{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T03:47:54Z","timestamp":1777002474664,"version":"3.51.4"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Comput. Methods Appl. Math."],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider elliptic PDE eigenvalue problems on a tensorized domain, discretized such that \n\t\t\tthe resulting matrix eigenvalue problem Ax=\u03bbx exhibits Kronecker product structure. In \n\t\t\tparticular, we are concerned with the case of high dimensions, where standard approaches to \n\t\t\tthe solution of matrix eigenvalue problems fail due to the exponentially growing degrees of \n\t\t\tfreedom. Recent work shows that this curse of dimensionality can in many cases be addressed \n\t\t\tby approximating the desired solution vector x in a low-rank tensor format. In this paper, we \n\t\t\tuse the hierarchical Tucker decomposition to develop a low-rank variant of LOBPCG, a classical \n\t\t\tpreconditioned eigenvalue solver. We also show how the ALS and MALS (DMRG) methods known from \n\t\t\tcomputational quantum physics can be adapted to the hierarchical Tucker decomposition. \n\t\t\tFinally, a combination of ALS and MALS with LOBPCG and with our low-rank variant is proposed. \n\t\t\tA number of numerical experiments indicate that such combinations represent the methods of \n\t\t\tchoice.<\/jats:p>","DOI":"10.2478\/cmam-2011-0020","type":"journal-article","created":{"date-parts":[[2013,4,15]],"date-time":"2013-04-15T15:48:24Z","timestamp":1366040904000},"page":"363-381","source":"Crossref","is-referenced-by-count":49,"title":["Preconditioned Low-Rank Methods for High-Dimensional Elliptic PDE Eigenvalue Problems"],"prefix":"10.2478","volume":"11","author":[{"given":"Daniel","family":"Kressner","sequence":"first","affiliation":[{"name":"1Chair of Numerical Algorithms and HPC, MATHICSE, EPF Lausanne, CH-1015 Lausanne, Switzerland."}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christine","family":"Tobler","sequence":"additional","affiliation":[{"name":"1Chair of Numerical Algorithms and HPC, MATHICSE, EPF Lausanne, CH-1015 Lausanne, Switzerland."}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/11\/3\/article-p363.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.2478\/cmam-2011-0020\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,4,22]],"date-time":"2021-04-22T14:21:58Z","timestamp":1619101318000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/11\/3\/article-p363.xml"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":0,"journal-issue":{"issue":"3"},"URL":"https:\/\/doi.org\/10.2478\/cmam-2011-0020","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2011]]}}}