning strategies using multilevel [Formula: see text] matrices for both symmetric and nonsymmetric multilevel Toeplitz systems. Our proposals constitute a general framework, as they are constructed solely based on the generating function of the multilevel Toeplitz coefficient matrix, when it can be defined. We begin with nonsymmetric systems, where we employ a symmetrization technique by permuting the coefficient matrix to produce a real symmetric multilevel Hankel structure. We propose a multilevel [Formula: see text] preconditioner tailored to the symmetrized system and prove that the eigenvalues of the preconditioned matrix sequence cluster at [Formula: see text], leading to rapid convergence when using the preconditioned minimal residual method. The high effectiveness of this approach is demonstrated through its application in solving space fractional diffusion equations. Next, for symmetric systems we introduce another multilevel [Formula: see text] preconditioner and show that the preconditioned conjugate gradient method can achieve an optimal convergence rate, namely a rate that is independent of the matrix size, when employed for a class of ill-conditioned multilevel Toeplitz systems. Numerical examples are provided to critically assess the effectiveness of our proposed preconditioners compared to several leading existing preconditioned solvers, highlighting their superior performance. (This article has been updated.)<\/jats:p>","DOI":"10.1137\/24m1702088","type":"journal-article","created":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T07:35:37Z","timestamp":1761118537000},"page":"2331-2359","source":"Crossref","is-referenced-by-count":7,"title":["Symbol-Based Multilevel Block \\(\\tau\\) Preconditioners for Multilevel Block Toeplitz Systems: GLT-Based Analysis and Applications"],"prefix":"10.1137","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5124-8776","authenticated-orcid":true,"given":"Sean Y.","family":"Hon","sequence":"first","affiliation":[{"name":"Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8654-2614","authenticated-orcid":true,"given":"Congcong","family":"Li","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rosita L.","family":"Sormani","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, University of Cagliari, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rolf","family":"Krause","sequence":"additional","affiliation":[{"name":"Center for Computational Medicine in Cardiology, University of Italian Switzerland, Switzerland."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Italy."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,10,22]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1553\/etna_vol53s28"},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1553\/etna_vol53s113"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2286"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(83)90009-5"},{"key":"ref5","doi-asserted-by":"crossref","unstructured":"D. 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