{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,23]],"date-time":"2026-09-23T23:53:17Z","timestamp":1790207597206,"version":"4.1.0"},"reference-count":36,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["23-FS-031"],"award-info":[{"award-number":["23-FS-031"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["24-ERD-033"],"award-info":[{"award-number":["24-ERD-033"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["DEAC52-07NA27344"],"award-info":[{"award-number":["DEAC52-07NA27344"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000105","name":"Office of Advanced Cyberinfrastructure","doi-asserted-by":"publisher","award":["OAC-2003720"],"award-info":[{"award-number":["OAC-2003720"]}],"id":[{"id":"10.13039\/100000105","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS-2208412"],"award-info":[{"award-number":["DMS-2208412"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["DMS 2038118"],"award-info":[{"award-number":["DMS 2038118"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2024,10,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in multigrid (MG) methods, addressing the well-known issue of increasing operator complexity. Guided by the MG theory on spectrally equivalent coarse-grid operators, we have developed novel machine learning algorithms that utilize neural networks combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall MG convergence for solving parametric partial differential equation problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and comparison with existing methods for computing non-Galerkin coarse-grid operators.<\/jats:p>\n                  <jats:p>Reproducibility of computational results. This paper has been awarded the \u201cSIAM Reproducibility Badge: Code and data available\u201d as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https:\/\/github.com\/liruipeng\/SparseCoarseOperator . [Formula: see text]<\/jats:p>","DOI":"10.1137\/23m1583533","type":"journal-article","created":{"date-parts":[[2024,9,6]],"date-time":"2024-09-06T04:51:05Z","timestamp":1725598265000},"page":"S296-S316","source":"Crossref","is-referenced-by-count":2,"title":["Reducing Operator Complexity of Galerkin Coarse-grid Operators with Machine Learning"],"prefix":"10.1137","volume":"46","author":[{"given":"Ru","family":"Huang","sequence":"first","affiliation":[{"name":"Department of Mathematics, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kai","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Huan","family":"He","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics 221 Parker Hall Auburn University Auburn, AL 36849 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2802-5763","authenticated-orcid":true,"given":"Ruipeng","family":"Li","sequence":"additional","affiliation":[{"name":"Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, L-561, Livermore, CA 94551 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0361-0931","authenticated-orcid":true,"given":"Yuanzhe","family":"Xi","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Emory University, Atlanta, GA 30322 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2024,9,6]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0069933"},{"key":"ref2","doi-asserted-by":"crossref","unstructured":"A. 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