{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,13]],"date-time":"2023-04-13T20:30:05Z","timestamp":1681417805661},"reference-count":48,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"Abstract<\/jats:title>\n In the present paper we propose and analyze a class of tensor approaches\nfor the efficient\nnumerical solution of a first order differential equation \n \n \n \n \n \n \n \u03c8<\/m:mi>\n \u2032<\/m:mo>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n +<\/m:mo>\n \n A<\/m:mi>\n \u2062<\/m:mo>\n \u03c8<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n =<\/m:mo>\n \n f<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n t<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\psi^{\\prime}(t)+A\\psi=f(t)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\nwith an unbounded operator coefficient A<\/jats:italic>.\nThese techniques are based on a Laguerre polynomial expansions with coefficients which are\npowers of the Cayley transform of the operator A<\/jats:italic>.\nThe Cayley transform under consideration is a useful tool to arrive at the following aims:\n(1) to separate time and spatial variables, (2) to switch from the\ncontinuous \u201ctime variable\u201d to \u201cthe discrete\ntime variable\u201d and from the study of functions of an unbounded operator to the ones of\na bounded operator, (3) to obtain exponentially accurate approximations.\nIn the earlier papers of the authors some approximations on the basis\nof the Cayley transform and the N<\/jats:italic>-term Laguerre expansions of the accuracy\norder \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n e<\/m:mi>\n \n -<\/m:mo>\n N<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(e^{-N})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> were proposed and\njustified provided that the initial value is analytical for A<\/jats:italic>.\nIn the present paper we combine the Cayley transform\nand the Chebyshev\u2013Gauss\u2013Lobatto interpolation and arrive at an approximation of the accuracy order\n\n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n e<\/m:mi>\n \n -<\/m:mo>\n N<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(e^{-N})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> without restrictions on the input data. The use of the Laguerre expansion or\nthe Chebyshev\u2013Gauss\u2013Lobatto interpolation allows to separate the time and space variables. The separation of\nthe multidimensional spatial variable can be achieved by the use of low-rank approximation to\nthe Cayley transform of the Laplace-like operator that is spectrally close to A<\/jats:italic>.\nAs a result a quasi-optimal numerical algorithm can be designed.<\/jats:p>","DOI":"10.1515\/cmam-2018-0021","type":"journal-article","created":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T11:10:23Z","timestamp":1531393823000},"page":"55-71","source":"Crossref","is-referenced-by-count":4,"title":["Quasi-Optimal Rank-Structured Approximation to Multidimensional Parabolic Problems by Cayley Transform and Chebyshev Interpolation"],"prefix":"10.1515","volume":"19","author":[{"given":"Ivan","family":"Gavrilyuk","sequence":"first","affiliation":[{"name":"University of Cooperative Education Gera\u2013Eisenach , Am Wartenberg 2, 99817 Eisenach , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boris N.","family":"Khoromskij","sequence":"additional","affiliation":[{"name":"Max-Planck-Institute for Mathematics in the Sciences , Inselstr. 22\u201326 , 04103 Leipzig , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,7,12]]},"reference":[{"key":"2023033110133761632_j_cmam-2018-0021_ref_001_w2aab3b7c18b1b6b1ab2ab1Aa","unstructured":"N. I. Akhiezer and I. M. Glazman,\nTheory of Linear Operators in Hilbert Space,\nDover, New York, 1993."},{"key":"2023033110133761632_j_cmam-2018-0021_ref_002_w2aab3b7c18b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"D. Z. Arov and I. P. Gavrilyuk,\nA method for solving initial value problems for linear differential equations in Hilbert space based on the Cayley transform,\nNumer. Funct. Anal. Optim. 14 (1993), no. 5\u20136, 459\u2013473.","DOI":"10.1080\/01630569308816534"},{"key":"2023033110133761632_j_cmam-2018-0021_ref_003_w2aab3b7c18b1b6b1ab2ab3Aa","unstructured":"D. Z. Arov, I. P. Gavrilyuk and V. L. Makarov,\nRepresentation and approximation of solutions of initial value problems for differential equations in Hilbert space based on the Cayley transform,\nElliptic and Parabolic Problems (Pont-\u00e0-Mousson 1994),\nPitman Res. Notes Math. Ser. 325,\nLongman Scientific & Technical, Harlow (1995), 40\u201350."},{"key":"2023033110133761632_j_cmam-2018-0021_ref_004_w2aab3b7c18b1b6b1ab2ab4Aa","unstructured":"H. Bateman and A. Erdelyi,\nHigher Transcendental Functions. Vol. 2,\nMc Graw-Hill, New York, 1988."},{"key":"2023033110133761632_j_cmam-2018-0021_ref_005_w2aab3b7c18b1b6b1ab2ab5Aa","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":"2023033110133761632_j_cmam-2018-0021_ref_006_w2aab3b7c18b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"P. Benner, V. Khoromskaia and B. N. Khoromskij,\nRange-separated tensor format for many-particle modeling,\nSIAM J. Sci. 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