{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,5]],"date-time":"2026-06-05T02:20:11Z","timestamp":1780626011039,"version":"3.54.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2014,12,2]],"date-time":"2014-12-02T00:00:00Z","timestamp":1417478400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2015,10]]},"DOI":"10.1007\/s00211-014-0689-y","type":"journal-article","created":{"date-parts":[[2014,12,1]],"date-time":"2014-12-01T19:33:26Z","timestamp":1417462406000},"page":"329-368","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":118,"title":["Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients"],"prefix":"10.1007","volume":"131","author":[{"given":"I. G.","family":"Graham","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F. Y.","family":"Kuo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"J. A.","family":"Nichols","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Scheichl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ch.","family":"Schwab","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"I. H.","family":"Sloan","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2014,12,2]]},"reference":[{"key":"689_CR1","volume-title":"The Geometry of Random Fields","author":"RJ Adler","year":"1981","unstructured":"Adler, R.J.: The Geometry of Random Fields. Wiley, London (1981)"},{"key":"689_CR2","doi-asserted-by":"crossref","unstructured":"Bogachev, V.I.: Gaussian Measures, AMS Monographs, vol. 62. American Mathematical Society, RI (1998)","DOI":"10.1090\/surv\/062"},{"key":"689_CR3","doi-asserted-by":"crossref","first-page":"216","DOI":"10.1137\/100800531","volume":"50","author":"J Charrier","year":"2012","unstructured":"Charrier, J.: Strong and weak error estimates for elliptic partial differential equations with random coefficients. SIAM J. Numer. Anal. 50, 216\u2013246 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"689_CR4","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/s40072-013-0006-2","volume":"1","author":"J Charrier","year":"2013","unstructured":"Charrier, J., Debussche, A.: Weak truncation error estimates for elliptic PDEs with lognormal coefficients. Stoch. PDEs: Anal. Comput. 1, 63\u201393 (2013)","journal-title":"Stoch. PDEs: Anal. Comput."},{"key":"689_CR5","doi-asserted-by":"crossref","first-page":"322","DOI":"10.1137\/110853054","volume":"51","author":"J Charrier","year":"2013","unstructured":"Charrier, J., Scheichl, R., Teckentrup, A.L.: Finite element error analysis of elliptic PDEs with random coefficients and its application to multilevel Monte Carlo methods. SIAM J. Numer. Anal. 51, 322\u2013352 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"689_CR6","doi-asserted-by":"crossref","first-page":"2162","DOI":"10.1137\/06065074X","volume":"28","author":"R Cools","year":"2006","unstructured":"Cools, R., Kuo, F.Y., Nuyens, D.: Constructing embedded lattice rules for multivariate integration. SIAM J. Sci. Comput. 28, 2162\u20132188 (2006)","journal-title":"SIAM J. Sci. Comput."},{"key":"689_CR7","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511666223","volume-title":"Stochastic Equations in Infinite Dimensions, Vol. 44 of Encyclopedia of Mathematics and its Applications","author":"G Prato Da","year":"1992","unstructured":"Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions, Vol. 44 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)"},{"key":"689_CR8","doi-asserted-by":"crossref","first-page":"493","DOI":"10.1016\/j.jco.2003.11.008","volume":"20","author":"J Dick","year":"2004","unstructured":"Dick, J.: On the convergence rate of the component-by-component construction of good lattice rules. J. Complexity 20, 493\u2013522 (2004)","journal-title":"J. Complexity"},{"key":"689_CR9","doi-asserted-by":"crossref","first-page":"133","DOI":"10.1017\/S0962492913000044","volume":"22","author":"J Dick","year":"2013","unstructured":"Dick, J., Kuo, F.Y., Sloan, I.H.: High dimensional integration\u2014the quasi-Monte Carlo way. Acta Numer. 22, 133\u2013288 (2013)","journal-title":"Acta Numer."},{"key":"689_CR10","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511761188","volume-title":"Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration","author":"J Dick","year":"2010","unstructured":"Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010)"},{"key":"689_CR11","doi-asserted-by":"crossref","first-page":"2345","DOI":"10.1090\/S0025-5718-08-02009-7","volume":"77","author":"J Dick","year":"2008","unstructured":"Dick, J., Pillichshammer, F., Waterhouse, B.J.: The construction of good extensible rank-1 lattices. Math. Comp. 77, 2345\u20132374 (2008)","journal-title":"Math. Comp."},{"key":"689_CR12","doi-asserted-by":"crossref","unstructured":"Frauenfelder, P., Schwab, Ch., Todor, R.A.: Finite elements for elliptic problems with stochastic coefficients. Compu. Methods Appl. Mech. Eng. 194, 205\u2013228 (2005)","DOI":"10.1016\/j.cma.2004.04.008"},{"key":"689_CR13","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic Finite Elements","author":"RG Ghanem","year":"1991","unstructured":"Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements. Dover, New York (1991)"},{"key":"689_CR14","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1142\/S0218202510004210","volume":"20","author":"CJ Gittelson","year":"2010","unstructured":"Gittelson, C.J.: Stochastic Galerkin discretization of the isotropic lognormal diffusion problem. Math. Models Methods Appl. Sci. 20, 237\u2013263 (2010)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"689_CR15","doi-asserted-by":"crossref","first-page":"3668","DOI":"10.1016\/j.jcp.2011.01.023","volume":"230","author":"IG Graham","year":"2011","unstructured":"Graham, I.G., Kuo, F.Y., Nuyens, D., Scheichl, R., Sloan, I.H.: Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications. J. Comput. Phys. 230, 3668\u20133694 (2011)","journal-title":"J. Comput. Phys."},{"key":"689_CR16","doi-asserted-by":"crossref","first-page":"51","DOI":"10.1007\/978-3-642-18743-8_3","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods 2002","author":"FJ Hickernell","year":"2004","unstructured":"Hickernell, F.J., Sloan, I.H., Wasilkowski, G.W.: On tractability of weighted integration for certain Banach spaces of functions. In: Niederreiter, H. (ed.) Monte Carlo and Quasi-Monte Carlo Methods 2002, pp. 51\u201371. Springer-Verlag, Berlin (2004)"},{"key":"689_CR17","doi-asserted-by":"crossref","unstructured":"Hoang, V.-H., Schwab, Ch.: $$N$$ N -term Wiener chaos approximation rates for elliptic PDEs with lognormal Gaussian random inputs. Math. Models Methods Appl. Sci. 24, 797\u2013826 (2014)","DOI":"10.1142\/S0218202513500681"},{"key":"689_CR18","doi-asserted-by":"crossref","first-page":"301","DOI":"10.1016\/S0885-064X(03)00006-2","volume":"19","author":"FY Kuo","year":"2003","unstructured":"Kuo, F.Y.: Component-by-component constructions achieve the optimal rate of convergence. J. Complexity 19, 301\u2013320 (2003)","journal-title":"J. Complexity"},{"key":"689_CR19","doi-asserted-by":"crossref","unstructured":"Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficient. SIAM J. Numer. Anal. 50, 3351\u20133374 (2012)","DOI":"10.1137\/110845537"},{"key":"689_CR20","doi-asserted-by":"crossref","unstructured":"Kuo, F.Y., Schwab, Ch., Sloan, I.H.: Quasi-Monte Carlo methods for very high dimensional integration: the standard weighted-space setting and beyond. ANZIAM J. 53, 1\u201337 (2011)","DOI":"10.1017\/S1446181112000077"},{"key":"689_CR21","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.jco.2009.07.005","volume":"26","author":"FY Kuo","year":"2010","unstructured":"Kuo, F.Y., Sloan, I.H., Wasilkowski, G.W., Waterhouse, B.J.: Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands. J. Complexity 26, 135\u2013160 (2010)","journal-title":"J. Complexity"},{"key":"689_CR22","doi-asserted-by":"crossref","first-page":"630","DOI":"10.1016\/j.jco.2006.04.006","volume":"22","author":"FY Kuo","year":"2006","unstructured":"Kuo, F.Y., Wasilkowski, G.W., Waterhouse, B.J.: Randomly shifted lattice rules for unbounded integrals. J. Complexity 22, 630\u2013651 (2006)","journal-title":"J. Complexity"},{"key":"689_CR23","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139017329","volume-title":"An Introduction to Computational Stochastic PDEs","author":"GJ Lord","year":"2014","unstructured":"Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge University Press, Cambridge (2014)"},{"key":"689_CR24","volume-title":"Strongly Elliptic Systems and Boundary Integral Equations","author":"W McLean","year":"2000","unstructured":"McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)"},{"key":"689_CR25","doi-asserted-by":"crossref","unstructured":"Naff, R.L., Haley, D.F., Sudicky, E.A.: High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 1. Methodology and flow results. Water Resour. Res. 34, 663\u2013677 (1998)","DOI":"10.1029\/97WR02712"},{"key":"689_CR26","doi-asserted-by":"crossref","first-page":"679","DOI":"10.1029\/97WR02711","volume":"34","author":"RL Naff","year":"1998","unstructured":"Naff, R.L., Haley, D.F., Sudicky, E.A.: High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media 2. Transport Results. Water Resour. Res. 34, 679\u2013697 (1998)","journal-title":"Water Resour. Res."},{"key":"689_CR27","unstructured":"Nichols, J.A.: Quasi-Monte Carlo methods with applications to partial differential equations with random coefficients. Ph.D. Thesis, University of New South Wales (2014)"},{"key":"689_CR28","doi-asserted-by":"crossref","first-page":"444","DOI":"10.1016\/j.jco.2014.02.004","volume":"30","author":"JA Nichols","year":"2014","unstructured":"Nichols, J.A., Kuo, F.Y.: Fast CBC construction of randomly shifted lattice rules achieving $${{\\cal O}}(N^{-1+\\delta })$$ O ( N - 1 + \u03b4 ) convergence for unbounded integrands in $${{\\mathbb{R}}}^s$$ R s in weighted spaces with POD weights. J. Complexity 30, 444\u2013468 (2014)","journal-title":"J. Complexity"},{"key":"689_CR29","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1090\/S0025-5718-06-01785-6","volume":"75","author":"D Nuyens","year":"2006","unstructured":"Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank- $$1$$ 1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comp. 75, 903\u2013920 (2006)","journal-title":"Math. Comp."},{"key":"689_CR30","doi-asserted-by":"crossref","first-page":"4","DOI":"10.1016\/j.jco.2005.07.002","volume":"22","author":"D Nuyens","year":"2006","unstructured":"Nuyens, D., Cools, R.: Fast component-by-component construction of rank- $$1$$ 1 lattice rules with a non-prime number of points. J. Complexity 22, 4\u201328 (2006)","journal-title":"J. Complexity"},{"key":"689_CR31","doi-asserted-by":"crossref","unstructured":"Schwab, Ch., Gittelson, C.J.: Sparse tensor discretizations of high dimensional and stochastic PDEs. Acta Numer. 20, 291\u2013467 (2011)","DOI":"10.1017\/S0962492911000055"},{"key":"689_CR32","doi-asserted-by":"crossref","unstructured":"Schwab, Ch., Todor, R.A.: Karhunen\u2013Lo\u00e8ve approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100\u2013122 (2006)","DOI":"10.1016\/j.jcp.2006.01.048"},{"key":"689_CR33","doi-asserted-by":"crossref","first-page":"1650","DOI":"10.1137\/S0036142901393942","volume":"40","author":"IH Sloan","year":"2002","unstructured":"Sloan, I.H., Kuo, F.Y., Joe, S.: Constructing randomly shifted lattice rules in weighted Sobolev spaces. SIAM J. Numer. Anal. 40, 1650\u20131665 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"689_CR34","doi-asserted-by":"crossref","unstructured":"Sloan, I.H., Wo\u017aniakowski, H.: When are quasi-Monte Carlo algorithms efficient for high-dimensional integrals? J. Complexity 14, 1\u201333 (1998), 125, 569\u2013600 (2013)","DOI":"10.1006\/jcom.1997.0463"},{"key":"689_CR35","doi-asserted-by":"crossref","first-page":"46","DOI":"10.1016\/j.jco.2003.11.003","volume":"20","author":"IH Sloan","year":"2004","unstructured":"Sloan, I.H., Wang, X., Wo\u017aniakowski, H.: Finite-order weights imply tractability of multivariate integration. J. Complexity 20, 46\u201374 (2004)","journal-title":"J. Complexity"},{"key":"689_CR36","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s00211-013-0546-4","volume":"125","author":"AL Teckentrup","year":"2013","unstructured":"Teckentrup, A.L., Scheichl, R., Giles, M.B., Ullmann, E.: Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients. Numer. Math. 125, 569\u2013600 (2013)","journal-title":"Numer. Math."},{"key":"689_CR37","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1006\/jath.1999.3435","volume":"103","author":"GW Wasilkowski","year":"2000","unstructured":"Wasilkowski, G.W., Wo\u017aniakowski, H.: Complexity of weighted approximation over $$\\mathbb{R}^1$$ R 1 . J. Approx. Theory. 103, 223\u2013251 (2000)","journal-title":"J. Approx. Theory."},{"key":"689_CR38","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1007\/978-3-642-56046-0_35","volume-title":"Monte Carlo and Quasi-Monte Carlo Methods 2000","author":"GW Wasilkowski","year":"2002","unstructured":"Wasilkowski, G.W., Wo\u017aniakowski, H.: Tractability of approximation and integration for weighted tensor product problems over unbounded domains. In: Fang, K.-T., Hickernell, F.J., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2000, pp. 497\u2013522. Springer, Berlin (2002)"},{"key":"689_CR39","doi-asserted-by":"crossref","first-page":"278","DOI":"10.1090\/S0002-9947-1963-0155161-0","volume":"109","author":"H Widom","year":"1963","unstructured":"Widom, H.: Asymptotic behaviour of the eigenvalues of certain integral operators. Trans. Am. Math. Soc. 109, 278\u2013295 (1963)","journal-title":"Trans. Am. Math. Soc."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0689-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-014-0689-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0689-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,23]],"date-time":"2019-05-23T16:22:51Z","timestamp":1558628571000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-014-0689-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,2]]},"references-count":39,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2015,10]]}},"alternative-id":["689"],"URL":"https:\/\/doi.org\/10.1007\/s00211-014-0689-y","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12,2]]}}}