{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:25:16Z","timestamp":1787333116628,"version":"build-2736575974"},"reference-count":24,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2010,1]]},"abstract":"<jats:p>We develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by Smolyak's construction based on a nested quadrature. More precisely, we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis; and by using the aforementioned fast transform, we construct two very efficient sparse spectral-Galerkin methods for a model elliptic equation. In particular, the Chebyshev\u2013Legendre\u2013Galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low-order sparse grid methods based on finite elements or wavelets, and can be efficiently solved by a suitable sparse solver. Ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms.<\/jats:p>","DOI":"10.1137\/100787842","type":"journal-article","created":{"date-parts":[[2010,11,9]],"date-time":"2010-11-09T18:51:20Z","timestamp":1289328680000},"page":"3228-3250","source":"Crossref","is-referenced-by-count":71,"title":["Efficient Spectral Sparse Grid Methods and Applications to High-Dimensional Elliptic Problems"],"prefix":"10.1137","volume":"32","author":[{"given":"Jie","family":"Shen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Haijun","family":"Yu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2010,11,9]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827593247035"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018977404843"},{"key":"R3","unstructured":"H.J. Bungartz,\n                      An adaptive Poisson solver using hierarchical bases and sparse grids\n                      , in Iterative Methods in Linear Algebra, R. Beauwens and P. de Groen, eds., North\u2013Holland, Amsterdam, 1992, pp. 293\u2013310."},{"key":"R4","first-page":"63","volume":"6","author":"Bungartz H.-J.","year":"1997","journal-title":"Electron. Trans. Numer. 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Klimke,\n                      Efficient Construction of Hierarchical Polynomial Sparse Grid Interpolants Using the Fast Discrete Cosine Transform\n                      , IANS preprint 2006\/007, Universit\u00e4t Stuttgart, Stuttgart, Germany, 2006."},{"key":"R16","first-page":"123","volume":"37","author":"Notay Y.","year":"2010","journal-title":"Electron. Trans. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/1097-4067","issn-type":"print"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-02019-4"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/0915089"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/0916006"},{"key":"R20","unstructured":"J. Shen,\n                      Efficient Chebyshev-Legendre Galerkin methods for elliptic problems\n                      , in Proceedings of ICOSAHOM'95, A. V. Ilin and R. Scott, eds., Houston J. Math., 1996, pp. 233\u2013240."},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1137\/090765547"},{"key":"R22","first-page":"1042","volume":"148","author":"Smolyak S. A.","year":"1963","journal-title":"Soviet Math. Dokl., 4 (1963), pp. 240\u2013243 (in English); Dokl. Akad. Nauk SSSR"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1137\/09074543X"},{"key":"R24","unstructured":"J. Xia and M. Gu,\n                      Robust Structured Multifrontal Factorization and Preconditioning for Discretized PDEs\n                      , preprint, 2009; available online at http:\/\/www.math.purdue.edu\/$\\!_{^{\\sim}}\\!$xiaj\/work\/mfhssr.pdf."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/100787842","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:33:41Z","timestamp":1787330021000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/100787842"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,1]]},"references-count":24,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2010,1]]}},"alternative-id":["10.1137\/100787842"],"URL":"https:\/\/doi.org\/10.1137\/100787842","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,1]]}}}