{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T23:24:12Z","timestamp":1768519452406,"version":"3.49.0"},"reference-count":55,"publisher":"Elsevier BV","issue":"1","license":[{"start":{"date-parts":[[2007,9,1]],"date-time":"2007-09-01T00:00:00Z","timestamp":1188604800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.elsevier.com\/tdm\/userlicense\/1.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Computational Physics"],"published-print":{"date-parts":[[2007,9]]},"DOI":"10.1016\/j.jcp.2007.04.009","type":"journal-article","created":{"date-parts":[[2007,4,25]],"date-time":"2007-04-25T11:10:09Z","timestamp":1177499409000},"page":"326-353","source":"Crossref","is-referenced-by-count":79,"title":["Modeling diffusion in random heterogeneous media: Data-driven models, stochastic collocation and the variational multiscale method"],"prefix":"10.1016","volume":"226","author":[{"given":"Baskar","family":"Ganapathysubramanian","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nicholas","family":"Zabaras","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"78","reference":[{"key":"10.1016\/j.jcp.2007.04.009_bib1","doi-asserted-by":"crossref","first-page":"271","DOI":"10.1146\/annurev.fl.19.010187.001415","article-title":"Viscous fingering in porous media","volume":"19","author":"Homsy","year":"1987","journal-title":"Ann. Rev. Fluid. Mech."},{"key":"10.1016\/j.jcp.2007.04.009_bib2","doi-asserted-by":"crossref","first-page":"446","DOI":"10.2118\/81909-PA","article-title":"A two-scale numerical subgrid technique for waterflood simulations","author":"Arbogast","year":"2002","journal-title":"SPE J."},{"key":"10.1016\/j.jcp.2007.04.009_bib3","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1016\/0045-7825(95)00844-9","article-title":"Multiscale phenomena: Green\u2019s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods","volume":"127","author":"Hughes","year":"1995","journal-title":"Comp. Meth. Appl. Mech. Eng."},{"issue":"1","key":"10.1016\/j.jcp.2007.04.009_bib4","doi-asserted-by":"crossref","first-page":"87","DOI":"10.4310\/CMS.2003.v1.n1.a8","article-title":"The heterogeneous multiscale methods","volume":"1","author":"Engquist","year":"2003","journal-title":"Comm. Math. Sci."},{"key":"10.1016\/j.jcp.2007.04.009_bib5","unstructured":"P.B. Ming, P.W. Zhang, Analysis of the heterogeneous multiscale method for elliptic homogenization problems, preprint, 2003."},{"key":"10.1016\/j.jcp.2007.04.009_bib6","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1006\/jcph.1997.5682","article-title":"A multiscale finite element method for elliptic problems in composite materials and porous media","volume":"134","author":"Hou","year":"1997","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib7","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1142\/S0219876204000083","article-title":"Generalized finite element methods: main ideas, results and perspective","volume":"1","author":"Babuska","year":"2004","journal-title":"Int. J. Comput. Meth."},{"key":"10.1016\/j.jcp.2007.04.009_bib8","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1137\/S1540345902411402","article-title":"Capturing small scales in elliptic problems using a residual-free bubbles finite element method","volume":"1","author":"Sangalli","year":"2003","journal-title":"SIAM MMS"},{"key":"10.1016\/j.jcp.2007.04.009_bib9","series-title":"Stochastic Finite Elements: A Spectral Approach","author":"Ghanem","year":"1991"},{"key":"10.1016\/j.jcp.2007.04.009_bib10","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1016\/S0045-7825(97)00250-8","article-title":"Probabilistic characterization of transport in heterogeneous porous media","volume":"158","author":"Ghanem","year":"1998","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.jcp.2007.04.009_bib11","doi-asserted-by":"crossref","first-page":"5499","DOI":"10.1016\/S0045-7825(02)00465-6","article-title":"Mid-frequency structural dynamics with parameter uncertainty","volume":"191","author":"Ghanem","year":"2002","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.jcp.2007.04.009_bib12","doi-asserted-by":"crossref","first-page":"4927","DOI":"10.1016\/S0045-7825(02)00421-8","article-title":"Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos","volume":"191","author":"Xiu","year":"2002","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.jcp.2007.04.009_bib13","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0021-9991(03)00092-5","article-title":"Modeling uncertainty in flow simulations via generalized polynomial chaos","volume":"187","author":"Xiu","year":"2003","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib14","doi-asserted-by":"crossref","first-page":"1251","DOI":"10.1016\/j.cma.2004.02.026","article-title":"Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation","volume":"194","author":"Babuska","year":"2005","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.jcp.2007.04.009_bib15","doi-asserted-by":"crossref","first-page":"800","DOI":"10.1137\/S0036142902418680","article-title":"Galerkin finite elements approximation of stochastic finite elements","volume":"42","author":"Babuska","year":"2004","journal-title":"SIAM J. Numer. Anal."},{"key":"10.1016\/j.jcp.2007.04.009_bib16","series-title":"International Conference on Computational Science","first-page":"346","article-title":"Performance evaluation of generalized polynomial chaos","volume":"vol. 2660","author":"Xiu","year":"2003"},{"key":"10.1016\/j.jcp.2007.04.009_bib17","doi-asserted-by":"crossref","unstructured":"B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comp. Physics (2007), doi:10.1016\/j.jcp.2006.12.014.","DOI":"10.1016\/j.jcp.2006.12.014"},{"key":"10.1016\/j.jcp.2007.04.009_bib18","doi-asserted-by":"crossref","first-page":"1118","DOI":"10.1137\/040615201","article-title":"High-order collocation methods for the differential equation with random inputs","volume":"27","author":"Xiu","year":"2005","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"10.1016\/j.jcp.2007.04.009_bib19","first-page":"293","article-title":"Efficient collocational approach for parametric uncertainty analysis","volume":"2","author":"Xiu","year":"2007","journal-title":"Comm. Comput. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib20","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1016\/j.jcp.2005.03.023","article-title":"An adaptive multi-element generalized polynomial chaos method for stochastic differential equations","volume":"209","author":"Wan","year":"2005","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib21","doi-asserted-by":"crossref","first-page":"134","DOI":"10.1016\/j.jcp.2005.02.007","article-title":"Using stochastic analysis to capture unstable equilibrium in natural convection","volume":"208","author":"Velamur Asokan","year":"2005","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib22","unstructured":"I. Babuska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, ICES Report 05-47, 2005."},{"key":"10.1016\/j.jcp.2007.04.009_bib23","unstructured":"F. Nobile, R. Tempone, C.G. Webster, A sparse grid stochastic collocation method for elliptic partial differential equations with random input data, preprint."},{"key":"10.1016\/j.jcp.2007.04.009_bib24","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1023\/A:1019129717644","article-title":"Numerical integration using sparse grids","volume":"18","author":"Gerstner","year":"1998","journal-title":"Numer. Algor."},{"key":"10.1016\/j.jcp.2007.04.009_bib25","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."},{"key":"10.1016\/j.jcp.2007.04.009_bib26","unstructured":"A. Klimke, Uncertainty Modeling using Fuzzy Arithmetic and Sparse Grids, PhD Thesis, Universit\u00e4t Stuttgart, Shaker Verlag, Aachen, 2006."},{"key":"10.1016\/j.jcp.2007.04.009_bib27","doi-asserted-by":"crossref","first-page":"654","DOI":"10.1016\/j.jcp.2006.02.026","article-title":"A stochastic variational multiscale method for diffusion in random heterogeneous media","volume":"218","author":"Velamur Asokan","year":"2006","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib28","doi-asserted-by":"crossref","first-page":"662","DOI":"10.1137\/03060268X","article-title":"A two-scale non-perturbative approach to uncertainty analysis of diffusion in random composites","volume":"2","author":"Xiu","year":"2004","journal-title":"SIAM J. Multiscale Model. Simul."},{"key":"10.1016\/j.jcp.2007.04.009_bib29","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/j.jcp.2006.01.037","article-title":"On the construction and analysis of stochastic models: characterization and propagation of the errors associated with limited data","volume":"217","author":"Doostan","year":"2006","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib30","doi-asserted-by":"crossref","first-page":"57","DOI":"10.1016\/j.jhydrol.2004.02.007","article-title":"Probabilistic reconstruction of geologic facies","volume":"294","author":"Guadagnini","year":"2004","journal-title":"J. Hydrol."},{"key":"10.1016\/j.jcp.2007.04.009_bib31","doi-asserted-by":"crossref","first-page":"1759","DOI":"10.1029\/1999GL011030","article-title":"Mean flow in composite porous media","volume":"27","author":"Winter","year":"2000","journal-title":"Geophys. Res. Lett."},{"key":"10.1016\/j.jcp.2007.04.009_bib32","doi-asserted-by":"crossref","first-page":"23.1","DOI":"10.1029\/2001WR000450","article-title":"Groundwater flow in heterogeneous composite aquifers","volume":"38","author":"Winter","year":"2002","journal-title":"Water Resour. Res."},{"key":"10.1016\/j.jcp.2007.04.009_bib33","doi-asserted-by":"crossref","first-page":"978","DOI":"10.1002\/nme.1576","article-title":"Maximum likelihood estimation of stochastic chaos representations from experimental data","volume":"66","author":"Desceliers","year":"2006","journal-title":"Int. J. Numer. Meth. Eng."},{"key":"10.1016\/j.jcp.2007.04.009_bib34","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.probengmech.2005.11.004","article-title":"Probabilistic characterization and simulation of multi-phase random media","volume":"21","author":"Koutsourelakis","year":"2006","journal-title":"Probabil. Eng. Mech."},{"key":"10.1016\/j.jcp.2007.04.009_bib35","doi-asserted-by":"crossref","first-page":"2279","DOI":"10.1016\/j.actamat.2006.11.025","article-title":"Computing property variability of polycrystals induced by grain size and orientation uncertainties","volume":"55","author":"Sankaran","year":"2007","journal-title":"Acta Mater."},{"key":"10.1016\/j.jcp.2007.04.009_bib36","series-title":"Random Heterogeneous Materials: Microstructure and Macroscopic Properties","author":"Torquato","year":"2002"},{"key":"10.1016\/j.jcp.2007.04.009_bib37","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1103\/PhysRevE.58.224","article-title":"Reconstructing random media. II. Three-dimensional media from two-dimensional cuts","volume":"58","author":"Yeong","year":"1998","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.jcp.2007.04.009_bib38","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1146\/annurev.matsci.32.110101.155324","article-title":"Statistical description of microstructures","volume":"32","author":"Torquato","year":"2002","journal-title":"Ann. Rev. Mater. Sci."},{"key":"10.1016\/j.jcp.2007.04.009_bib39","doi-asserted-by":"crossref","first-page":"1847","DOI":"10.1063\/1.102184","article-title":"Efficient simulation technique to compute effective properties of heterogeneous media","volume":"55","author":"Torquato","year":"1989","journal-title":"Appl. Phys. Lett."},{"issue":"8","key":"10.1016\/j.jcp.2007.04.009_bib40","doi-asserted-by":"crossref","first-page":"3923","DOI":"10.1063\/1.1388558","article-title":"Gaussian random fields with two level-cuts \u2013 model for asymmetric microemulsions with nonzero spontaneous curvature","volume":"115","author":"Arleth","year":"2001","journal-title":"J. Chem. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib41","doi-asserted-by":"crossref","first-page":"2313","DOI":"10.1103\/PhysRevE.54.2313","article-title":"Structure property correlations in model composite materials","volume":"54","author":"Roberts","year":"1996","journal-title":"Phys. Rev. E"},{"key":"10.1016\/j.jcp.2007.04.009_bib42","doi-asserted-by":"crossref","first-page":"2029","DOI":"10.1016\/S0022-5096(99)00016-2","article-title":"Elastic properties of a tungsten\u2013silver composite by reconstruction and computation","volume":"47","author":"Roberts","year":"1999","journal-title":"J. Mech. Phys. Solids"},{"key":"10.1016\/j.jcp.2007.04.009_bib43","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1061\/(ASCE)0733-9399(2006)132:6(619)","article-title":"Simulation of multi-dimensional binary random fields with application to modeling of two phase random media","volume":"132","author":"Koutsourelakis","year":"2006","journal-title":"J. Eng. Mech."},{"key":"10.1016\/j.jcp.2007.04.009_bib44","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1063\/1.1327609","article-title":"Generating microstructures with specified correlation functions","volume":"89","author":"Sheehan","year":"2001","journal-title":"J. Appl. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib45","doi-asserted-by":"crossref","first-page":"2265","DOI":"10.1016\/j.actamat.2006.01.015","article-title":"A maximum entropy approach for property prediction of random microstructures","volume":"54","author":"Sankaran","year":"2006","journal-title":"Acta Mater."},{"key":"10.1016\/j.jcp.2007.04.009_bib46","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/j.msea.2003.09.021","article-title":"Simulation of liquid phase sintering using the Monte-Carlo method","volume":"365","author":"Aldazabal","year":"2004","journal-title":"Mater. Sci. Eng. A"},{"key":"10.1016\/j.jcp.2007.04.009_bib47","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.jcp.2005.05.013","article-title":"A level set simulation of dendritic solidification with combined features of front tracking and fixed domain methods","volume":"211","author":"Tan","year":"2006","journal-title":"J. Comp. Phys."},{"issue":"1","key":"10.1016\/j.jcp.2007.04.009_bib48","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1016\/j.jcp.2006.06.003","article-title":"A level set simulation of dendritic solidification of multi-component alloys","volume":"221","author":"Tan","year":"2007","journal-title":"J. Comp. Phys."},{"key":"10.1016\/j.jcp.2007.04.009_bib49","series-title":"Turbulence, Coherent Structures, Dynamical Systems and Symmetry","author":"Holmes","year":"1998"},{"key":"10.1016\/j.jcp.2007.04.009_bib50","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1016\/j.commatsci.2004.07.004","article-title":"Classification of three-dimensional microstructures using support vector machines","volume":"32","author":"Sundararaghavan","year":"2005","journal-title":"Comp. Mat. Sci."},{"key":"10.1016\/j.jcp.2007.04.009_bib51","series-title":"Numerical Optimization","author":"Nocedal","year":"2000"},{"key":"10.1016\/j.jcp.2007.04.009_bib52","unstructured":"<https:\/\/2.zoppoz.workers.dev:443\/http\/www.mathworks.com\/products\/matlab\/>."},{"key":"10.1016\/j.jcp.2007.04.009_bib53","doi-asserted-by":"crossref","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":"ACM Transactions on Mathematical Software"},{"key":"10.1016\/j.jcp.2007.04.009_bib54","unstructured":"A. Klimke, Sparse Grid Interpolation Toolbox \u2013 User\u2019s Guide, IANS Report 2006\/001, University of Stuttgart, 2006."},{"key":"10.1016\/j.jcp.2007.04.009_bib55","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1016\/0022-5096(65)90012-8","article-title":"Elastic properties of a tungsten\u2013silver composite above and below the melting point of silver","volume":"13","author":"Umekawa","year":"1965","journal-title":"J. Mech. Phys. Solids"}],"container-title":["Journal of Computational Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/api.elsevier.com\/content\/article\/PII:S002199910700160X?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/api.elsevier.com\/content\/article\/PII:S002199910700160X?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2019,4,27]],"date-time":"2019-04-27T13:40:19Z","timestamp":1556372419000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/linkinghub.elsevier.com\/retrieve\/pii\/S002199910700160X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,9]]},"references-count":55,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2007,9]]}},"alternative-id":["S002199910700160X"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1016\/j.jcp.2007.04.009","relation":{},"ISSN":["0021-9991"],"issn-type":[{"value":"0021-9991","type":"print"}],"subject":[],"published":{"date-parts":[[2007,9]]}}}