{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,17]],"date-time":"2026-02-17T12:11:11Z","timestamp":1771330271968,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2014,12,6]],"date-time":"2014-12-06T00:00:00Z","timestamp":1417824000000},"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":[[2015,8]]},"DOI":"10.1007\/s00211-014-0676-3","type":"journal-article","created":{"date-parts":[[2014,12,5]],"date-time":"2014-12-05T05:30:05Z","timestamp":1417757405000},"page":"579-613","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":48,"title":["Convergence analysis of multilevel Monte Carlo variance estimators and application for random obstacle problems"],"prefix":"10.1007","volume":"130","author":[{"given":"Claudio","family":"Bierig","sequence":"first","affiliation":[]},{"given":"Alexey","family":"Chernov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,12,6]]},"reference":[{"key":"676_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"1975","unstructured":"Adams, R.A.: Sobolev Spaces, vol. 65. Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York, London (1975)"},{"issue":"3","key":"676_CR2","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1137\/050645142","volume":"45","author":"I Babu\u0161ka","year":"2007","unstructured":"Babu\u0161ka, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005\u20131034 (2007)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"676_CR3","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s00211-011-0377-0","volume":"119","author":"A Barth","year":"2011","unstructured":"Barth, A., Schwab, C., Zollinger, N.: Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math. 119(1), 123\u2013161 (2011)","journal-title":"Numer. Math."},{"key":"676_CR4","unstructured":"Brezis, H.: Nouveaux th\u00e9or\u00e8mes de r\u00e9gularit\u00e9 pour les probl\u00e8mes unilat\u00e9raux. Rencontre entre physiciens th\u00e9oriciens et math\u00e9maticiens 12, pp. 14. Strasbourg,(1971)"},{"issue":"4","key":"676_CR5","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1007\/BF01404345","volume":"28","author":"F Brezzi","year":"1977","unstructured":"Brezzi, F., Hager, W.W., Raviart, P.-A.: Error estimates for the finite element solution of variational inequalities. Numer. Math. 28(4), 431\u2013443 (1977)","journal-title":"Numer. Math."},{"key":"676_CR6","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1017\/S0962492904000182","volume":"13","author":"H-J Bungartz","year":"2004","unstructured":"Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 147\u2013269 (2004)","journal-title":"Acta Numer."},{"issue":"284","key":"676_CR7","doi-asserted-by":"crossref","first-page":"1859","DOI":"10.1090\/S0025-5718-2013-02692-0","volume":"82","author":"A Chernov","year":"2013","unstructured":"Chernov, A., Schwab, C.: First order $$k$$ k -th moment finite element analysis of nonlinear operator equations with stochastic data. Math. Comput. 82(284), 1859\u20131888 (2013)","journal-title":"Math. Comput."},{"key":"676_CR8","volume-title":"The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and its Applications, vol. 4. North-Holland Publishing Co., Amsterdam (1978)"},{"issue":"1","key":"676_CR9","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00791-011-0160-x","volume":"14","author":"KA Cliffe","year":"2011","unstructured":"Cliffe, K.A., Giles, M.B., Scheichl, R., Teckentrup, A.L.: Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients. Comput. Vis. Sci. 14(1), 3\u201315 (2011)","journal-title":"Comput. Vis. Sci."},{"key":"676_CR10","doi-asserted-by":"crossref","unstructured":"Da Prato, G.: An introduction to infinite-dimensional analysis. Universitext, Springer, Berlin (2006), (Revised and extended from the 2001 original by Da Prato)","DOI":"10.1007\/3-540-29021-4"},{"key":"676_CR11","doi-asserted-by":"crossref","unstructured":"Forster, R., Kornhuber, R.: A polynomial chaos approach to stochastic variational inequalities. J. Numer. Math. 18(4), 235\u2013255 (2010)","DOI":"10.1515\/jnum.2010.012"},{"key":"676_CR12","doi-asserted-by":"crossref","unstructured":"Giles, M.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Monte Carlo and quasi-Monte Carlo methods 2006, pp. 343\u2013358. Springer, Berlin, Heidelberg (2008)","DOI":"10.1007\/978-3-540-74496-2_20"},{"issue":"3","key":"676_CR13","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1287\/opre.1070.0496","volume":"56","author":"MB Giles","year":"2008","unstructured":"Giles, M.B.: Multilevel Monte Carlo path simulation. Oper. Res. 56(3), 607\u2013617 (2008)","journal-title":"Oper. Res."},{"key":"676_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-12613-4","volume-title":"Numerical Methods for Nonlinear Variational Problems","author":"R Glowinski","year":"1984","unstructured":"Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer Series in Computational Physics, Springer, New York (1984)"},{"issue":"1","key":"676_CR15","first-page":"1","volume":"27","author":"C Gr\u00e4ser","year":"2009","unstructured":"Gr\u00e4ser, C., Kornhuber, R.: Multigrid methods for obstacle problems. J. Comput. Math. 27(1), 1\u201344 (2009)","journal-title":"J. Comput. Math."},{"key":"676_CR16","volume-title":"Multigrid Methods and Applications, Volume 4 of Springer Series in Computational Mathematics","author":"W Hackbusch","year":"1985","unstructured":"Hackbusch, W.: Multigrid Methods and Applications, Volume 4 of Springer Series in Computational Mathematics. Springer, Berlin (1985)"},{"issue":"2","key":"676_CR17","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1007\/BF03325426","volume":"69","author":"R Kornhuber","year":"1994","unstructured":"Kornhuber, R.: Monotone multigrid methods for elliptic variational inequalities. I. Numer. Math. 69(2), 167\u2013184 (1994)","journal-title":"Numer. Math."},{"issue":"3","key":"676_CR18","doi-asserted-by":"crossref","first-page":"1243","DOI":"10.1137\/130916126","volume":"52","author":"R Kornhuber","year":"2014","unstructured":"Kornhuber, R., Schwab, C., Wolf, M.-W.: Multi-level Monte-Carlo finite element methods for stochastic elliptic variational inequalities. SIAM J. Numer. Anal. 52(3), 1243\u20131268 (2014)","journal-title":"SIAM J. Numer. Anal."},{"key":"676_CR19","doi-asserted-by":"crossref","unstructured":"Ledoux, M., Talagrand, M.: Probability in Banach Spaces, Volume 23 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer, Berlin (1991). (Isoperimetry and processes)","DOI":"10.1007\/978-3-642-20212-4"},{"key":"676_CR20","doi-asserted-by":"crossref","DOI":"10.1007\/BFb0075391","volume-title":"Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics","author":"WA Light","year":"1985","unstructured":"Light, W.A., Cheney, E.W.: Approximation Theory in Tensor Product Spaces. Lecture Notes in Mathematics, vol. 1169. Springer, Berlin (1985)"},{"issue":"8","key":"676_CR21","doi-asserted-by":"crossref","first-page":"3365","DOI":"10.1016\/j.jcp.2012.01.011","volume":"231","author":"S Mishra","year":"2012","unstructured":"Mishra, S., Schwab, C., \u0160ukys, J.: Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions. J. Comput. Phys. 231(8), 3365\u20133388 (2012)","journal-title":"J. Comput. Phys."},{"key":"676_CR22","doi-asserted-by":"crossref","unstructured":"P\u00e9bay, P.: Formulas for robust, one-pass parallel computation of covariances and arbitrary-order statistical moments. Sandia Report SAND2008-6212, Sandia National Laboratories (2008)","DOI":"10.2172\/1028931"},{"key":"676_CR23","doi-asserted-by":"crossref","first-page":"R1","DOI":"10.1088\/0953-8984\/17\/1\/R01","volume":"17","author":"BNJ Persson","year":"2005","unstructured":"Persson, B.N.J., Albohr, O., Tartaglino, U., Volokitin, A.I., Tosatti, E.: On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion. J. Phys. Condens. Matter 17, R1\u2013R62 (2005)","journal-title":"J. Phys. Condens. Matter"},{"key":"676_CR24","unstructured":"Pr\u00e9v\u00f4t, C., R\u00f6ckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2007)"},{"key":"676_CR25","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1017\/S0962492911000055","volume":"20","author":"C Schwab","year":"2011","unstructured":"Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291\u2013467 (2011)","journal-title":"Acta Numer."},{"issue":"4","key":"676_CR26","doi-asserted-by":"crossref","first-page":"755","DOI":"10.1007\/s002110200404","volume":"93","author":"X-C Tai","year":"2003","unstructured":"Tai, X.-C.: Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities. Numer. Math. 93(4), 755\u2013786 (2003)","journal-title":"Numer. Math."},{"key":"676_CR27","doi-asserted-by":"crossref","first-page":"419","DOI":"10.1080\/00401706.1962.10490022","volume":"4","author":"BP Welford","year":"1962","unstructured":"Welford, B.P.: Note on a method for calculating corrected sums of squares and products. Technometrics 4, 419\u2013420 (1962)","journal-title":"Technometrics"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0676-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-014-0676-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-014-0676-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,18]],"date-time":"2019-08-18T04:47:01Z","timestamp":1566103621000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-014-0676-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,6]]},"references-count":27,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2015,8]]}},"alternative-id":["676"],"URL":"https:\/\/doi.org\/10.1007\/s00211-014-0676-3","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,12,6]]}}}