{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:44Z","timestamp":1787337224411,"version":"build-2736575974"},"reference-count":44,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We give a convergence proof for the approximation by sparse collocation of Hilbert-space-valued functions depending on countably many Gaussian random variables. Such functions appear as solutions of elliptic PDEs with lognormal diffusion coefficients. We outline a general $L^2$-convergence theory based on previous work by Bachmayr et al. [ ESAIM Math. Model. Numer. Anal., 51 (2017), pp. 341--363] and Chen [ ESAIM Math. Model. Numer. Anal., in press, 2018, https:\/\/doi.org\/10.1051\/m2an\/2018012] and establish an algebraic convergence rate for sufficiently smooth functions assuming a mild growth bound for the univariate hierarchical surpluses of the interpolation scheme applied to Hermite polynomials. We specifically verify for Gauss--Hermite nodes that this assumption holds and also show algebraic convergence with respect to the resulting number of sparse grid points for this case. Numerical experiments illustrate the dimension-independent convergence rate.<\/jats:p>","DOI":"10.1137\/17m1123079","type":"journal-article","created":{"date-parts":[[2018,4,10]],"date-time":"2018-04-10T12:20:36Z","timestamp":1523362836000},"page":"877-905","source":"Crossref","is-referenced-by-count":30,"title":["Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs)"],"prefix":"10.1137","volume":"56","author":[{"given":"Oliver G.","family":"Ernst","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bj\u00f6rn","family":"Sprungk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lorenzo","family":"Tamellini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,4,10]]},"reference":[{"key":"atypb1","volume-title":"Handbook of Mathematical Functions","author":"Abramowitz M.","edition":"10"},{"key":"atypb2","volume-title":"The Geometry of Random Fields","author":"Adler R. J.","year":"1981"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/100786356"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016051"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016045"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-15337-2_3"},{"key":"atypb7","first-page":"1","volume-title":"Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM","author":"Beck J.","year":"2012"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/100800531"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2018012"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jat.2012.11.005"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-013-9154-z"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.matpur.2014.04.009"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492915000033"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-010-9072-2"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001728"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2012.03.019"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/S0096-3003(03)00580-0"},{"key":"atypb18","volume-title":"Convergence of Sparse Collocation for Functions of Countably Many Gaussian Random Variables (with Application to Elliptic PDEs), preprint, https:\/\/arxiv.org\/abs\/1611.07239","author":"Ernst O. G.","year":"2017"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s10596-012-9311-5"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/080717924"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(95)00232-4"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0015-5"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3094-6"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61798-0"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202510004210"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-016-9327-7"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202513500681"},{"key":"atypb28","volume-title":"IMA J. Numer. Anal., in press, https:\/\/doi.org\/10.1093\/imanum\/dry002","author":"Jantsch P.","year":"2017"},{"key":"atypb29","volume-title":"Scientific Computation","author":"Le Maitre O. P.","year":"2010"},{"key":"atypb30","first-page":"230","volume":"23","author":"Leja F.","year":"1950","journal-title":"Ann. Soc. Math. Polon."},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2013066"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/140966368"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(80)90030-1"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0773-y"},{"key":"atypb35","first-page":"191","volume-title":"Sparse Grids and Applications -- Stuttgart","author":"Nobile F.","year":"2014"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1137\/070680540"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1137\/060663660"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1002\/2014WR015838"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/29\/6\/065011"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492911000055"},{"key":"atypb41","first-page":"37","volume":"102","author":"Stechkin S. B.","year":"1955","journal-title":"Doklady Akademii Nauk SSSR"},{"key":"atypb42","volume-title":"Orthogonal Polynomials","author":"Szeg\u00f6 G.","year":"1975","edition":"4"},{"key":"atypb43","unstructured":"L. Tamellini,\n                      Polynomial Approximation of PDEs with Stochastic Coefficients\n                      , Ph.D. thesis, Politecnico di Milano, Milano, Italy, 2012."},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1137\/040615201"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1123079","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:50:10Z","timestamp":1787334610000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1123079"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":44,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/17M1123079"],"URL":"https:\/\/doi.org\/10.1137\/17m1123079","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}