{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,5]],"date-time":"2026-05-05T02:05:43Z","timestamp":1777946743034,"version":"3.51.4"},"reference-count":38,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T00:00:00Z","timestamp":1782864000000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/100000121","name":"National Science Foundation Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["NSF-DMS-2137305"],"award-info":[{"award-number":["NSF-DMS-2137305"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100006602","name":"Air Force Research Laboratory Information Directorate","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100006602","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-22-1-0004"],"award-info":[{"award-number":["FA9550-22-1-0004"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computer Physics Communications"],"published-print":{"date-parts":[[2026,7]]},"DOI":"10.1016\/j.cpc.2026.110133","type":"journal-article","created":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T03:40:57Z","timestamp":1774582857000},"page":"110133","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem"],"prefix":"10.1016","volume":"324","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8886-9901","authenticated-orcid":false,"given":"Howard C.","family":"Elman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0910-5557","authenticated-orcid":false,"given":"Jiaxing","family":"Liang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8930-2798","authenticated-orcid":false,"given":"Tonatiuh","family":"S\u00e1nchez-Vizuet","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/j.cpc.2026.110133_bib0001","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110699","article-title":"Surrogate approximation of the Grad-Shafranov free boundary problem via stochastic collocation on sparse grids","volume":"448","author":"Elman","year":"2022","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.cpc.2026.110133_bib0002","doi-asserted-by":"crossref","DOI":"10.1016\/j.cpc.2024.109099","article-title":"Multilevel Monte Carlo methods for the Grad-Shafranov free boundary problem","volume":"298","author":"Elman","year":"2024","journal-title":"Comput. Phys. Commun."},{"issue":"1","key":"10.1016\/j.cpc.2026.110133_bib0003","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1007\/s00211-011-0377-0","article-title":"Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients","volume":"119","author":"Barth","year":"2011","journal-title":"Numer. Math."},{"issue":"3","key":"10.1016\/j.cpc.2026.110133_bib0004","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1287\/opre.1070.0496","article-title":"Multilevel Monte Carlo path simulation","volume":"56","author":"Giles","year":"2008","journal-title":"Oper. Res."},{"key":"10.1016\/j.cpc.2026.110133_bib0005","series-title":"Proc. Second International Conference on the Peaceful Uses of Atomic Energy, Geneva","article-title":"Hydromagnetic equilibria and force-free fields","volume":"31,190","author":"Grad","year":"1958"},{"key":"10.1016\/j.cpc.2026.110133_bib0006","doi-asserted-by":"crossref","first-page":"850","DOI":"10.1515\/zna-1957-1014","article-title":"Axialsymmetrische magnetohydrodynamische Gleichgewichtskonfigurationen","volume":"12a","author":"L\u00fcst","year":"1957","journal-title":"Z. Naturf."},{"key":"10.1016\/j.cpc.2026.110133_bib0007","first-page":"545","article-title":"On magnetohydrodynamical equilibrium configurations","volume":"6","author":"Shafranov","year":"1958","journal-title":"Soviet Phys. JETP"},{"issue":"6","key":"10.1016\/j.cpc.2026.110133_bib0008","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1088\/0029-5515\/22\/6\/009","article-title":"Magnetic analysis of non-circular cross-section tokamaks","volume":"22","author":"Luxon","year":"1982","journal-title":"Nucl. Fusion"},{"issue":"1","key":"10.1016\/j.cpc.2026.110133_bib0009","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00791-011-0160-x","article-title":"Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients","volume":"14","author":"Cliffe","year":"2011","journal-title":"Comput. Vis. Sci."},{"issue":"3","key":"10.1016\/j.cpc.2026.110133_bib0010","doi-asserted-by":"crossref","first-page":"569","DOI":"10.1007\/s00211-013-0546-4","article-title":"Further analysis of multilevel Monte Carlo methods for elliptic PDEs with random coefficients","volume":"125","author":"Teckentrup","year":"2013","journal-title":"Numer. Math."},{"key":"10.1016\/j.cpc.2026.110133_bib0011","doi-asserted-by":"crossref","first-page":"273","DOI":"10.1023\/A:1018977404843","article-title":"High dimensional polynomial interpolation on sparse grids","volume":"12","author":"Barthelmann","year":"2000","journal-title":"Adv. Comput. Math."},{"issue":"4","key":"10.1016\/j.cpc.2026.110133_bib0012","doi-asserted-by":"crossref","first-page":"561","DOI":"10.1145\/1114268.1114275","article-title":"Algorithm 847: spinterp: piecewise multilinear hierarchical sparse grid interpolation in MATLAB","volume":"31","author":"Klimke","year":"2005","journal-title":"Assoc. Comput. Mach. Trans. Math. Softw."},{"issue":"8","key":"10.1016\/j.cpc.2026.110133_bib0013","doi-asserted-by":"crossref","first-page":"3084","DOI":"10.1016\/j.jcp.2009.01.006","article-title":"An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations","volume":"228","author":"Ma","year":"2009","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.cpc.2026.110133_bib0014","first-page":"1042","article-title":"Quadrature and interpolation formulae on tensor products of certain function classes","volume":"148","author":"Smolyak","year":"1963","journal-title":"Dokl. Akad. Nauk SSSR"},{"issue":"5","key":"10.1016\/j.cpc.2026.110133_bib0015","doi-asserted-by":"crossref","first-page":"2309","DOI":"10.1137\/060663660","article-title":"A sparse grid stochastic collocation method for partial differential equations with random input data","volume":"46","author":"Nobile","year":"2008","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"10.1016\/j.cpc.2026.110133_bib0016","doi-asserted-by":"crossref","first-page":"1046","DOI":"10.1137\/140969002","article-title":"A multilevel stochastic collocation method for partial differential equations with random input data","volume":"3","author":"Teckentrup","year":"2015","journal-title":"SIAM\/ASA J. Uncertainty Quant."},{"key":"10.1016\/j.cpc.2026.110133_bib0017","series-title":"Technical Report","article-title":"Modelisation de l\u2019equilibre d\u2019un plasma de tokamak","author":"Grandgirard","year":"1999"},{"key":"10.1016\/j.cpc.2026.110133_bib0018","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/BF01386223","article-title":"A method for numerical integration on an automatic computer","volume":"2","author":"Clenshaw","year":"1960","journal-title":"Numer. Math."},{"key":"10.1016\/j.cpc.2026.110133_bib0019","series-title":"Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics","author":"Braess","year":"2007"},{"issue":"3","key":"10.1016\/j.cpc.2026.110133_bib0020","doi-asserted-by":"crossref","first-page":"1005","DOI":"10.1137\/050645142","article-title":"A stochastic collocation method for elliptic partial differential equations with random input data","volume":"45","author":"Babu\u0161ka","year":"2007","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.cpc.2026.110133_bib0021","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1017\/S096249291500001X","article-title":"Multilevel Monte Carlo methods","volume":"24","author":"Giles","year":"2015","journal-title":"Acta Numer."},{"issue":"3","key":"10.1016\/j.cpc.2026.110133_bib0022","first-page":"398","article-title":"A multi level Monte Carlo method with control variate for elliptic PDEs with log-normal coefficients","volume":"3","author":"Nobile","year":"2015","journal-title":"Stochastic Partial Differ. Eq. Anal. Comput."},{"key":"10.1016\/j.cpc.2026.110133_bib0023","doi-asserted-by":"crossref","first-page":"308","DOI":"10.1098\/rstl.1819.0023","article-title":"XXI. A new method of solving numerical equations of all orders, by continuous approximation","volume":"109","author":"Horner","year":"1819","journal-title":"Philosoph. Trans. R. Soc. London"},{"issue":"51","key":"10.1016\/j.cpc.2026.110133_bib0024","doi-asserted-by":"crossref","first-page":"118","DOI":"10.1090\/S0025-5718-1955-0071856-0","article-title":"A note on the summation of Chebyshev series","volume":"9","author":"Clenshaw","year":"1955","journal-title":"Math. Comput."},{"issue":"189","key":"10.1016\/j.cpc.2026.110133_bib0025","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1090\/S0025-5718-1990-0993925-1","article-title":"Evaluation of multivariate polynomials and their derivatives","volume":"54","author":"Carnicer","year":"1990","journal-title":"Math. Comput."},{"key":"10.1016\/j.cpc.2026.110133_bib0026","series-title":"2003 International Conference on Geometric Modeling and Graphics, 2003. Proceedings","first-page":"162","article-title":"A linear complexity algorithm for the Bernstein basis","author":"Delgado","year":"2003"},{"issue":"4","key":"10.1016\/j.cpc.2026.110133_bib0027","doi-asserted-by":"crossref","first-page":"1668","DOI":"10.1137\/060649148","article-title":"A corner cutting algorithm for evaluating rational b\u00e9zier surfaces and the optimal stability of the basis","volume":"29","author":"Delgado","year":"2007","journal-title":"SIAM J. Sci. Comput."},{"key":"10.1016\/j.cpc.2026.110133_bib0028","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/j.jcp.2017.04.047","article-title":"FEM-BEM coupling methods for Tokamak plasma axisymmetric free-boundary equilibrium computations in unbounded domains","volume":"343","author":"Faugeras","year":"2017","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.cpc.2026.110133_bib0029","unstructured":"H. Heumann, FEEQS.M, http:\/\/www-sop.inria.fr\/members\/Holger.Heumann\/Software.html."},{"issue":"3","key":"10.1016\/j.cpc.2026.110133_bib0030","doi-asserted-by":"crossref","DOI":"10.1017\/S0022377814001251","article-title":"Quasi-static free-boundary equilibrium of toroidal plasma with CEDRES++: computational methods and applications","volume":"81","author":"Heumann","year":"2015","journal-title":"J. Plasma Phys."},{"issue":"1\u20133","key":"10.1016\/j.cpc.2026.110133_bib0031","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/S0925-7721(01)00047-5","article-title":"Delaunay refinement algorithms for triangular mesh generation","volume":"22","author":"Shewchuk","year":"2002","journal-title":"Comput. Geometry"},{"issue":"1","key":"10.1016\/j.cpc.2026.110133_bib0032","doi-asserted-by":"crossref","first-page":"1219","DOI":"10.1137\/15M1016448","article-title":"An adaptive multilevel Monte Carlo method with stochastic bounds for quantities of interest with uncertain data","volume":"4","author":"Eigel","year":"2016","journal-title":"SIAM\/ASA J. Uncertainty Quant."},{"key":"10.1016\/j.cpc.2026.110133_bib0033","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2020.113163","article-title":"An adaptive multilevel Monte Carlo algorithm for the stochastic drift-diffusion-Poisson system","volume":"368","author":"Khodadadian","year":"2020","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"6","key":"10.1016\/j.cpc.2026.110133_bib0034","doi-asserted-by":"crossref","first-page":"745","DOI":"10.1142\/S0218488504003181","article-title":"Uncertainty modeling using fuzzy arithmetic based on sparse grids: applications to dynamic systems","volume":"12","author":"Klimke","year":"2004","journal-title":"Int. J. Uncertainty Fuzziness Knowledge Based Syst."},{"key":"10.1016\/j.cpc.2026.110133_bib0035","series-title":"Efficient Computational Algorithms for Magnetic Equilibrium in a Fusion Reactor","author":"Liang","year":"2024"},{"key":"10.1016\/j.cpc.2026.110133_bib0036","article-title":"Direct Methods for Sparse Linear Systems","volume":"2","author":"Davis","year":"2006"},{"key":"10.1016\/j.cpc.2026.110133_bib0037","article-title":"Matrix Computations","author":"Golub","year":"2013"},{"key":"10.1016\/j.cpc.2026.110133_bib0038","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1137\/0710032","article-title":"Nested dissection of a regular finite element mesh","volume":"10","author":"George","year":"1973","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Computer Physics Communications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0010465526001153?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0010465526001153?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,5,2]],"date-time":"2026-05-02T20:18:32Z","timestamp":1777753112000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0010465526001153"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,7]]},"references-count":38,"alternative-id":["S0010465526001153"],"URL":"https:\/\/doi.org\/10.1016\/j.cpc.2026.110133","relation":{},"ISSN":["0010-4655"],"issn-type":[{"value":"0010-4655","type":"print"}],"subject":[],"published":{"date-parts":[[2026,7]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Surrogate-based multilevel Monte Carlo methods for uncertainty quantification in the Grad-Shafranov free boundary problem","name":"articletitle","label":"Article Title"},{"value":"Computer Physics Communications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.cpc.2026.110133","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}],"article-number":"110133"}}