{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,11]],"date-time":"2026-06-11T08:22:12Z","timestamp":1781166132913,"version":"3.54.1"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Most important computational problems nowadays are those related to processing of the\nlarge data sets and to numerical solution of the high-dimensional integral-differential\nequations. These problems arise in numerical modeling in quantum chemistry, material science,\nand multiparticle dynamics, as well as in machine learning, computer simulation of stochastic\nprocesses and many other applications related to big data analysis.\nModern tensor numerical methods enable solution of the multidimensional\npartial differential equations (PDE) in <jats:inline-formula id=\"j_cmam-2018-0014_ineq_9999_w2aab3b7b1b1b6b1aab1c14b1b1Aa\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:msup>\n                              <m:mi>\u211d<\/m:mi>\n                              <m:mi>d<\/m:mi>\n                           <\/m:msup>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0014_eq_0009.png\"\/>\n                        <jats:tex-math>{\\mathbb{R}^{d}}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> by reducing them to one-dimensional\ncalculations.\nThus, they allow to avoid the so-called \u201ccurse of dimensionality\u201d, i.e. exponential\ngrowth of the computational complexity in the dimension size <jats:italic>d<\/jats:italic>, in the course of numerical\nsolution of high-dimensional problems.\nAt present, both tensor numerical methods and multilinear algebra\nof big data continue\nto expand actively to further theoretical and applied research topics.\nThis issue of CMAM is devoted to the recent developments in the theory of tensor\nnumerical methods and their applications in scientific computing and data analysis.\nCurrent activities in this emerging field on the effective numerical modeling\nof temporal and stationary multidimensional PDEs and beyond are presented in the following\nten articles, and some future trends are highlighted therein.<\/jats:p>","DOI":"10.1515\/cmam-2018-0014","type":"journal-article","created":{"date-parts":[[2018,6,26]],"date-time":"2018-06-26T22:15:44Z","timestamp":1530051344000},"page":"1-4","source":"Crossref","is-referenced-by-count":5,"title":["Tensor Numerical Methods: Actual Theory and Recent Applications"],"prefix":"10.1515","volume":"19","author":[{"given":"Ivan","family":"Gavrilyuk","sequence":"first","affiliation":[{"name":"University of Cooperative Education Gera-Eisenach , Am Wartenberg 2, 99817 , Eisenach , Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Boris N.","family":"Khoromskij","sequence":"additional","affiliation":[{"name":"Max-Planck-Institute for Mathematics in the Sciences , Inselstr. 22\u201326, 04103 Leipzig , Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2018,6,26]]},"reference":[{"key":"2023033110133785681_j_cmam-2018-0014_ref_001_w2aab3b7b1b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"P.  Benner, A.  Onwunta and M.  Stoll,\nAn low-rank inexact Newton\u2013Krylov method for stochastic eigenvalue problems,\nComput. Methods Appl. Math. 19 (2019), no. 1, 5\u201322.","DOI":"10.1515\/cmam-2018-0030"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_002_w2aab3b7b1b1b6b1ab2b1b2Aa","doi-asserted-by":"crossref","unstructured":"S.  Dolgov,\nA tensor decomposition algorithm for large ODEs with conservation laws,\nComput. Methods Appl. Math. 19 (2019), no. 1, 23\u201338.","DOI":"10.1515\/cmam-2018-0023"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_003_w2aab3b7b1b1b6b1ab2b1b3Aa","doi-asserted-by":"crossref","unstructured":"M.  Eigel, J.  Neumann, R.  Schneider and S.  Wolf,\nNon-intrusive tensor reconstruction for high dimensional random PDEs,\nComput. Methods Appl. Math. 19 (2019), no. 1, 39\u201353.","DOI":"10.1515\/cmam-2018-0028"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_004_w2aab3b7b1b1b6b1ab2b1b4Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk,\nSuper exponentially convergent approximation to the solution of the Schr\u00f6dinger equation in abstract setting,\nComput. Methods Appl. Math. 10 (2010), no. 4, 345\u2013358.","DOI":"10.2478\/cmam-2010-0020"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_005_w2aab3b7b1b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk, W.  Hackbusch and B.  Khoromskij,\nData-sparse approximation to the operator-valued functions of elliptic operators,\nMath. Comp. 73 (2004), 1297\u20131324.","DOI":"10.1090\/S0025-5718-03-01590-4"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_006_w2aab3b7b1b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk, W.  Hackbusch and B.  Khoromskij,\nTensor-product approximation to elliptic and parabolic solution operators in higher dimensions,\nComputing 74 (2005), 131\u2013157.","DOI":"10.1007\/s00607-004-0086-y"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_007_w2aab3b7b1b1b6b1ab2b1b7Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk and B.  Khoromskij,\nQuantized-TT-Cayley transform for computing the dynamics and the spectrum of high-dimensional Hamiltonians,\nComput. Methods Appl. Math. 11 (2011), no. 3, 273\u2013290.","DOI":"10.2478\/cmam-2011-0015"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_008_w2aab3b7b1b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk and B.  Khoromskij,\nQuasi-optimal rank-structured approximation to multidimensional parabolic problems by Cayley transform and Chebyshev interpolation,\nComput. Methods Appl. Math. 19 (2019), no. 1, 55\u201371.","DOI":"10.1515\/cmam-2018-0021"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_009_w2aab3b7b1b1b6b1ab2b1b9Aa","doi-asserted-by":"crossref","unstructured":"I.  Gavrilyuk, V.  Makarov and V.  Vasylyk,\nExponentially Convergent Algorithms for Abstract Differential Equations,\nBirkh\u00e4user, Basel, 2011.","DOI":"10.1007\/978-3-0348-0119-5"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_010_w2aab3b7b1b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"L.  Grasedyck, D.  Kressner and C.  Tobler,\nA literature survey of low-rank tensor approximation techniques,\nGAMM-Mitt. 36 (2013), no. 1, 53\u201378.","DOI":"10.1002\/gamm.201310004"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_011_w2aab3b7b1b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"W.  Hackbusch,\nTensor Spaces and Numerical Tensor Calculus,\nSpringer, Berlin, 2012.","DOI":"10.1007\/978-3-642-28027-6"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_012_w2aab3b7b1b1b6b1ab2b1c12Aa","doi-asserted-by":"crossref","unstructured":"V.  Khoromskaia and B. N.  Khoromskij,\nTensor Numerical Methods in Computational Quantum Chemistry,\nDe Gruyter, Berlin, 2018.","DOI":"10.1515\/9783110365832"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_013_w2aab3b7b1b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"B. N.  Khoromskij,\nO\u2062(d\u2062log\u2061N){O(d\\log N)}-quantics approximation of N-d tensors in high-dimensional numerical modeling,\nConstr. Approx. 34 (2011), no. 2, 257\u2013289.","DOI":"10.1007\/s00365-011-9131-1"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_014_w2aab3b7b1b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"B. N.  Khoromskij,\nTensors-structured numerical methods in scientific computing: Survey on recent advances,\nChemometr. Intell. Lab. Syst. 110 (2012), 1\u201319.","DOI":"10.1016\/j.chemolab.2011.09.001"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_015_w2aab3b7b1b1b6b1ab2b1c15Aa","doi-asserted-by":"crossref","unstructured":"B. N.  Khoromskij,\nTensor Numerical Methods in Scientific Computing,\nDe Gruyter, Berlin, 2018.","DOI":"10.1515\/9783110365917"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_016_w2aab3b7b1b1b6b1ab2b1c16Aa","doi-asserted-by":"crossref","unstructured":"E.  Kieri and B.  Vandereycken,\nProjection methods for dynamical low-rank approximation of high-dimensional problems,\nComput. Methods Appl. Math. 19 (2019), no. 1, 73\u201392.","DOI":"10.1515\/cmam-2018-0029"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_017_w2aab3b7b1b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"M. A.  Kuznetsov and I. V.  Oseledets,\nTensor train spectral method for learning of hidden Markov models (HMM),\nComput. Methods Appl. Math. 19 (2019), no. 1, 93\u201399.","DOI":"10.1515\/cmam-2018-0027"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_018_w2aab3b7b1b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"A.  Litvinenko, D.  Keyes, V.  Khoromskaia, B.  Khoromskij and H.  Matthies,\nTucker tensor analysis of Mat\u00e9rn functions in spatial statistics,\nComput. Methods Appl. Math. 19 (2019), no. 1, 101\u2013122.","DOI":"10.1515\/cmam-2018-0022"},{"key":"2023033110133785681_j_cmam-2018-0014_ref_019_w2aab3b7b1b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"A.  Mantzaflaris, F.  Scholz and I.  Toulopoulos,\nLow-rank space-time decoupled isogeometric analysis for parabolic problems with varying coefficients,\nComput. Methods Appl. 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