{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:47:44Z","timestamp":1776836864249,"version":"3.51.2"},"reference-count":37,"publisher":"American Mathematical Society (AMS)","issue":"342","license":[{"start":{"date-parts":[[2024,3,7]],"date-time":"2024-03-07T00:00:00Z","timestamp":1709769600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["EXC-2047\/1-390685813"],"award-info":[{"award-number":["EXC-2047\/1-390685813"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["SFB 1060"],"award-info":[{"award-number":["SFB 1060"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper Omega Subscript i Baseline subset-of double-struck upper R Superscript n Super Subscript i\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u03a9\n                                \n                              <\/mml:mi>\n                              <mml:mi>i<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2282\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msub>\n                                  <mml:mi>n<\/mml:mi>\n                                  <mml:mi>i<\/mml:mi>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\Omega _i\\subset \\mathbb {R}^{n_i}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"i equals 1 comma ellipsis comma m\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">i=1,\\ldots ,m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , be given domains. In this article, we study the low-rank approximation with respect to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L squared left-parenthesis normal upper Omega 1 times midline-horizontal-ellipsis times normal upper Omega Subscript m Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u03a9\n                                \n                              <\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>\n                              \u00d7\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">\n                                \u03a9\n                                \n                              <\/mml:mi>\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^2(\\Omega _1\\times \\dots \\times \\Omega _m)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    of functions from Sobolev spaces with dominating mixed smoothness. To this end, we first estimate the rank of a bivariate approximation, i.e., the rank of the continuous singular value decomposition. In comparison to the case of functions from Sobolev spaces with isotropic smoothness, compare Griebel and Harbrecht [IMA J. Numer. Anal. 34 (2014), pp. 28\u201354] and Griebel and Harbrecht [IMA J. Numer. Anal. 39 (2019), pp. 1652\u20131671], we obtain improved results due to the additional mixed smoothness. This convergence result is then used to study the tensor train decomposition as a method to construct multivariate low-rank approximations of functions from Sobolev spaces with dominating mixed smoothness. We show that this approach is able to beat the curse of dimension.\n                  <\/p>","DOI":"10.1090\/mcom\/3813","type":"journal-article","created":{"date-parts":[[2023,3,7]],"date-time":"2023-03-07T14:56:12Z","timestamp":1678200972000},"page":"1729-1746","source":"Crossref","is-referenced-by-count":4,"title":["Low-rank approximation of continuous functions in Sobolev spaces with dominating mixed smoothness"],"prefix":"10.1090","volume":"92","author":[{"given":"Michael","family":"Griebel","sequence":"first","affiliation":[]},{"given":"Helmut","family":"Harbrecht","sequence":"additional","affiliation":[]},{"given":"Reinhold","family":"Schneider","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,3,7]]},"reference":[{"key":"1","unstructured":"M. Ali and A. Nouy, Approximation with tensor networks. Part III: multivariate approximation,  arXiv:2101.11932, 2021."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"297","DOI":"10.1017\/S0308210500022757","article-title":"Multiscale convergence and reiterated homogenisation","volume":"126","author":"Allaire, G.","year":"1996","journal-title":"Proc. Roy. Soc. Edinburgh Sect. A","ISSN":"https:\/\/id.crossref.org\/issn\/0308-2105","issn-type":"print"},{"key":"3","unstructured":"M. Bachmayr, A. Nouy, and R. Schneider, Approximation by tree tensor networks in high dimensions: Sobolev and compositional functions,  arXiv:2112.01474, 2021."},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"A2405--A2439","DOI":"10.1137\/15M1036919","article-title":"Spectral tensor-train decomposition","volume":"38","author":"Bigoni, Daniele","year":"2016","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1017\/S0962492904000182","article-title":"Sparse grids","volume":"13","author":"Bungartz, Hans-Joachim","year":"2004","journal-title":"Acta Numer.","ISSN":"https:\/\/id.crossref.org\/issn\/0962-4929","issn-type":"print"},{"key":"6","unstructured":"J. Barrett, D. Knezevic, and E. S\u00fcli, Kinetic models of dilute polymers. Analysis, approximation and computation, 11th School on Mathematical Theory in Fluid Mechanics (22\u201329 May 2009, Kacov, Czech Republic), Necas Center for Mathematical Modeling, Prague, 2009."},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.1137\/080713148","article-title":"The periodic unfolding method in homogenization","volume":"40","author":"Cioranescu, D.","year":"2008","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1017\/S0962492900002713","article-title":"Wavelet and multiscale methods for operator equations","author":"Dahmen, Wolfgang","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0521591066"},{"issue":"3","key":"9","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/s00211-011-0391-2","article-title":"Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators","volume":"119","author":"Fasshauer, Gregory E.","year":"2011","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"10","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic finite elements: a spectral approach","author":"Ghanem, Roger G.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0387974563"},{"issue":"282","key":"11","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1090\/S0025-5718-2012-02638-X","article-title":"On the construction of sparse tensor product spaces","volume":"82","author":"Griebel, Michael","year":"2013","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/s00365-012-9178-7","article-title":"A note on the construction of \ud835\udc3f-fold sparse tensor product spaces","volume":"38","author":"Griebel, Michael","year":"2013","journal-title":"Constr. Approx.","ISSN":"https:\/\/id.crossref.org\/issn\/0176-4276","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1093\/imanum\/drs047","article-title":"Approximation of bi-variate functions: singular value decomposition versus sparse grids","volume":"34","author":"Griebel, Michael","year":"2014","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"1652","DOI":"10.1093\/imanum\/dry039","article-title":"Singular value decomposition versus sparse grids: refined complexity estimates","volume":"39","author":"Griebel, Michael","year":"2019","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"1","key":"15","doi-asserted-by":"publisher","first-page":"219","DOI":"10.1007\/s10208-021-09544-6","article-title":"Analysis of tensor approximation schemes for continuous functions","volume":"23","author":"Griebel, Michael","year":"2023","journal-title":"Found. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1615-3375","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00607-007-0241-3","article-title":"A sparse grid space-time discretization scheme for parabolic problems","volume":"81","author":"Griebel, M.","year":"2007","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"279","DOI":"10.1007\/s002110050450","article-title":"Sparse grids for boundary integral equations","volume":"83","author":"Griebel, M.","year":"1999","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"18","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-35554-8","volume-title":"Tensor spaces and numerical tensor calculus","volume":"56","author":"Hackbusch, Wolfgang","year":"[2019] \\copyright2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030355531"},{"issue":"5","key":"19","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1007\/s00041-009-9094-9","article-title":"A new scheme for the tensor representation","volume":"15","author":"Hackbusch, W.","year":"2009","journal-title":"J. Fourier Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1069-5869","issn-type":"print"},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1016\/j.apnum.2009.12.002","article-title":"A finite element method for elliptic problems with stochastic input data","volume":"60","author":"Harbrecht, Helmut","year":"2010","journal-title":"Appl. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"3","key":"21","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s00211-008-0147-9","article-title":"Sparse second moment analysis for elliptic problems in stochastic domains","volume":"109","author":"Harbrecht, Helmut","year":"2008","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"22","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1137\/030601077","article-title":"High-dimensional finite elements for elliptic problems with multiple scales","volume":"3","author":"Hoang, Viet Ha","year":"2004","journal-title":"Multiscale Model. Simul.","ISSN":"https:\/\/id.crossref.org\/issn\/1540-3459","issn-type":"print"},{"key":"23","isbn-type":"print","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1007\/978-3-540-88857-4_2","article-title":"Multiscale modelling of complex fluids: a mathematical initiation","author":"Le Bris, Claude","year":"2009","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540888567"},{"key":"24","doi-asserted-by":"crossref","unstructured":"A. Lozinski, R. Owens, and T. Phillips, The Langevin and Fokker-Planck equations in polymer rheology, Handbook of Numerical Analysis XVI\/XVII, Elsevier North-Holland, 2011, pp. 211\u2013303.","DOI":"10.1016\/B978-0-444-53047-9.00002-2"},{"key":"25","series-title":"Lecture Notes in Statistics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4615-7892-5","volume-title":"Spatial variation","volume":"36","author":"Mat\u00e9rn, Bertil","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540963650","edition":"2"},{"issue":"6","key":"26","doi-asserted-by":"publisher","first-page":"829","DOI":"10.1142\/S0219530516400042","article-title":"Deep vs. shallow networks: an approximation theory perspective","volume":"14","author":"Mhaskar, H. N.","year":"2016","journal-title":"Anal. Appl. (Singap.)","ISSN":"https:\/\/id.crossref.org\/issn\/0219-5305","issn-type":"print"},{"issue":"5","key":"27","doi-asserted-by":"publisher","first-page":"2295","DOI":"10.1137\/090752286","article-title":"Tensor-train decomposition","volume":"33","author":"Oseledets, I. V.","year":"2011","journal-title":"SIAM J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1051\/m2an:2004005","article-title":"Numerical solution of parabolic equations in high dimensions","volume":"38","author":"von Petersdorff, Tobias","year":"2004","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"2","key":"29","doi-asserted-by":"publisher","first-page":"56","DOI":"10.1016\/j.jco.2013.10.001","article-title":"Approximation rates for the hierarchical tensor format in periodic Sobolev spaces","volume":"30","author":"Schneider, Reinhold","year":"2014","journal-title":"J. Complexity","ISSN":"https:\/\/id.crossref.org\/issn\/0885-064X","issn-type":"print"},{"issue":"4","key":"30","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1007\/s00211-003-0455-z","article-title":"Sparse finite elements for elliptic problems with stochastic loading","volume":"95","author":"Schwab, Christoph","year":"2003","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"31","first-page":"113","article-title":"Approximations of functions with bounded mixed derivative","volume":"178","author":"Temlyakov, V. N.","year":"1986","journal-title":"Trudy Mat. Inst. Steklov.","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"32","first-page":"250","article-title":"Estimates for the best bilinear approximations of periodic functions","volume":"181","author":"Temlyakov, V. N.","year":"1988","journal-title":"Trudy Mat. Inst. Steklov.","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"33","first-page":"229","article-title":"Bilinear approximation and related questions","volume":"194","author":"Temlyakov, V. N.","year":"1992","journal-title":"Trudy Mat. Inst. Steklov.","ISSN":"https:\/\/id.crossref.org\/issn\/0371-9685","issn-type":"print"},{"key":"34","unstructured":"H. Weyl, \u00dcber die asymptotische Verteilung der Eigenwerte, Nachrichten der K\u00f6niglichen Gesellschaft der Wissenschaften zu G\u00f6ttingen, 1911, pp. 110\u2013117."},{"issue":"12","key":"35","doi-asserted-by":"publisher","first-page":"6071","DOI":"10.1016\/j.jcp.2008.02.025","article-title":"Sparse adaptive finite elements for radiative transfer","volume":"227","author":"Widmer, G.","year":"2008","journal-title":"J. Comput. Phys.","ISSN":"https:\/\/id.crossref.org\/issn\/0021-9991","issn-type":"print"},{"key":"36","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139171755","volume-title":"Partial differential equations","author":"Wloka, J.","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0521259142"},{"key":"37","doi-asserted-by":"crossref","unstructured":"D. Yarotsky, Error bounds for approximations with deep ReLU networks, Neural Netw. 94 (2017), 103\u2013114.","DOI":"10.1016\/j.neunet.2017.07.002"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-342\/S0025-5718-2023-03813-3\/S0025-5718-2023-03813-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:58:27Z","timestamp":1776833907000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2023-92-342\/S0025-5718-2023-03813-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,7]]},"references-count":37,"journal-issue":{"issue":"342","published-print":{"date-parts":[[2023,7]]}},"alternative-id":["S0025-5718-2023-03813-3"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3813","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2023,3,7]]}}}