{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,27]],"date-time":"2026-09-27T18:08:54Z","timestamp":1790532534651,"version":"4.1.0"},"reference-count":75,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["F65"],"award-info":[{"award-number":["F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012190","name":"Ministry of Science and Higher Education","doi-asserted-by":"publisher","award":["075-10-2021-068"],"award-info":[{"award-number":["075-10-2021-068"]}],"id":[{"id":"10.13039\/501100012190","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Multiscale Model. Simul."],"published-print":{"date-parts":[[2022,9,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Homogenization in terms of multiscale limits transforms a multiscale problem with [Formula: see text] asymptotically separated microscales posed on a physical domain [Formula: see text] into a one-scale problem posed on a product domain of dimension [Formula: see text] by introducing [Formula: see text] so-called fast variables. This procedure allows one to convert [Formula: see text] scales in [Formula: see text] physical dimensions into a single-scale structure in [Formula: see text] dimensions. We prove here that both the original, physical multiscale problem and the corresponding high-dimensional, one-scale limiting problem can be efficiently treated numerically with the recently developed quantized tensor-train finite-element method (QTT-FEM). The QTT-FE approximation consists in restricting approximation and computation to sequences of nested subspaces, each of which is a tensor product of two factors of low dimension (rank), within a vast, but generic \u201cvirtual\u201d (background) discretization space. In practice, these subspaces are determined iteratively and data-adaptively at runtime bypassing any \u201coffline precomputation.\u201d For theoretical analysis, low-dimensional subspaces are constructed analytically to bound the tensor ranks against the error tolerance. We consider a model linear elliptic multiscale problem in several physical dimensions and show, theoretically and experimentally, that both (i) the solution of the associated high-dimensional one-scale problem and (ii) the approximation to the solution of the multiscale problem induced thereby admit efficient QTT-FE approximations. These problems can therefore be numerically solved in a scale-robust fashion by standard (low-order) PDE discretizations combined with state-of-the-art general-purpose solvers for tensor-structured linear systems. Specifically, we prove the existence of QTT-FE approximations with an upper bound on the tensor ranks growing no faster than algebraically with respect to [Formula: see text] and independent of the scale parameters, where [Formula: see text] is the target accuracy for the approximation of the solution and of its gradient. In numerical experiments, we verify the theoretical rank bounds and computationally investigate the dependence of the complexity of the solutions on the number [Formula: see text] of microscales.<\/jats:p>","DOI":"10.1137\/20m1341659","type":"journal-article","created":{"date-parts":[[2022,8,1]],"date-time":"2022-08-01T19:31:46Z","timestamp":1659382306000},"page":"893-935","source":"Crossref","is-referenced-by-count":11,"title":["Quantized Tensor FEM for Multiscale Problems: Diffusion Problems in Two and Three Dimensions"],"prefix":"10.1137","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9445-3267","authenticated-orcid":true,"given":"Vladimir","family":"Kazeev","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, University of Vienna, Kolingasse 14\u201316, 1090 Vienna, Austria ()."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ivan","family":"Oseledets","sequence":"additional","affiliation":[{"name":"Skolkovo Institute of Science and Technology, Novaya Ul. 100, 143025 Skolkovo, Moscow Region, Russia and Institute of Numerical Mathematics, Russian Academy of Sciences, Gubkina St. 8, 119333, Moscow, Russia ()."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Maxim V.","family":"Rakhuba","sequence":"additional","affiliation":[{"name":"HSE University, Pokrovsky Boulevard 11, Moscow, 109028 Russia ()."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Schwab","sequence":"additional","affiliation":[{"name":"Seminar for Applied Mathematics, ETH Zurich, R\u00e4mistrasse 101, 8092 Zurich, Switzerland ()."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2022,6,28]]},"reference":[{"key":"ref1","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492912000025"},{"key":"ref2","unstructured":"M. Ali  and \nA. Nouy , Approximation with Tensor Networks. Part I: Approximation Spaces, preprint, http:\/\/arxiv.org\/abs\/2007.00118, 2020."},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1137\/0523084"},{"key":"ref4","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500022757"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492921000015"},{"key":"ref6","doi-asserted-by":"publisher","DOI":"10.1137\/100791051"},{"key":"ref7","doi-asserted-by":"publisher","DOI":"10.1137\/0725048"},{"key":"ref8","doi-asserted-by":"publisher","DOI":"10.1137\/0519014"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/0520054"},{"key":"ref10","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-020-09446-z"},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-016-9314-z"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-016-9317-9"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-2247-1"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1818"},{"key":"ref15","volume-title":"Stud. Math. Appl. 5","author":"Bensoussan A.","year":"1978"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9045(72)90075-5"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970869"},{"key":"ref18","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2020.112756"},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1007\/BF01396415"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1137\/100817942"},{"key":"ref21","doi-asserted-by":"publisher","DOI":"10.1137\/080713148"},{"key":"ref22","series-title":"GAKUTO Internat. Ser.","first-page":"1","volume-title":"Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics","author":"Cioranescu D.","year":"2009"},{"key":"ref23","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492906360011"},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1137\/090764189"},{"key":"ref25","unstructured":"L. Grasedyck , Polynomial Approximation in Hierarchical Tucker Format by Vector-Tensorization, Preprint 308, Institut f\u00fcr Geometrie und Praktische Mathematik, RWTH Aachen, 2010, http:\/\/www.igpm.rwth-aachen.de\/Download\/reports\/pdf\/IGPM308_k.pdf."},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-28027-6"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s00041-009-9094-9"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2011.07.001"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/070698683"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/030601077"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1137\/100818893"},{"key":"ref32","doi-asserted-by":"publisher","DOI":"10.1142\/9789814273268_0003"},{"key":"ref33","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01077-7"},{"key":"ref34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-84659-5"},{"key":"ref35","unstructured":"V. Kazeev , Quantized Tensor-Structured Finite Elements for Second-order Elliptic PDEs in Two Dimensions, Ph.D. thesis, SAM, ETH Zurich, ETH Dissertation 23002, 2015, https:\/\/doi.org\/10.3929\/ethz-a-010554062."},{"key":"ref36","unstructured":"V. Kazeev , TensorRefinement.jl, a Julia Package for Low-Rank Computations with Functions, 2022, http:\/\/github.com\/TensorRefinement\/TensorRefinement.jl."},{"key":"ref37","unstructured":"V. Kazeev , Low-Rank Tensor Frames, in preparation, http:\/\/mat.univie.ac.at\/\u223ckazeev\/pub."},{"key":"ref38","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-016-9491-y"},{"key":"ref39","doi-asserted-by":"publisher","DOI":"10.1002\/pamm.201510353"},{"key":"ref40","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-017-0899-1"},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1137\/100820479"},{"key":"ref42","doi-asserted-by":"publisher","DOI":"10.1515\/cmam-2017-0014"},{"key":"ref43","unstructured":"B. N. Khoromskij ,   \\(\\mathcal{O}(d \\log n)\\)  -Quantics Approximation of   \\(n\\)  -  \\(d\\)   Tensors in High-Dimensional Numerical Modeling, Preprint 55, Max Planck Institute for Mathematics in the Sciences, 2009, http:\/\/www.mis.mpg.de\/publications\/preprints\/2009\/prepr2009-55.html."},{"key":"ref44","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-011-9131-1"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1515\/9783110365917"},{"key":"ref46","first-page":"329","volume":"30","author":"Khoromskij B. N.","year":"2015","journal-title":"Russian J. Numer. Anal. Model."},{"key":"ref47","first-page":"1600","volume":"4","author":"Kondrat\u2019ev V. A.","year":"1963","journal-title":"Soviet Math."},{"key":"ref48","first-page":"209","volume":"16","author":"Kondrat\u2019ev V. A.","year":"1967","journal-title":"Trans. Moscow Math. Soc."},{"key":"ref49","doi-asserted-by":"publisher","DOI":"10.1137\/15M1032909"},{"key":"ref50","doi-asserted-by":"publisher","DOI":"10.1137\/100799010"},{"key":"ref51","first-page":"727","volume":"8","author":"Lehman R. S.","year":"1959","journal-title":"J. Math. Mech."},{"key":"ref52","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-022-09925-7"},{"key":"ref53","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202503002817"},{"key":"ref54","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2032-9_3"},{"key":"ref55","doi-asserted-by":"publisher","DOI":"10.1137\/0520043"},{"key":"ref56","doi-asserted-by":"publisher","DOI":"10.4171\/ZAA\/1133"},{"key":"ref57","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2014.06.013"},{"key":"ref58","doi-asserted-by":"publisher","DOI":"10.1134\/S1064562409050056"},{"key":"ref59","doi-asserted-by":"publisher","DOI":"10.1137\/090757861"},{"key":"ref60","doi-asserted-by":"publisher","DOI":"10.1137\/090752286"},{"key":"ref61","doi-asserted-by":"publisher","DOI":"10.1137\/110833142"},{"key":"ref62","doi-asserted-by":"publisher","DOI":"10.1137\/090748330"},{"key":"ref63","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.75.3537"},{"key":"ref64","unstructured":"T. Rohwedder  and \nA. Uschmajew , Local Convergence of Alternating Schemes for Optimization of Convex Problems in the TT Format, Preprint 112, DFG-Schwerpunktprogramm 1324, 2012, http:\/\/www.dfg-spp1324.de\/download\/preprints\/preprint112.pdf."},{"key":"ref65","doi-asserted-by":"publisher","DOI":"10.1016\/j.aop.2010.09.012"},{"key":"ref66","unstructured":"C. Schwab , High-dimensional finite elements for elliptic problems with multiple scales and stochastic data, in Proceedings of the International Congress of Mathematicians, Vol. III, Higher Education Press, Beijing, 2002, pp. 727\u2013734."},{"key":"ref67","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-018-09657-7"},{"key":"ref68","doi-asserted-by":"publisher","DOI":"10.1070\/SM2003v194n06ABEH000747"},{"key":"ref69","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.03.016"},{"key":"ref70","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevB.73.094423"},{"key":"ref71","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.91.147902"},{"key":"ref72","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.69.2863"},{"key":"ref73","doi-asserted-by":"publisher","DOI":"10.1137\/120902409"},{"key":"ref74","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dru033"},{"key":"ref75","doi-asserted-by":"publisher","DOI":"10.1016\/j.cam.2015.01.001"}],"container-title":["Multiscale Modeling &amp; Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M1341659","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T12:19:58Z","timestamp":1787228398000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M1341659"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,6,28]]},"references-count":75,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2022,9,30]]}},"alternative-id":["10.1137\/20M1341659"],"URL":"https:\/\/doi.org\/10.1137\/20m1341659","relation":{},"ISSN":["1540-3459","1540-3467"],"issn-type":[{"value":"1540-3459","type":"print"},{"value":"1540-3467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,6,28]]}}}