{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T15:30:21Z","timestamp":1772119821796,"version":"3.50.1"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T00:00:00Z","timestamp":1676937600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2023,10]]},"DOI":"10.1007\/s11075-023-01514-z","type":"journal-article","created":{"date-parts":[[2023,2,21]],"date-time":"2023-02-21T14:58:00Z","timestamp":1676991480000},"page":"643-668","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["An ensemble scheme for the numerical solution of a random transient heat equation with uncertain inputs"],"prefix":"10.1007","volume":"94","author":[{"given":"Tingfu","family":"Yao","sequence":"first","affiliation":[]},{"given":"Changlun","family":"Ye","sequence":"additional","affiliation":[]},{"given":"Xianbing","family":"Luo","sequence":"additional","affiliation":[]},{"given":"Shuwen","family":"Xiang","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,2,21]]},"reference":[{"issue":"2","key":"1514_CR1","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1137\/100786356","volume":"52","author":"I Babuska","year":"2010","unstructured":"Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52(2), 317\u2013355 (2010)","journal-title":"SIAM Rev."},{"key":"1514_CR2","volume-title":"The Mathematical Theory of Finite Element Methods","author":"SS Brenner","year":"2007","unstructured":"Brenner, S.S., Scott, R.: The Mathematical Theory of Finite Element Methods. Springer, New York (2007)"},{"key":"1514_CR3","doi-asserted-by":"publisher","DOI":"10.5772\/50994","volume-title":"Stochastic Analysis of Heat Conduction and Thermal Stresses in Solids: A Review","author":"R Chiba","year":"2012","unstructured":"Chiba, R.: Stochastic Analysis of Heat Conduction and Thermal Stresses in Solids: A Review. Chapter 9 in Heat Transfer Phenomena and Applications, IntechOpen, London (2012)"},{"issue":"1","key":"1514_CR4","doi-asserted-by":"publisher","first-page":"507","DOI":"10.1016\/j.apm.2007.11.014","volume":"33","author":"R Chiba","year":"2009","unstructured":"Chiba, R.: Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients. Appl. Math. Model. 33(1), 507\u2013523 (2009)","journal-title":"Appl. Math. Model."},{"key":"1514_CR5","doi-asserted-by":"publisher","first-page":"72","DOI":"10.1007\/s10915-021-01737-z","volume":"90","author":"XB Feng","year":"2022","unstructured":"Feng, X.B., Luo, Y., Vo, L., Wang, Z.: An efficient iterative method for solving parameter-dependent and random convection-diffusion problems. J. Sci. Comput. 90, 72 (2022)","journal-title":"J. Sci. Comput."},{"key":"1514_CR6","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-2553-7","volume-title":"Monte Carlo: Concepts, Algorithms, and Applications","author":"G Fishman","year":"1996","unstructured":"Fishman, G.: Monte Carlo: Concepts, Algorithms, and Applications. Springer, New York (1996)"},{"key":"1514_CR7","doi-asserted-by":"publisher","first-page":"652","DOI":"10.1016\/j.jcp.2006.12.014","volume":"225","author":"B Ganapathysubramanian","year":"2007","unstructured":"Ganapathysubramanian, B., Zabaras, N.: Sparse grid collocation schemes for stochastic natural convection problems. J. Comput. Phys. 225, 652\u2013685 (2007)","journal-title":"J. Comput. Phys."},{"key":"1514_CR8","doi-asserted-by":"publisher","first-page":"259","DOI":"10.1017\/S096249291500001X","volume":"24","author":"MB Giles","year":"2015","unstructured":"Giles, M.B.: Multilevel Monte Carlo methods. Acta Numerica 24, 259\u2013328 (2015)","journal-title":"Acta Numerica"},{"key":"1514_CR9","doi-asserted-by":"publisher","first-page":"521","DOI":"10.1017\/S0962492914000075","volume":"23","author":"MD Gunzburger","year":"2014","unstructured":"Gunzburger, M.D., Webster, C.G., Zhang, G.: Stochastic finite element methods for partial differential equations with random input data. Acta Numer. 23, 521\u2013650 (2014)","journal-title":"Acta Numer."},{"key":"1514_CR10","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/S0951-8320(03)00058-9","volume":"81","author":"JC Helton","year":"2003","unstructured":"Helton, J.C., Davis, F.J.: Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliab. Eng. Syst. Safe. 81, 23\u201369 (2003)","journal-title":"Reliab. Eng. Syst. Safe."},{"key":"1514_CR11","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1615\/Int.J.UncertaintyQuantification.2014007691","volume":"4","author":"N Jiang","year":"2014","unstructured":"Jiang, N., Layton, W.: An algorithm for fast calculation of flow ensembles. Int. J. Uncertain. Quan. 4, 273\u2013301 (2014)","journal-title":"Int. J. Uncertain. Quan."},{"issue":"3","key":"1514_CR12","doi-asserted-by":"publisher","first-page":"677","DOI":"10.1093\/imanum\/drn066","volume":"30","author":"B Jin","year":"2010","unstructured":"Jin, B., Zou, J.: Numerical estimation of the Robin coefficient in a stationary diffusion equation. IMA J. Numer. Anal. 30(3), 677\u2013701 (2010)","journal-title":"IMA J. Numer. Anal."},{"key":"1514_CR13","doi-asserted-by":"publisher","first-page":"1369","DOI":"10.1090\/S0025-5718-2012-02559-2","volume":"81","author":"B Jin","year":"2012","unstructured":"Jin, B., Zou, J.: Numerical identification of a Robin coefficient in parabolic problems. Math. Comp. 81, 1369\u20131398 (2012)","journal-title":"Math. Comp."},{"issue":"12","key":"1514_CR14","first-page":"4905","volume":"25","author":"L Ju","year":"2020","unstructured":"Ju, L., Leng, W., Wang, Z., Yuan, S.: Numerical investigation of ensemble methods with block iterative solvers for evolution problems. Discrete Contin. Dyn. Syst. Ser. B 25(12), 4905\u20134923 (2020)","journal-title":"Discrete Contin. Dyn. Syst. Ser. B"},{"key":"1514_CR15","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1007\/s11009-017-9579-y","volume":"20","author":"A Kebaier","year":"2018","unstructured":"Kebaier, A., Lelong, J.: Coupling importance sampling and multilevel Monte Carlo using sample average approximation. Methodology and Computing in Applied Probability 20, 611\u2013641 (2018)","journal-title":"Methodology and Computing in Applied Probability"},{"key":"1514_CR16","doi-asserted-by":"publisher","first-page":"1755","DOI":"10.1007\/s11075-021-01250-2","volume":"90","author":"M Li","year":"2022","unstructured":"Li, M., Luo, X.: An EMC-HDG scheme for the convection-diffusion equation with random diffusivity. Numer. Algorithms 90, 1755\u20131776 (2022)","journal-title":"Numer. Algorithms"},{"key":"1514_CR17","doi-asserted-by":"publisher","first-page":"127","DOI":"10.1016\/j.camwa.2022.10.002","volume":"127","author":"M Li","year":"2022","unstructured":"Li, M., Luo, X.: An MLMCE-HDG method for the convection diffusion equation with random diffusivity. Comput. Math. Applic. 127, 127\u2013143 (2022)","journal-title":"Comput. Math. Applic."},{"key":"1514_CR18","doi-asserted-by":"publisher","first-page":"2477","DOI":"10.1080\/00207160.2013.784280","volume":"90","author":"K Liu","year":"2013","unstructured":"Liu, K., Riviere, B.M.: Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients. Int. J. Comput. Math. 90, 2477\u20132490 (2013)","journal-title":"Int. J. Comput. Math."},{"key":"1514_CR19","doi-asserted-by":"publisher","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, New York (2014)"},{"key":"1514_CR20","doi-asserted-by":"publisher","first-page":"859","DOI":"10.1137\/17M1131489","volume":"56","author":"Y Luo","year":"2018","unstructured":"Luo, Y., Wang, Z.: An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs. SIAM J. Numer. Anal. 56, 859\u2013876 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1514_CR21","doi-asserted-by":"publisher","first-page":"A622","DOI":"10.1137\/18M1174635","volume":"41","author":"Y Luo","year":"2019","unstructured":"Luo, Y., Wang, Z.: A multilevel Monte Carlo ensemble scheme for solving random parabolic PDEs. SIAM J. Sci. Comput. 41, A622\u2013A642 (2019)","journal-title":"SIAM J. Sci. Comput."},{"key":"1514_CR22","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1002\/nme.5210","volume":"108","author":"J Martinez-Frutos","year":"2016","unstructured":"Martinez-Frutos, J., Kessler, M., Mnch, A., Periago, F.: Robust optimal Robin boundary control for the transient heat equation with random input data. Int. J. Numer. Methods Eng. 108, 116\u2013135 (2016)","journal-title":"Int. J. Numer. Methods Eng."},{"key":"1514_CR23","doi-asserted-by":"crossref","unstructured":"Martinez-Frutos, J., Esparza, F.P.: Optimal Control of PDEs Under Uncertainty: An Introduction with Application to Optimal Shape Design of Structures. Springer (bcam) (2018)","DOI":"10.1007\/978-3-319-98210-6"},{"key":"1514_CR24","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1007\/s11075-004-2866-z","volume":"38","author":"L Mathelin","year":"2005","unstructured":"Mathelin, L., Hussaini, M.Y., Zang, T.A.: Stochastic approaches to uncertainty quantification in CFD simulations. Numer. Algorithms 38, 209\u2013236 (2005)","journal-title":"Numer. Algorithms"},{"key":"1514_CR25","doi-asserted-by":"publisher","first-page":"544","DOI":"10.1016\/j.cam.2013.06.014","volume":"255","author":"J Meng","year":"2014","unstructured":"Meng, J., Zhu, P.Y., Li, H.B.: A block method for linear systems with multiple right-hand sides. J. Comput. Appl. Math. 255, 544\u2013554 (2014)","journal-title":"J. Comput. Appl. Math."},{"key":"1514_CR26","doi-asserted-by":"publisher","first-page":"1118","DOI":"10.1137\/040615201","volume":"27","author":"D Xiu","year":"2005","unstructured":"Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27, 1118\u20131139 (2005)","journal-title":"SIAM J. Sci. Comput."},{"key":"1514_CR27","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1016\/j.jcp.2017.04.022","volume":"341","author":"X Zhu","year":"2017","unstructured":"Zhu, X., Linebarger, E.M., Xiu, D.: Multi-fidelity stochastic collocation method for computation of statistical moments. J. Comput. Phys. 341, 386\u2013396 (2017)","journal-title":"J. Comput. Phys."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01514-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-023-01514-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-023-01514-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,5]],"date-time":"2023-09-05T05:15:00Z","timestamp":1693890900000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-023-01514-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,21]]},"references-count":27,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,10]]}},"alternative-id":["1514"],"URL":"https:\/\/doi.org\/10.1007\/s11075-023-01514-z","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2349931\/v1","asserted-by":"object"}]},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"value":"1017-1398","type":"print"},{"value":"1572-9265","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,2,21]]},"assertion":[{"value":"6 December 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 February 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"Not applicable.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Ethics approval"}},{"value":"The authors declare no competing interests.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Competing interests"}}]}}