{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,14]],"date-time":"2026-04-14T05:44:30Z","timestamp":1776145470929,"version":"3.50.1"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We propose a new method for the efficient approximation of a class of\nhighly oscillatory weighted integrals where the oscillatory function\ndepends on the frequency parameter <jats:inline-formula id=\"eq1_w2aab3b7c15b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2015-0033_787f60e25dad9bdaa354645bdac4de8d.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>\u03c9<\/m:mi>\n                              <m:mo>\u2265<\/m:mo>\n                              <m:mn>0<\/m:mn>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>${\\omega \\ge 0}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, typically varying in a large interval.\nOur approach is based, for a fixed but arbitrary oscillator,\non the pre-computation and low-parametric approximation of certain \u03c9-dependent\nprototype functions whose evaluation leads in a straightforward way to recover\nthe target integral.\nThe difficulty that arises is that these prototype\nfunctions consist of oscillatory integrals which makes them difficult to evaluate.\nFurthermore, they have to be approximated typically in large intervals.\nHere we use the quantized-tensor train (QTT) approximation method for functional <jats:italic>M<\/jats:italic>-vectors\nof logarithmic complexity in <jats:italic>M<\/jats:italic> in combination with a cross-approximation scheme for TT tensors.\nThis allows the accurate approximation and efficient storage of these functions in\nthe wide range of grid and frequency parameters.\nNumerical examples illustrate the efficiency of the QTT-based numerical integration\nscheme on various examples in one and several spatial dimensions.<\/jats:p>","DOI":"10.1515\/cmam-2015-0033","type":"journal-article","created":{"date-parts":[[2015,11,26]],"date-time":"2015-11-26T17:01:30Z","timestamp":1448557290000},"page":"145-159","source":"Crossref","is-referenced-by-count":8,"title":["Efficient Computation of Highly Oscillatory Integrals by Using QTT Tensor Approximation"],"prefix":"10.1515","volume":"16","author":[{"given":"Boris","family":"Khoromskij","sequence":"first","affiliation":[{"name":"Max-Planck-Institut f\u00fcr Mathematik in den Naturwissenschaften, Inselstr. 22\u201326, 04103 Leipzig, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Veit","sequence":"additional","affiliation":[{"name":"Department of Computer Science, University of Chicago, 1100 East 58th Street, Chicago, IL 60637, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2015,11,26]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/16\/1\/article-p145.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0033\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0033\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:34:09Z","timestamp":1680298449000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2015-0033\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,11,26]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2016,1,1]]},"published-print":{"date-parts":[[2016,1,1]]}},"alternative-id":["10.1515\/cmam-2015-0033"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2015-0033","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2015,11,26]]}}}