{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,9]],"date-time":"2026-03-09T15:16:50Z","timestamp":1773069410462,"version":"3.50.1"},"reference-count":26,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2012,1,26]],"date-time":"2012-01-26T00:00:00Z","timestamp":1327536000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numerical Linear Algebra App"],"published-print":{"date-parts":[[2012,3]]},"abstract":"<jats:title>SUMMARY<\/jats:title><jats:p>A smoothed aggregation\u2010based algebraic multigrid solver for anisotropic diffusion problems is presented. Algebraic multigrid is a popular and effective method for solving sparse linear systems that arise from discretizing partial differential equations. However, although algebraic multigrid was designed for elliptic problems, the case of non\u2010grid\u2010aligned anisotropic diffusion is not adequately addressed by existing methods. To achieve scalable performance, it is shown that neither new coarsening nor new relaxation strategies are necessary. Instead, a novel smoothed aggregation approach is developed that combines long\u2010distance interpolation, coarse\u2010grid injection, and an energy\u2010minimization strategy that finds the interpolation weights. Previously developed theory by Falgout and Vassilevski is used to discern that existing coarsening strategies are sufficient, but that existing interpolation methods are not. In particular, an interpolation quality measure tracks \u2018closeness\u2019 to the ideal interpolant and guides the interpolation sparsity pattern choice. Although the interpolation quality measure is computable for only small model problems, an inexact, but computable, measure is proposed for larger problems. This paper concludes with encouraging numerical results that also potentially show broad applicability (e.g., for linear elasticity). Copyright \u00a9 2012 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.1805","type":"journal-article","created":{"date-parts":[[2012,1,26]],"date-time":"2012-01-26T06:17:25Z","timestamp":1327558645000},"page":"296-312","source":"Crossref","is-referenced-by-count":24,"title":["Smoothed aggregation solvers for anisotropic diffusion"],"prefix":"10.1002","volume":"19","author":[{"given":"Jacob B.","family":"Schroder","sequence":"first","affiliation":[{"name":"Center for Applied Scientific Computing Lawrence Livermore National Laboratory  L\u2010561 Livermore CA 94551 U.S.A."}]}],"member":"311","published-online":{"date-parts":[[2012,1,26]]},"reference":[{"key":"e_1_2_10_2_1","first-page":"257","volume-title":"Sparsity and Its Applications","author":"Brandt A","year":"1984"},{"key":"e_1_2_10_3_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611971057.ch4"},{"key":"e_1_2_10_4_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02238511"},{"key":"e_1_2_10_5_1","doi-asserted-by":"publisher","DOI":"10.1007\/s211-001-8015-y"},{"key":"e_1_2_10_6_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.480"},{"key":"e_1_2_10_7_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.669"},{"key":"e_1_2_10_8_1","volume-title":"Multigrid","author":"Trottenberg U","year":"2001"},{"key":"e_1_2_10_9_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719505"},{"key":"e_1_2_10_10_1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903429742"},{"key":"e_1_2_10_11_1","first-page":"1","article-title":"General highly accurate algebraic coarsening","volume":"10","author":"Brandt A","year":"2000","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"e_1_2_10_12_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.378"},{"key":"e_1_2_10_13_1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-540-34469-8_2"},{"key":"e_1_2_10_14_1","doi-asserted-by":"publisher","DOI":"10.1137\/100803031"},{"key":"e_1_2_10_15_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827596310998"},{"issue":"195","key":"e_1_2_10_16_1","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1090\/S0025-5718-1991-1079008-4","article-title":"Convergence estimates for multigrid algorithms without regularity assumptions","volume":"57","author":"Bramble JH","year":"1991","journal-title":"Mathematics of Computation"},{"key":"e_1_2_10_17_1","first-page":"106","article-title":"Multigrid algorithm with conditional coarsening for non\u2010aligned sonic flow","volume":"6","author":"Diskin B","year":"1997","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"e_1_2_10_18_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599361047"},{"key":"e_1_2_10_19_1","first-page":"591","volume-title":"Seventh Copper Mountain Conference on Multigrid Methods","author":"Morano E","year":"1996"},{"key":"e_1_2_10_20_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1998.6036"},{"key":"e_1_2_10_21_1","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1996.5442"},{"key":"e_1_2_10_22_1","doi-asserted-by":"publisher","DOI":"10.1007\/s006070050033"},{"key":"e_1_2_10_23_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.686"},{"key":"e_1_2_10_24_1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827594276552"},{"key":"e_1_2_10_25_1","unstructured":"BellWN OlsonLN SchroderJ. 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