{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,23]],"date-time":"2026-08-23T02:19:01Z","timestamp":1787451541556,"version":"build-2736575974"},"reference-count":38,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>This work proposes and analyzes a Smolyak-type sparse grid stochastic collocation method for the approximation of statistical quantities related to the solution of partial differential equations with random coefficients and forcing terms (input data of the model). To compute solution statistics, the sparse grid stochastic collocation method uses approximate solutions, produced here by finite elements, corresponding to a deterministic set of points in the random input space. This naturally requires solving uncoupled deterministic problems as in the Monte Carlo method. If the number of random variables needed to describe the input data is moderately large, full tensor product spaces are computationally expensive to use due to the curse of dimensionality. In this case the sparse grid approach is still expected to be competitive with the classical Monte Carlo method. Therefore, it is of major practical relevance to understand in which situations the sparse grid stochastic collocation method is more efficient than Monte Carlo. This work provides error estimates for the fully discrete solution using $L^q$ norms and analyzes the computational efficiency of the proposed method. In particular, it demonstrates algebraic convergence with respect to the total number of collocation points and quantifies the effect of the dimension of the problem (number of input random variables) in the final estimates. The derived estimates are then used to compare the method with Monte Carlo, indicating for which problems the former is more efficient than the latter. Computational evidence complements the present theory and shows the effectiveness of the sparse grid stochastic collocation method compared to full tensor and Monte Carlo approaches.<\/jats:p>","DOI":"10.1137\/060663660","type":"journal-article","created":{"date-parts":[[2008,5,16]],"date-time":"2008-05-16T18:04:06Z","timestamp":1210961046000},"page":"2309-2345","source":"Crossref","is-referenced-by-count":745,"title":["A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data"],"prefix":"10.1137","volume":"46","author":[{"given":"F.","family":"Nobile","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Tempone","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. G.","family":"Webster","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,16]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00354-7"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250300257X"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1137\/050645142"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142902418680"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.02.026"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018977404843"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Springer-Verlag, New York, 1994.","DOI":"10.1007\/978-1-4757-4338-8"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000182"},{"key":"R9","first-page":"368","volume":"4","author":"Burkardt J.","year":"2007","journal-title":"Int. J. Numer. Anal. Model."},{"key":"R10","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, New York, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1007\/BF01386223"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"R. A. DeVore and G. G. Lorentz,\n                      Constructive Approximation\n                      , Grundlehren Math. Wiss. 303, Springer-Verlag, Berlin, 1993.","DOI":"10.1007\/978-3-662-02888-9_10"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/BF01982005"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.2307\/1968516"},{"key":"R15","unstructured":"G. S. Fishman,\n                      Monte Carlo: Concepts, Algorithms, and Applications\n                      , Springer Ser. Oper. Res., Springer-Verlag, New York, 1996."},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.04.008"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1029\/1999WR900160"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019129717644"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0015-5"},{"key":"R20","doi-asserted-by":"crossref","unstructured":"R. G. Ghanem and P. D. Spanos,\n                      Stochastic Finite Elements: A Spectral Approach\n                      , Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-3094-6"},{"key":"R21","doi-asserted-by":"crossref","unstructured":"M. Grigoriu,\n                      Stochastic Calculus\n                      , Appl. Sci. Engrg., Birkh\u00e4user, Boston, MA, 2002.","DOI":"10.1007\/978-0-8176-8228-6"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.11.033"},{"key":"R23","doi-asserted-by":"crossref","unstructured":"M. Lo\u00e8ve,\n                      Probability Theory\n                      , I, 4th ed., Grad. Texts in Math. 45, Springer-Verlag, New York, 1977.","DOI":"10.1007\/978-1-4757-6288-4"},{"key":"R24","doi-asserted-by":"crossref","unstructured":"M. Lo\u00e8ve,\n                      Probability Theory\n                      , II, 4th ed., Grad. Texts in Math. 46, Springer-Verlag, New York, 1978.","DOI":"10.1007\/978-1-4612-6257-2"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/s11075-004-2866-z"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2004.05.027"},{"key":"R27","unstructured":"F. Riesz and B. Sz-Nagy,\n                      Functional Analysis\n                      , Dover, New York, 1990."},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2006.6.941"},{"key":"R29","first-page":"240","volume":"4","author":"Smolyak S. A.","year":"1963","journal-title":"Dokl. Akad. Nauk SSSR","ISSN":"https:\/\/id.crossref.org\/issn\/0002-3264","issn-type":"print"},{"key":"R30","unstructured":"M. A. Tatang,\n                      Direct Incorporation of Uncertainty in Chemical and Environmental Engineering Systems\n                      , Ph.D. thesis, MIT, Cambridge, MA, 1995."},{"key":"R31","unstructured":"R. A. Todor,\n                      Sparse Perturbation Algorithms for Elliptic PDE's with Stochastic Data\n                      , Dissertation 16192, ETH Zurich, Switzerland, 2005."},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1137\/060659831"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1006\/jcom.1995.1001"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.2307\/2371268"},{"key":"R35","doi-asserted-by":"crossref","unstructured":"C. L. Winter and D. M. Tartakovsky,\n                      Groundwater flow in heterogeneous composite aquifers\n                      , Water Resources Res., 38 (8) (2002), article 1148.","DOI":"10.1029\/2001WR000450"},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/040615201"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00421-8"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827501387826"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060663660","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:07Z","timestamp":1787325247000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060663660"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":38,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/060663660"],"URL":"https:\/\/doi.org\/10.1137\/060663660","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}