{"id":211,"date":"2016-07-01T15:56:09","date_gmt":"2016-07-01T13:56:09","guid":{"rendered":"http:\/\/project.inria.fr\/gatipor\/?p=211"},"modified":"2021-12-02T19:07:38","modified_gmt":"2021-12-02T18:07:38","slug":"laplace-eigenvalue-and-eigenvector-bounds","status":"publish","type":"post","link":"https:\/\/project.inria.fr\/gatipor\/laplace-eigenvalue-and-eigenvector-bounds\/","title":{"rendered":"Guaranteed eigenvalue and eigenvector bounds"},"content":{"rendered":"<table style=\"width:100%\">\n<tr>\n<td><div id=\"attachment_140\" style=\"width: 510px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_conf_str.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_conf_str.jpg\" alt=\"eigs_conf_str\" width=\"740\" height=\"757\" class=\"alignnone size-full wp-image-222\" srcset=\"https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_conf_str.jpg 740w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_conf_str-293x300.jpg 293w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_conf_str-147x150.jpg 147w\" sizes=\"auto, (max-width: 740px) 100vw, 740px\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\">First <strong>eigenvalue<\/strong> and <strong>eigenvector bounds<\/strong> on a square (structured mesh, 1st order conforming finite elements)<\/p><\/div><\/td>\n<td><div id=\"attachment_142\" style=\"width: 510px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/DG_12.png\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-142\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/DG_12.png\" alt=\"eigs_DG_adapt\" width=\"740\" height=\"755\" class=\"alignnone size-full wp-image-223\" \/><\/a><p id=\"caption-attachment-142\" class=\"wp-caption-text\">First <strong>eigenvector bounds<\/strong> on an L-shaped domain (adaptively refined mesh, 1st and 2nd order discontinuous Galerkin elements)<\/p><\/div><\/td>\n<\/tr>\n<tr>\n<td><div id=\"attachment_140\" style=\"width: 510px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh-1022x1024.jpg\" alt=\"eigs_hole_mesh\" width=\"900\" height=\"902\" class=\"alignnone size-large wp-image-216\" srcset=\"https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh-1022x1024.jpg 1022w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh-150x150.jpg 150w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh-300x300.jpg 300w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh-768x770.jpg 768w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_mesh.jpg 1641w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\">Adaptively refined mesh, domain with a hole<\/p><\/div><\/td>\n<td><div id=\"attachment_140\" style=\"width: 510px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl-1024x1024.jpg\" alt=\"eigs_hole_incl\" width=\"900\" height=\"900\" class=\"alignnone size-large wp-image-215\" srcset=\"https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl-1024x1024.jpg 1024w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl-150x150.jpg 150w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl-300x300.jpg 300w, https:\/\/project.inria.fr\/gatipor\/files\/2016\/07\/eigs_hole_incl-768x768.jpg 768w\" sizes=\"auto, (max-width: 900px) 100vw, 900px\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\"><strong>Guaranteed inclusion<\/strong> of the first eigenvalue, domain with a hole<\/p><\/div><\/td>\n<\/tr>\n<\/table>\n<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td><div id=\"attachment_140\" style=\"width: 1110px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/clusters.png\"><img decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/clusters.png\" alt=\"clusters\" width=\"1100\" class=\"alignnone size-large wp-image-452\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\"><strong>Eigenvalue<\/strong> and <strong>eigenvector bounds<\/strong>, <strong>2-cluster<\/strong> on a square (left) and <strong>3-cluster<\/strong> on an L-shaped domain (right), 1st order conforming finite elements<\/p><\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Main results<\/strong>:<\/p>\n<ul>\n<li><strong>guaranteed<\/strong> upper and lower <strong>bounds<\/strong> for an <strong>arbitrary simple eigenvalue<\/strong>, <strong>multiple eigenvalue<\/strong>, or a <strong>cluster of eigenvalues<\/strong>;<\/li>\n<li><strong>guaranteed energy error<\/strong> bounds on the associated <strong>eigenvector(s)<\/strong>;<\/li>\n<li>efficiency (optimal convergence rate) and <strong>polynomial-degree robustness<\/strong>;<\/li>\n<li>multiplicative constant tending to one under an elliptic regularity assumption on the corresponding<br \/>\nsource problem;<\/li>\n<li>inexact algebraic solvers taken into account;<\/li>\n<li><strong>unified framework<\/strong> covering basically any numerical method (conforming, nonconforming, discontinuous Galerkin, and mixed finite elements);<\/li>\n<li>abstract framework for any <strong>second-order self-adjoint elliptic linear operator<\/strong> with <strong>compact resolvent<\/strong>.<\/li>\n<\/ul>\n<p>Details: <\/p>\n<ul>\n<li>with Eric Canc\u00e8s, Genevi\u00e8ve Dusson, Yvon Maday, and Benjamin Stamm;<\/li>\n<li>conforming setting, simple eigenvalue: <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Pub_SINUM_17_a.pdf\"> paper<\/a>;<\/li>\n<li>unified framework, simple eigenvalue: <a href=\"https:\/\/link.springer.com\/article\/10.1007%2Fs00211-018-0984-0\">paper<\/a> (<a href=\"https:\/\/hal.inria.fr\/hal-01483461\">preprint<\/a>) and <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Exp_FoCM_17.pdf\">presentation<\/a>;<\/li>\n<li>conforming setting, multiple eigenvalues and clusters: <a href=\"https:\/\/www.ams.org\/journals\/mcom\/2020-89-326\/S0025-5718-2020-03549-2\/\">paper<\/a> (<a href=\"https:\/\/hal.inria.fr\/hal-02127954\">preprint<\/a>), <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Exp_Berlin_19.pdf\">presentation<\/a>.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Main results: guaranteed upper and lower bounds for an arbitrary simple eigenvalue, multiple eigenvalue, or a cluster of eigenvalues; guaranteed energy error bounds on the associated eigenvector(s); efficiency (optimal convergence rate) and polynomial-degree robustness; multiplicative constant tending to one under an elliptic regularity assumption on the corresponding source problem; inexact\u2026<\/p>\n<p> <a class=\"continue-reading-link\" href=\"https:\/\/project.inria.fr\/gatipor\/laplace-eigenvalue-and-eigenvector-bounds\/\"><span>Continue reading<\/span><i class=\"crycon-right-dir\"><\/i><\/a> <\/p>\n","protected":false},"author":932,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-211","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/211","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=211"}],"version-history":[{"count":25,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/211\/revisions"}],"predecessor-version":[{"id":1045,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/211\/revisions\/1045"}],"wp:attachment":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/categories?post=211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/tags?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}