{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T19:25:43Z","timestamp":1787340343082,"version":"build-2736575974"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM\/ASA J. Uncertainty Quantification"],"published-print":{"date-parts":[[2015,1]]},"abstract":"<jats:p>Gaussian random fields are widely used as building blocks for modeling stochastic processes. This paper is concerned with the efficient representation of d-point correlations for such fields, which in turn enables the representation of more general stochastic processes that can be expressed as a function of one (or several) Gaussian random fields. Our representation consists of two ingredients. In the first step, we replace the random field by a truncated Karhunen--Lo\u00e8ve expansion and analyze the resulting error. The parameters describing the d-point correlation can be arranged in a tensor, but its storage grows exponentially in d. To avoid this, the second step consists of approximating the tensor in a low-rank tensor format, the so-called tensor train decomposition. By exploiting the particular structure of the tensor, an approximation algorithm is derived that does not need to form this tensor explicitly and allows processing correlations of order as high as d=20. The resulting representation is very compact, and its use is illustrated for elliptic partial differential equations with random Gaussian forcing terms.<\/jats:p>","DOI":"10.1137\/140968938","type":"journal-article","created":{"date-parts":[[2015,5,27]],"date-time":"2015-05-27T11:11:09Z","timestamp":1432725069000},"page":"393-416","source":"Crossref","is-referenced-by-count":11,"title":["Low-Rank Tensor Approximation for High-Order Correlation Functions of Gaussian Random Fields"],"prefix":"10.1137","volume":"3","author":[{"given":"Daniel","family":"Kressner","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rajesh","family":"Kumar","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Fabio","family":"Nobile","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christine","family":"Tobler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2015,5,27]]},"reference":[{"key":"atypb1","unstructured":"R. J. Adler and J. E. Taylor,\n                      Random Fields and Geometry\n                      , Springer Monogr. Math., Springer, New York, 2007."},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2013.03.004"},{"key":"atypb3","unstructured":"F. Bonizzoni,\n                      Analysis and Approximation of Moment Equations for PDEs with Stochastic Data\n                      , Ph.D. thesis, Department of Mathematics, Politecnico di Milano, Milano, Italy, 2013."},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drt041"},{"key":"atypb5","unstructured":"F. Bonizzoni, F. Nobile, and D. Kressner,\n                      Tensor Train Approximation of Moment Equations for the Log-Normal Darcy Problem\n                      , Technical Report 39.2014, MATHICSE, EPFL, Lausanne, Switzerland, 2014."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2013-02692-0"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1142\/S0219530511001728"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"P. J. Diggle and P. J. Ribeiro,\n                      Model-Based Geostatistics\n                      , Springer Ser. Statist., Springer, New York, 2007.","DOI":"10.1007\/978-0-387-48536-2"},{"key":"atypb9","unstructured":"J. Do\u0308lz, H. Harbrecht, and C. Schwab,\n                      Covariance Regularity and $\\mathcal{H}$-Matrix Approximation for Rough Random Fields\n                      , Preprint 2014-11, Mathematisches Institut, Universita\u0308t Basel, Basel, Switzerland, 2014."},{"key":"atypb10","doi-asserted-by":"crossref","unstructured":"I. Graham, F. Kuo, J. Nichols, R. Scheichl, C. Schwab, and I. Sloan,\n                      Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients\n                      , Numer. Math., 2014, DOI 10.1007\/s00211-014-0689-y.","DOI":"10.1007\/s00211-014-0689-y"},{"key":"atypb11","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Tensor Spaces and Numerical Tensor Calculus\n                      , Springer Ser. Comput. 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Loe\u0300ve,\n                      Probability Theory. \\textupII\n                      , 4th ed., Springer-Verlag, New York, 1978."},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/090752286"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/110833142"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.07.024"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"D. V. Savostyanov and I. V. Oseledets,\n                      Fast adaptive interpolation of multi-dimensional arrays in tensor train format\n                      , in Proceedings of the 7th International IEEE Workshop on Multidimensional (nD) Systems, 2011.","DOI":"10.1109\/nDS.2011.6076873"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.jco.2013.10.001"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0455-z"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0024-4"},{"key":"atypb26","unstructured":"C. Tobler,\n                      \n                        Matlab\n                        Software Package constr_tt\n                      \n                      , http:\/\/anchp.epfl.ch\/software\/misc (2013)."},{"key":"atypb27","unstructured":"R. A. Todor,\n                      Sparse Perturbation Algorithms for Elliptic PDEs with Stochastic Data\n                      , Ph.D. thesis, ETH Zurich, Zurich, Switzerland, 2005."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1007\/s10492-006-0010-1"}],"container-title":["SIAM\/ASA Journal on Uncertainty Quantification"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/140968938","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:31:08Z","timestamp":1787337068000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/140968938"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,1]]},"references-count":28,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2015,1]]}},"alternative-id":["10.1137\/140968938"],"URL":"https:\/\/doi.org\/10.1137\/140968938","relation":{},"ISSN":["2166-2525"],"issn-type":[{"value":"2166-2525","type":"electronic"}],"subject":[],"published":{"date-parts":[[2015,1]]}}}