{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T18:13:47Z","timestamp":1770488027184,"version":"3.49.0"},"reference-count":66,"publisher":"Walter de Gruyter GmbH","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,1,1]]},"abstract":"Abstract<\/jats:title>\n In this work, we describe advanced numerical tools for working with multivariate functions and for the\nanalysis of large data sets. These tools will drastically reduce the required computing time and the\nstorage cost, and, therefore, will allow us to consider much larger data sets or finer meshes.\nCovariance matrices are crucial in spatio-temporal statistical tasks, but are often very expensive to\ncompute and store, especially in three dimensions. Therefore, we approximate covariance functions by cheap surrogates\nin a low-rank tensor format. We apply the Tucker and canonical tensor decompositions to a family of\nMat\u00e9rn- and Slater-type functions with varying parameters and demonstrate numerically\nthat their approximations exhibit exponentially fast convergence.\nWe prove the exponential convergence of the Tucker and canonical approximations in tensor\nrank parameters.\nSeveral statistical operations are performed in this low-rank tensor format, including evaluating the\nconditional covariance matrix, spatially averaged\nestimation variance, computing a quadratic form, determinant, trace, loglikelihood, inverse,\nand Cholesky decomposition of a large covariance matrix.\nLow-rank tensor approximations reduce the computing and storage costs essentially.\nFor example, the storage cost\nis reduced from an exponential \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n n<\/m:mi>\n d<\/m:mi>\n <\/m:msup>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(n^{d})}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to a linear scaling \n \n \n \n \ud835\udcaa<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \n d<\/m:mi>\n \u2062<\/m:mo>\n r<\/m:mi>\n \u2062<\/m:mo>\n n<\/m:mi>\n <\/m:mrow>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\mathcal{O}(drn)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>,\nwhere d<\/jats:italic> is the spatial dimension, n<\/jats:italic> is the number of mesh points in one direction,\nand r<\/jats:italic> is the tensor rank.\nPrerequisites for applicability of the proposed techniques are the assumptions that the data, locations,\nand measurements lie on a tensor (axes-parallel) grid and that the covariance\nfunction depends on a distance, \n \n \n \n \u2225<\/m:mo>\n \n x<\/m:mi>\n -<\/m:mo>\n y<\/m:mi>\n <\/m:mrow>\n \u2225<\/m:mo>\n <\/m:mrow>\n <\/m:math>\n \n {\\|x-y\\|}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/cmam-2018-0022","type":"journal-article","created":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T16:51:18Z","timestamp":1531414278000},"page":"101-122","source":"Crossref","is-referenced-by-count":11,"title":["Tucker Tensor Analysis of Mat\u00e9rn Functions in Spatial Statistics"],"prefix":"10.1515","volume":"19","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5427-3598","authenticated-orcid":false,"given":"Alexander","family":"Litvinenko","sequence":"first","affiliation":[{"name":"Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division , King Abdullah University of Science and Technology , Thuwal-Jeddah , Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"David","family":"Keyes","sequence":"additional","affiliation":[{"name":"Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division , King Abdullah University of Science and Technology , Thuwal-Jeddah , Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Venera","family":"Khoromskaia","sequence":"additional","affiliation":[{"name":"Max-Planck Institute for Mathematics in the Sciences , 04103 Leipzig ; and Max-Planck Institute for Dynamics of Complex Technical Systems, 39106 Magdeburg , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Boris N.","family":"Khoromskij","sequence":"additional","affiliation":[{"name":"Max-Planck Institute for Mathematics in the Sciences , 04103 Leipzig , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hermann G.","family":"Matthies","sequence":"additional","affiliation":[{"name":"Institute of Scientific Computing , TU Braunschweig , 38106 Braunschweig , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2018,7,7]]},"reference":[{"key":"2023033110133742831_j_cmam-2018-0022_ref_001_w2aab3b7d369b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"S. Ambikasaran, J. Y. Li, P. K. Kitanidis and E. Darve,\nLarge-scale stochastic linear inversion using hierarchical matrices,\nComput. Geosci. 17 (2013), no. 6, 913\u2013927.","DOI":"10.1007\/s10596-013-9364-0"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_002_w2aab3b7d369b1b6b1ab2ab2Aa","unstructured":"J. Ballani and D. Kressner,\nSparse inverse covariance estimation with hierarchical matrices,\npreprint (2015), http:\/\/sma.epfl.ch\/~anchpcommon\/publications\/quic_ballani_kressner_2014.pdf."},{"key":"2023033110133742831_j_cmam-2018-0022_ref_003_w2aab3b7d369b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"C. Bertoglio and B. N. Khoromskij,\nLow-rank quadrature-based tensor approximation of the Galerkin projected Newton\/Yukawa kernels,\nComput. Phys. Commun. 183 (2012), no. 4, 904\u2013912.","DOI":"10.1016\/j.cpc.2011.12.016"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_004_w2aab3b7d369b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"S. B\u00f6rm and J. Garcke,\nApproximating gaussian processes with H2{{H^{2}}}-matrices,\nProceedings of 18th European Conference on Machine Learning\u2014ECML 2007,\nLecture Notes in Artificial Intelligence 4701,\nSpringer, Berlin (2007), 42\u201353.","DOI":"10.1007\/978-3-540-74958-5_8"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_005_w2aab3b7d369b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"S. F. Boys, G. B. Cook, C. M. Reeves and I. Shavitt,\nAutomatic fundamental calculations of molecular structure,\nNature 178 (1956), 1207\u20131209.","DOI":"10.1038\/1781207a0"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_006_w2aab3b7d369b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"D. Braess,\nNonlinear Approximation Theory,\nSpringer Ser. Comput. Math. 7,\nSpringer, Berlin, 1986.","DOI":"10.1007\/978-3-642-61609-9"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_007_w2aab3b7d369b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"J.-P. Chil\u00e8s and P. Delfiner,\nGeostatistics,\nWiley Ser. Probab. Stat.,\nJohn Wiley & Sons, New York, 1999.","DOI":"10.1002\/9780470316993"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_008_w2aab3b7d369b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"A. Cichocki and S. Amari,\nAdaptive Blind Signal and Image Processing: Learning Algorithms and Applications,\nWiley, New York, 2002.","DOI":"10.1002\/0470845899"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_009_w2aab3b7d369b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"S. De Iaco, S. Maggio, M. Palma and D. Posa,\nToward an automatic procedure for modeling multivariate space-time data,\nComput. Geosci. 41 (2011), 10.1016\/j.cageo.2011.08.008.","DOI":"10.1016\/j.cageo.2011.08.008"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_010_w2aab3b7d369b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"L. De Lathauwer, B. De Moor and J. Vandewalle,\nA multilinear singular value decomposition,\nSIAM J. Matrix Anal. 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Savostyanov,\nSuperfast Fourier transform using QTT approximation,\nJ. Fourier Anal. Appl. 18 (2012), no. 5, 915\u2013953.","DOI":"10.1007\/s00041-012-9227-4"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_014_w2aab3b7d369b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"P. A. Finke, D. J. Brus, M. F. P. Bierkens, T. Hoogland, M. Knotters and F. De Vries,\nMapping groundwater dynamics using multiple sources of exhaustive high resolution data,\nGeoderma 123 (2004), no. 1, 23\u201339.","DOI":"10.1016\/j.geoderma.2004.01.025"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_015_w2aab3b7d369b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"R. Furrer and M. G. Genton,\nAggregation-cokriging for highly multivariate spatial data,\nBiometrika 98 (2011), no. 3, 615\u2013631.","DOI":"10.1093\/biomet\/asr029"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_016_w2aab3b7d369b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"I. P. Gavrilyuk, W. 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Huijbregts,\nMining Geostatistics,\nAcademic Press, New York, 1978."},{"key":"2023033110133742831_j_cmam-2018-0022_ref_031_w2aab3b7d369b1b6b1ab2ac31Aa","doi-asserted-by":"crossref","unstructured":"V. Khoromskaia,\nComputation of the Hartree\u2013Fock exchange by the tensor-structured methods,\nComput. Methods Appl. Math. 10 (2010), no. 2, 204\u2013218.","DOI":"10.2478\/cmam-2010-0012"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_032_w2aab3b7d369b1b6b1ab2ac32Aa","doi-asserted-by":"crossref","unstructured":"V. Khoromskaia and B. N. Khoromskij,\nFast tensor method for summation of long-range potentials on 3D lattices with defects,\nNumer. Linear Algebra Appl. 23 (2016), no. 2, 249\u2013271.","DOI":"10.1002\/nla.2023"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_033_w2aab3b7d369b1b6b1ab2ac33Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij,\nStructured rank-(R1,\u2026,RD)(R_{1},\\dots,R_{D}) decomposition of function-related tensors in \u211dD\\mathbb{R}^{D},\nComput. Methods Appl. Math. 6 (2006), no. 2, 194\u2013220.","DOI":"10.2478\/cmam-2006-0010"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_034_w2aab3b7d369b1b6b1ab2ac34Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij,\nTensors-structured numerical methods in scientific computing: Survey on recent advances,\nChemometr. Intell. Laboratory Syst. 110 (2011), no. 1, 1\u201319.","DOI":"10.1016\/j.chemolab.2011.09.001"},{"key":"2023033110133742831_j_cmam-2018-0022_ref_035_w2aab3b7d369b1b6b1ab2ac35Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij,\nTensor numerical methods for multidimensional PDEs: Theoretical analysis and initial applications,\nCEMRACS 2013\u2014Modelling and Simulation of Complex Systems: Stochastic and Deterministic Approaches,\nESAIM Proc. 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