{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:07:22Z","timestamp":1776802042998,"version":"3.51.2"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,3,28]],"date-time":"2017-03-28T00:00:00Z","timestamp":1490659200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["Sonderforschungsbereich SFB 611"],"award-info":[{"award-number":["Sonderforschungsbereich SFB 611"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We will discuss the spectral equivalence of hierarchical matrix approximations for second order elliptic problems. Our theory will show that a modified variant of the hierarchical matrix Cholesky decomposition which preserves test vectors while truncating blocks to lower rank will lead to a spectrally equivalent approximation when using an adapted truncation threshold. Our theory also covers the usual hierarchical Cholesky decomposition which does not preserve test vectors but expects a significantly more restrictive threshold adaption to obtain a spectrally equivalent approximation. Numerical experiments indicate that the adaption of the truncation parameter seems to be necessary for the traditional hierarchical Cholesky preconditioner to obtain mesh-independent convergence while the variant which preserves test vectors works in practice quite well even with a fixed parameter.<\/p>","DOI":"10.1090\/mcom\/3086","type":"journal-article","created":{"date-parts":[[2016,3,28]],"date-time":"2016-03-28T15:04:06Z","timestamp":1459177446000},"page":"2839-2861","source":"Crossref","is-referenced-by-count":6,"title":["On the spectral equivalence of hierarchical matrix preconditioners for elliptic problems"],"prefix":"10.1090","volume":"85","author":[{"given":"M.","family":"Bebendorf","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Bollh\u00f6fer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Bratsch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,3,28]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511624100","volume-title":"Iterative solution methods","author":"Axelsson, Owe","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0521445248"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"751","DOI":"10.4171\/ZAA\/1170","article-title":"A note on the Poincar\u00e9 inequality for convex domains","volume":"22","author":"Bebendorf, M.","year":"2003","journal-title":"Z. 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