{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,11]],"date-time":"2026-02-11T18:50:45Z","timestamp":1770835845065,"version":"3.50.1"},"reference-count":43,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2018,2,24]],"date-time":"2018-02-24T00:00:00Z","timestamp":1519430400000},"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":["J Sci Comput"],"published-print":{"date-parts":[[2018,8]]},"DOI":"10.1007\/s10915-018-0652-7","type":"journal-article","created":{"date-parts":[[2018,2,24]],"date-time":"2018-02-24T15:53:48Z","timestamp":1519487628000},"page":"1105-1126","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Galerkin Methods for Stationary Radiative Transfer Equations with Uncertain Coefficients"],"prefix":"10.1007","volume":"76","author":[{"given":"Xinghui","family":"Zhong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qin","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2018,2,24]]},"reference":[{"key":"652_CR1","doi-asserted-by":"crossref","unstructured":"Albi, G., Pareschi, L., Zanella, M.: Uncertainty quantification in control problems for flocking models. Math. Probl. Eng.(ID 850124), 14 (2015)","DOI":"10.1155\/2015\/850124"},{"issue":"3","key":"652_CR2","doi-asserted-by":"publisher","first-page":"1005","DOI":"10.1137\/050645142","volume":"45","author":"I Babuska","year":"2007","unstructured":"Babuska, 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":"2","key":"652_CR3","doi-asserted-by":"publisher","first-page":"800","DOI":"10.1137\/S0036142902418680","volume":"42","author":"I Babuska","year":"2004","unstructured":"Babuska, I., Tempone, R., Zouraris, G.: Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J. Numer. Anal. 42(2), 800\u2013825 (2004)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1","key":"652_CR4","doi-asserted-by":"publisher","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. Numerische Mathematik 119(1), 123\u2013161 (2011)","journal-title":"Numerische Mathematik"},{"issue":"1","key":"652_CR5","doi-asserted-by":"publisher","first-page":"55","DOI":"10.4310\/CMS.2013.v11.n1.a3","volume":"11","author":"M Branicki","year":"2013","unstructured":"Branicki, M., Majda, A.J.: Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities. Commun. Math. Sci. 11(1), 55\u2013103 (2013)","journal-title":"Commun. Math. Sci."},{"issue":"11","key":"652_CR6","doi-asserted-by":"publisher","first-page":"1376","DOI":"10.1063\/1.1703916","volume":"4","author":"KM Case","year":"1963","unstructured":"Case, K.M., Zweifel, P.F.: Existence and uniqueness theorems for the neutron transport equation. J. Math. Phys. 4(11), 1376\u20131385 (1963)","journal-title":"J. Math. Phys."},{"key":"652_CR7","volume-title":"Linear Transport Theory","author":"KM Case","year":"1967","unstructured":"Case, K.M., Zweifel, P.F.: Linear Transport Theory. Addison-Wesley Pub. Co., Boston (1967)"},{"key":"652_CR8","volume-title":"The Boltzmann Equation and its Applications","author":"C Cercignani","year":"2012","unstructured":"Cercignani, C.: The Boltzmann Equation and its Applications. Springer, Berlin (2012)"},{"key":"652_CR9","volume-title":"Radiative Transfer. Dover Books on Intermediate and Advanced Mathematics","author":"S Chandrasekhar","year":"1960","unstructured":"Chandrasekhar, S.: Radiative Transfer. Dover Books on Intermediate and Advanced Mathematics. Dover Publications, Mineola (1960)"},{"issue":"1","key":"652_CR10","doi-asserted-by":"publisher","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(1), 322\u2013352 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"652_CR11","doi-asserted-by":"crossref","unstructured":"Chkifa, A., Cohen, A., Schwab, C.: Breaking the curse of dimensionality in sparse polynomial approximation of parametric PDEs. Journal de Math\u00e9matiques Pures et Appliqu\u00e9es (2014)","DOI":"10.1016\/j.matpur.2014.04.009"},{"issue":"1","key":"652_CR12","first-page":"87","volume":"36","author":"M Choulli","year":"1999","unstructured":"Choulli, M., Stefanov, P.: An inverse boundary value problem for the stationary transport equation. Osaka J. Math. 36(1), 87\u2013104 (1999)","journal-title":"Osaka J. Math."},{"issue":"6","key":"652_CR13","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1007\/s10208-010-9072-2","volume":"10","author":"A Cohen","year":"2010","unstructured":"Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10(6), 615\u2013646 (2010)","journal-title":"Found. Comput. Math."},{"issue":"01","key":"652_CR14","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1142\/S0219530511001728","volume":"09","author":"A Cohen","year":"2011","unstructured":"Cohen, A., Devore, R., Schwab, C.: Analytic regularity and polynomial approximation of parametric and stochastic elliptic pde\u2019s. Anal. Appl. 09(01), 11\u201347 (2011)","journal-title":"Anal. Appl."},{"key":"652_CR15","volume-title":"Mathematical Analysis and Numerical Methods for Science and Technology","author":"R Dautray","year":"1993","unstructured":"Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology. Springer, Berlin (1993)"},{"issue":"1","key":"652_CR16","doi-asserted-by":"publisher","first-page":"980","DOI":"10.1137\/15M1018861","volume":"4","author":"B Despres","year":"2016","unstructured":"Despres, B., Perthame, B.: Uncertainty propagation; intrusive kinetic formulations of scalar conservation laws. SIAM\/ASA J. Uncerta. Quantif. 4(1), 980\u20131013 (2016)","journal-title":"SIAM\/ASA J. Uncerta. Quantif."},{"issue":"6","key":"652_CR17","doi-asserted-by":"publisher","first-page":"1283","DOI":"10.1080\/00036811.2013.826798","volume":"93","author":"H Egger","year":"2014","unstructured":"Egger, H., Schlottbom, M.: An Lp theory for stationary radiative transfer. Appl. Anal. 93(6), 1283\u20131296 (2014)","journal-title":"Appl. Anal."},{"key":"652_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic Finite Elements: A Spectral Approach","author":"RG Ghanem","year":"1991","unstructured":"Ghanem, R.G., Spanos, P.D.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)"},{"issue":"3","key":"652_CR19","doi-asserted-by":"publisher","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."},{"issue":"2","key":"652_CR20","first-page":"505","volume":"3","author":"D Gottlieb","year":"2008","unstructured":"Gottlieb, D., Xiu, D.: Galerkin method for wave equations with uncertain coefficients. Commun. Comput. Phys 3(2), 505\u2013518 (2008)","journal-title":"Commun. Comput. Phys"},{"issue":"6","key":"652_CR21","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1007\/BF01389733","volume":"49","author":"W Gui","year":"1986","unstructured":"Gui, W., Babuska, I.: The h, p and h\u2013p versions of the finite element method in 1 dimension. Part 1. The error analysis of the p-version. Numer. Math. 49(6), 577\u2013612 (1986)","journal-title":"Numer. Math."},{"issue":"2","key":"652_CR22","doi-asserted-by":"publisher","first-page":"661","DOI":"10.1137\/16M1077611","volume":"15","author":"YT Hou","year":"2016","unstructured":"Hou, Y.T., Li, Q., Zhang, P.: Exploring the locally low dimensional structure in solving random elliptic PDEs. SIAM Multisc. Model. Simul. 15(2), 661\u2013695 (2016)","journal-title":"SIAM Multisc. Model. Simul."},{"issue":"1","key":"652_CR23","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1137\/16M107760X","volume":"15","author":"YT Hou","year":"2016","unstructured":"Hou, Y.T., Li, Q., Zhang, P.: A sparse decomposition of low rank symmetric positive semi-definite matrices. SIAM Multisc. Model. Simul. 15(1), 410\u2013444 (2016)","journal-title":"SIAM Multisc. Model. Simul."},{"key":"652_CR24","doi-asserted-by":"crossref","unstructured":"Hu, J., Jin, S., Li, Q.: Asymptotic-preserving schemes for multiscale hyperbolic and kinetic equations. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. North Holland\/Elsevier (2017)","DOI":"10.1016\/bs.hna.2016.09.001"},{"key":"652_CR25","first-page":"177","volume":"3","author":"S Jin","year":"2012","unstructured":"Jin, S.: Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Rivista di Matematica della Universita di Parma 3, 177\u2013216 (2012)","journal-title":"Rivista di Matematica della Universita di Parma"},{"key":"652_CR26","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1016\/j.jcp.2015.02.023","volume":"289","author":"S Jin","year":"2015","unstructured":"Jin, S., Xiu, D., Zhu, X.: Asymptotic-preserving methods for hyperbolic and transport equations with random inputs and diffusive scalings. J. Comput. Phys. 289, 35\u201352 (2015)","journal-title":"J. Comput. Phys."},{"key":"652_CR27","unstructured":"Knutson, A., Tao, T.: Honeycombs and sums of hermitian matrices. arXiv:math\/0009048v1 [math.RT] (2000)"},{"key":"652_CR28","unstructured":"Li, Q., Wang, L.: Polynomial interpolation of burgers\u2019 equation with randomness. arXiv:1708.04332 [math] (2017)"},{"key":"652_CR29","doi-asserted-by":"crossref","unstructured":"Li, Q., Wang, L.: Uniform regularity for linear kinetic equations with random input based on hypocoercivity. SIAM\/ASA J. Uncertain. Quantif. (2017)","DOI":"10.1137\/16M1106675"},{"key":"652_CR30","doi-asserted-by":"publisher","first-page":"235207","DOI":"10.1103\/PhysRevB.84.235207","volume":"84","author":"AJ Minnich","year":"2011","unstructured":"Minnich, A.J., Chen, G., Mansoor, S., Yilbas, B.S.: Quasiballistic heat transfer studied using the frequency-dependent boltzmann transport equation. Phys. Rev. B 84, 235207 (2011)","journal-title":"Phys. Rev. B"},{"key":"652_CR31","doi-asserted-by":"publisher","first-page":"552","DOI":"10.1137\/120896967","volume":"4","author":"S Mishra","year":"2016","unstructured":"Mishra, S., Risebro, N.H., Schwab, C., Tokareva, S.: Numerical solution of scalar conservation laws with random flux functions. SIAM\/ASA J. Uncertain. Quantif. 4, 552\u2013591 (2016)","journal-title":"SIAM\/ASA J. Uncertain. Quantif."},{"key":"652_CR32","doi-asserted-by":"publisher","first-page":"1979","DOI":"10.1090\/S0025-5718-2012-02574-9","volume":"81","author":"S Mishra","year":"2012","unstructured":"Mishra, S., Schwab, C.: Sparse tensor multi-level monte carlo finite volume methods for hyperbolic conservation laws with random initial data. Math. Comp. 81, 1979\u20132018 (2012)","journal-title":"Math. Comp."},{"key":"652_CR33","first-page":"225294","volume-title":"Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws","author":"S Mishra","year":"2013","unstructured":"Mishra, S., Schwab, C., \u0160ukys, J.: Multi-level Monte Carlo Finite Volume Methods for Uncertainty Quantification in Nonlinear Systems of Balance Laws, p. 225294. Springer, Cham (2013)"},{"issue":"5","key":"652_CR34","doi-asserted-by":"publisher","first-page":"2411","DOI":"10.1137\/070680540","volume":"46","author":"F Nobile","year":"2008","unstructured":"Nobile, F., Tempone, R., Webster, C.G.: An anisotropic sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2411\u20132442 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"5","key":"652_CR35","doi-asserted-by":"publisher","first-page":"2309","DOI":"10.1137\/060663660","volume":"46","author":"F Nobile","year":"2008","unstructured":"Nobile, F., Tempone, R., Webster, C.G.: A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM J. Numer. Anal. 46(5), 2309\u20132345 (2008)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"652_CR36","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1007\/s00211-003-0455-z","volume":"95","author":"C Schwab","year":"2003","unstructured":"Schwab, C., Todor, R.-A.: Sparse finite elements for elliptic problems with stochastic loading. Numerische Mathematik 95(4), 707\u2013734 (2003)","journal-title":"Numerische Mathematik"},{"issue":"4","key":"652_CR37","doi-asserted-by":"publisher","first-page":"897","DOI":"10.2307\/2371268","volume":"60","author":"N Wiener","year":"1938","unstructured":"Wiener, N.: The homogeneous chaos. Am. J. Math. 60(4), 897\u2013936 (1938)","journal-title":"Am. J. Math."},{"key":"652_CR38","unstructured":"Xiu, D.: Generalized (Weiner-Askey) Polynomial Chaos. Ph.D. thesis, Brown University (2004)"},{"key":"652_CR39","doi-asserted-by":"crossref","DOI":"10.2307\/j.ctv7h0skv","volume-title":"Numerical Methods for Stochastic Computations: A Spectral Method Approach","author":"D Xiu","year":"2010","unstructured":"Xiu, D.: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010)"},{"issue":"3","key":"652_CR40","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(3), 1118\u20131139 (2005)","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"652_CR41","doi-asserted-by":"publisher","first-page":"619","DOI":"10.1137\/S1064827501387826","volume":"24","author":"D Xiu","year":"2002","unstructured":"Xiu, D., Karniadakis, G.: The Wiener\u2013Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619\u2013644 (2002)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"652_CR42","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1016\/S0021-9991(03)00092-5","volume":"187","author":"D Xiu","year":"2003","unstructured":"Xiu, D., Karniadakis, G.E.: Modeling uncertainty in flow simulations via generalized polynomial chaos. J. Comput. Phys. 187(1), 137\u2013167 (2003)","journal-title":"J. Comput. Phys."},{"issue":"4","key":"652_CR43","doi-asserted-by":"publisher","first-page":"1922","DOI":"10.1137\/11084306X","volume":"50","author":"G Zhang","year":"2012","unstructured":"Zhang, G., Gunzburger, M.: Error analysis of a stochastic collocation method for parabolic partial differential equations with random input data. SIAM J. Numer. Anal. 50(4), 1922\u20131940 (2012)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0652-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0652-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0652-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,10,11]],"date-time":"2019-10-11T20:14:14Z","timestamp":1570824854000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0652-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,2,24]]},"references-count":43,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2018,8]]}},"alternative-id":["652"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0652-7","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,2,24]]},"assertion":[{"value":"8 April 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 November 2017","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 January 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 February 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}