{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:48:19Z","timestamp":1760143699846,"version":"build-2065373602"},"reference-count":21,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T00:00:00Z","timestamp":1708905600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"European High-Performance Computing Joint Undertaking (JU)","award":["955701"],"award-info":[{"award-number":["955701"]}]},{"name":"Laboratory of Theory, Economics and Systems\u2014Department of Computer Science at Athens University of Economics and Business","award":["955701"],"award-info":[{"award-number":["955701"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We consider the Helmholtz equation and the fractional Laplacian in the case of the complex-valued unbounded variable coefficient wave number \u03bc, approximated by finite differences. In a recent analysis, singular value clustering and eigenvalue clustering have been proposed for a \u03c4 preconditioning when the variable coefficient wave number \u03bc is uniformly bounded. Here, we extend the analysis to the unbounded case by focusing on the case of a power singularity. Several numerical experiments concerning the spectral behavior and convergence of the related preconditioned GMRES are presented.<\/jats:p>","DOI":"10.3390\/a17030100","type":"journal-article","created":{"date-parts":[[2024,2,26]],"date-time":"2024-02-26T06:50:23Z","timestamp":1708930223000},"page":"100","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Clustering\/Distribution Analysis and Preconditioned Krylov Solvers for the Approximated Helmholtz Equation and Fractional Laplacian in the Case of Complex-Valued, Unbounded Variable Coefficient Wave Number \u03bc"],"prefix":"10.3390","volume":"17","author":[{"given":"Andrea","family":"Adriani","sequence":"first","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[{"name":"Department of Science and High Technology, University of Insubria, Via Valleggio 11, 22100 Como, Italy"},{"name":"Division of Scientific Computing, Department of Information Technology, Uppsala University, L\u00e4gerhyddsv 2, hus 2, SE-751 05 Uppsala, Sweden"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cristina","family":"Tablino-Possio","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Applications, University of Milano-Bicocca, Via Cozzi 53, 20125 Milano, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,2,26]]},"reference":[{"unstructured":"Adriani, A., Sormani, R.L., Tablino-Possio, C., Krause, R., and Serra-Capizzano, S. (2024). Asymptotic spectral properties and preconditioning of an approximated nonlocal Helmholtz equation with Caputo fractional Laplacian and variable coefficient wave number \u03bc. arXiv.","key":"ref_1"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1007\/s10915-023-02332-0","article-title":"Preconditioning technique based on sine transformation for nonlocal Helmholtz equations with fractional Laplacian","volume":"97","author":"Li","year":"2023","journal-title":"J. Sci. Comput."},{"doi-asserted-by":"crossref","unstructured":"Garoni, C., and Serra-Capizzano, S. (2018). 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