{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:55:38Z","timestamp":1776848138447,"version":"3.51.2"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"255","license":[{"start":{"date-parts":[[2007,3,8]],"date-time":"2007-03-08T00:00:00Z","timestamp":1173312000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    While there exist effective methods for univariate highly oscillatory quadrature, this is not the case in a multivariate setting. In this paper we embark on a project, extending univariate theory to more variables.\n                    <italic>Inter alia,<\/italic>\n                    we demonstrate that, in the absence of critical points and subject to a nonresonance condition, an integral over a simplex can be expanded asymptotically using only function values and derivatives at the vertices, a direct counterpart of the univariate case. This provides a convenient avenue towards the generalization of asymptotic and Filon-type methods, as formerly introduced by the authors in a single dimension, to simplices and, more generally, to polytopes. The nonresonance condition is bound to be violated once the boundary of the domain of integration is smooth: in effect, its violation is equivalent to the presence of stationary points in a single dimension. We further explore this issue and propose a technique that often can be used in this situation. Yet, much remains to be done to understand more comprehensively the influence of resonance on the asymptotics of highly oscillatory integrals.\n                  <\/p>","DOI":"10.1090\/s0025-5718-06-01854-0","type":"journal-article","created":{"date-parts":[[2006,5,24]],"date-time":"2006-05-24T14:43:01Z","timestamp":1148481781000},"page":"1233-1258","source":"Crossref","is-referenced-by-count":54,"title":["Quadrature methods for multivariate highly oscillatory integrals using derivatives"],"prefix":"10.1090","volume":"75","author":[{"given":"Arieh","family":"Iserles","sequence":"first","affiliation":[]},{"given":"Syvert","family":"N\u00f8rsett","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2006,3,8]]},"reference":[{"key":"1","unstructured":"[DS03] I. Degani and J. Schiff, RCMS: Right correction Magnus series approach for integration of linear ordinary differential equations with highly oscillatory terms, Tech. report, Weizmann Institute of Science, 2003."},{"issue":"2057","key":"2","doi-asserted-by":"publisher","first-page":"1383","DOI":"10.1098\/rspa.2004.1401","article-title":"Efficient quadrature of highly oscillatory integrals using derivatives","volume":"461","author":"Iserles, Arieh","year":"2005","journal-title":"Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1364-5021","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"[IN05b] \\bysame, On quadrature methods for highly oscillatory integrals and their implementation, BIT 44 (2005), 755\u2013772.","DOI":"10.1007\/s10543-004-5243-3"},{"key":"4","series-title":"Cambridge Texts in Applied Mathematics","isbn-type":"print","volume-title":"A first course in the numerical analysis of differential equations","author":"Iserles, Arieh","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0521553768"},{"issue":"1-2","key":"5","doi-asserted-by":"publisher","first-page":"145","DOI":"10.1016\/S0168-9274(02)00122-8","article-title":"Think globally, act locally: solving highly-oscillatory ordinary differential equations","volume":"43","author":"Iserles, Arieh","year":"2002","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1023\/B:BITN.0000046810.25353.95","article-title":"On the method of Neumann series for highly oscillatory equations","volume":"44","author":"Iserles, A.","year":"2004","journal-title":"BIT","ISSN":"https:\/\/id.crossref.org\/issn\/0006-3835","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"365","DOI":"10.1093\/imanum\/24.3.365","article-title":"On the numerical quadrature of highly-oscillating integrals. I. Fourier transforms","volume":"24","author":"Iserles, Arieh","year":"2004","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1093\/imanum\/drh022","article-title":"On the numerical quadrature of highly-oscillating integrals. II. Irregular oscillators","volume":"25","author":"Iserles, Arieh","year":"2005","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1016\/0377-0427(94)00118-9","article-title":"Fast integration of rapidly oscillatory functions","volume":"67","author":"Levin, David","year":"1996","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"10","isbn-type":"print","volume-title":"Analysis on manifolds","author":"Munkres, James R.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0201510359"},{"key":"11","series-title":"Computer Science and Applied Mathematics","volume-title":"Asymptotics and special functions","author":"Olver, F. W. J.","year":"1974"},{"key":"12","unstructured":"[Olv05] S. Olver, Moment-free numerical integration of highly oscillatory functions, Tech. Report NA2005\/04, DAMTP, University of Cambridge, 2005."},{"key":"13","series-title":"Princeton Mathematical Series","isbn-type":"print","volume-title":"Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals","volume":"43","author":"Stein, Elias M.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0691032165"},{"key":"14","series-title":"DMV Seminar","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-0348-7630-8","volume-title":"Mathematical theory of finite and boundary element methods","volume":"15","author":"Schatz, Albert H.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/376432211X"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01854-0\/S0025-5718-06-01854-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01854-0\/S0025-5718-06-01854-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T14:36:15Z","timestamp":1776782175000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2006-75-255\/S0025-5718-06-01854-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2006,3,8]]},"references-count":14,"journal-issue":{"issue":"255","published-print":{"date-parts":[[2006,7]]}},"alternative-id":["S0025-5718-06-01854-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-06-01854-0","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2006,3,8]]}}}