{"id":764,"date":"2019-10-02T11:09:46","date_gmt":"2019-10-02T09:09:46","guid":{"rendered":"http:\/\/project.inria.fr\/gatipor\/?p=764"},"modified":"2021-12-02T18:42:07","modified_gmt":"2021-12-02T17:42:07","slug":"p-robust-multilevel-algebraic-error-estimator-solver","status":"publish","type":"post","link":"https:\/\/project.inria.fr\/gatipor\/p-robust-multilevel-algebraic-error-estimator-solver\/","title":{"rendered":"<em>p<\/em>-robust multilevel algebraic error estimator &#038; solver"},"content":{"rendered":"<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td>\n<p><div id=\"attachment_140\" style=\"width: 910px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/MG.png\"><img decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/MG.png\" alt=\"MG\" width=\"900\" class=\"alignnone size-large wp-image-452\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\">Comparative performance (number of iterations and elapsed time) of the a-posteriori-steered <em>p<\/em>-robust multilevel solvers<\/p><\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Main results<\/strong>:<\/p>\n<ul>\n<li>multilevel <strong>a posteriori estimator<\/strong> of the algebraic error;<\/li>\n<li><strong>guaranteed<\/strong> (reliable) and efficient:<br \/>\n<a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_rel.jpg\"><img decoding=\"async\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_rel.jpg\" alt=\"reliablity\" width=\"400\" class=\"alignnone size-large wp-image-452\" \/><\/a>,<br \/>\n<a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_eff.jpg\"><img decoding=\"async\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_eff.jpg\" alt=\"efficiency\" width=\"400\" class=\"alignnone size-large wp-image-452\" \/><\/a>;\n<\/li>\n<li><strong>polynomial-degree robust<\/strong>: <em>&beta;<\/em> independent of <em>p<\/em>;<\/li>\n<li>gives rise to a multilevel <strong>iterative algebraic solver<\/strong> with <strong>contraction independent<\/strong> of the polynomial degree <em>p<\/em>:<br \/>\n<a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_contr.jpg\"><img decoding=\"async\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/alg_contr.jpg\" alt=\"contraction\" width=\"650\" class=\"alignnone size-large wp-image-452\" \/><\/a>;\n<\/li>\n<li>equivalence estimator &ndash; solver: <em>&alpha;<\/em> = <span>&#8730; (1 &#8211; <em>&beta;<\/em><sup>2<\/sup>)<\/span>;<\/li>\n<li>adaptive number of smoothing steps \/ adaptive smoothing possible.<\/li>\n<\/ul>\n<p><strong>Construction<\/strong>:<\/p>\n<ul>\n<li><strong>global<\/strong> coarsest-level solve of <strong>lowest-order<\/strong> (<em>p<\/em>=1);<\/li>\n<li><strong>local<\/strong> patchwise contributions from other levels;<\/li>\n<li>corresponds to <em>V<\/em>-cycle <strong>geometric multigrid<\/strong> with <strong>zero pre<\/strong>&#8211; and <strong>one<\/strong> (or <strong>adaptive<\/strong> number of) <strong>post-smoothing<\/strong> steps by block-Jacobi;<\/li>\n<li><strong>optimal step size<\/strong> for the<strong> descent direction<\/strong>.<\/li>\n<\/ul>\n<p>Details in <a href=\"https:\/\/tel.archives-ouvertes.fr\/tel-03152913\/document\"> Ph.D. thesis of Ani Mira\u00e7i<\/a>, <a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2021\/12\/Thesis_defense_Miraci.pdf\"> Ph.D. defense presentation<\/a>; with Jan Pape\u017e:<\/p>\n<ul>\n<li>basic theory: <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Pub_SINUM_20.pdf\">paper<\/a>;<\/li>\n<li>adaptive number of smoothing steps: <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Pub_SISC_21.pdf\">paper<\/a>;<\/li>\n<li>adaptive smoothing: <a href=\"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2020-0024\/html\">paper<\/a> (<a href=\"https:\/\/hal.inria.fr\/hal-02498247\">preprint<\/a>).<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Main results: multilevel a posteriori estimator of the algebraic error; guaranteed (reliable) and efficient: , ; polynomial-degree robust: &beta; independent of p; gives rise to a multilevel iterative algebraic solver with contraction independent of the polynomial degree p: ; equivalence estimator &ndash; solver: &alpha; = &#8730; (1 &#8211; &beta;2); adaptive\u2026<\/p>\n<p> <a class=\"continue-reading-link\" href=\"https:\/\/project.inria.fr\/gatipor\/p-robust-multilevel-algebraic-error-estimator-solver\/\"><span>Continue reading<\/span><i class=\"crycon-right-dir\"><\/i><\/a> <\/p>\n","protected":false},"author":932,"featured_media":1010,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-764","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/764","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/users\/932"}],"replies":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/comments?post=764"}],"version-history":[{"count":44,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/764\/revisions"}],"predecessor-version":[{"id":1031,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/764\/revisions\/1031"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/media\/1010"}],"wp:attachment":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/media?parent=764"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/categories?post=764"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/tags?post=764"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}