{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T19:22:14Z","timestamp":1776280934540,"version":"3.50.1"},"reference-count":33,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We consider dynamical low-rank approximation on the manifold of fixed-rank matrices and tensor trains (also called matrix product states), and analyse projection methods for the time integration of such problems. First, under suitable approximability assumptions, we prove error estimates for the explicit Euler method equipped with quasi-optimal projections to the manifold. Then we discuss the possibilities and difficulties with higher-order explicit methods. In particular, we discuss ways for limiting rank growth in the increments, and robustness with respect to small singular values.<\/jats:p>","DOI":"10.1515\/cmam-2018-0029","type":"journal-article","created":{"date-parts":[[2018,7,21]],"date-time":"2018-07-21T22:15:48Z","timestamp":1532211348000},"page":"73-92","source":"Crossref","is-referenced-by-count":34,"title":["Projection Methods for Dynamical Low-Rank Approximation of High-Dimensional Problems"],"prefix":"10.1515","volume":"19","author":[{"given":"Emil","family":"Kieri","sequence":"first","affiliation":[{"name":"Hausdorff Center for Mathematics & Institute for Numerical Simulation , University of Bonn , Bonn , Germany"}]},{"given":"Bart","family":"Vandereycken","sequence":"additional","affiliation":[{"name":"Section of Mathematics , University of Geneva , Geneva , Switzerland"}]}],"member":"374","published-online":{"date-parts":[[2018,7,21]]},"reference":[{"key":"2023033110133784867_j_cmam-2018-0029_ref_001_w2aab3b7e2044b1b6b1ab2b2b1Aa","doi-asserted-by":"crossref","unstructured":"P.-A.  Absil and I. V.  Oseledets,\nLow-rank retractions: A survey and new results,\nComput. Optim. Appl. 62 (2015), no. 1, 5\u201329.","DOI":"10.1007\/s10589-014-9714-4"},{"key":"2023033110133784867_j_cmam-2018-0029_ref_002_w2aab3b7e2044b1b6b1ab2b2b2Aa","doi-asserted-by":"crossref","unstructured":"M. H.  Beck, A.  J\u00e4ckle, G. A.  Worth and H.-D.  Meyer,\nThe multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets,\nPhys. Rep. 324 (2000), 1\u2013105.","DOI":"10.1016\/S0370-1573(99)00047-2"},{"key":"2023033110133784867_j_cmam-2018-0029_ref_003_w2aab3b7e2044b1b6b1ab2b2b3Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess and W.  Hackbusch,\nApproximation of 1\/x1\/x by exponential sums in [1,\u221e)[1,\\infty),\nIMA J. Numer. 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