{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:09Z","timestamp":1787333649299,"version":"build-2736575974"},"reference-count":47,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","funder":[{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-17-1-0190"],"award-info":[{"award-number":["FA9550-17-1-0190"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["ACI-1550593"],"award-info":[{"award-number":["ACI-1550593"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"publisher","award":["560651"],"award-info":[{"award-number":["560651"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0019303"],"award-info":[{"award-number":["DE-SC0019303"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2020,1]]},"abstract":"<jats:p>We present a method for converting tensors into the tensor train format based on actions of the tensor as a vector-valued multilinear function. Existing methods for constructing tensor trains require access to \u201carray entries\u201d of the tensor and are therefore inefficient or computationally prohibitive if the tensor is accessible only through its action, especially for high order tensors. Our method permits efficient tensor train compression of large high order derivative tensors for nonlinear mappings that are implicitly defined through the solution of a system of equations. Array entries of these derivative tensors are not directly accessible, but actions of these tensors can be computed efficiently via a procedure that we discuss. Such tensors are often amenable to tensor train compression in theory, but until now no efficient algorithm existed to convert them into tensor train format. We demonstrate our method by compressing a Hilbert tensor of size 41 x 42 x 43 x 44 x 45, and by forming high order (up to fifth order derivatives\/sixth order tensors) Taylor series surrogates of the noise-whitened parameter-to-output map for a stochastic partial differential equation with boundary output.<\/jats:p>","DOI":"10.1137\/20m131936x","type":"journal-article","created":{"date-parts":[[2020,10,27]],"date-time":"2020-10-27T14:21:26Z","timestamp":1603808486000},"page":"A3516-A3539","source":"Crossref","is-referenced-by-count":14,"title":["Tensor Train Construction From Tensor Actions, With Application to Compression of Large High Order Derivative Tensors"],"prefix":"10.1137","volume":"42","author":[{"given":"Nick","family":"Alger","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peng","family":"Chen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Omar","family":"Ghattas","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2020,10,27]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/16M106306X"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1002\/nme.6009"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.08.010"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1002\/nme.2100"},{"key":"atypb5","volume-title":"Greedy Inference with Structure-Exploiting Lazy Maps, preprint, https:\/\/arxiv.org\/abs\/1906.00031","author":"Bigoni D.","year":"2019"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/130949415"},{"key":"atypb7","unstructured":"F. Bonizzoni, F. Nobile, and D. Kressner,\n                      Tensor Train Approximation of Moment Equations for the Log-normal Darcy Problem\n                      , Tech. rep., Mathicse Technical Report, Lausanne, Switzerland, 2014."},{"key":"atypb8","volume-title":"A scaled stochastic Newton algorithm for Markov chain Monte Carlo simulations, unpublished","author":"Bui-Thanh T.","year":"2012"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/12089586X"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-018-9622-8"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1615\/Int.J.UncertaintyQuantification.2019028753"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2017.08.016"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2019.01.047"},{"key":"atypb14","volume-title":"Proceedings of the 33rd Conference on Neural Information Processing Systems","author":"Chen P.","year":"2019"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1561\/2200000059"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.12.051"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.10.008"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1145\/3307650.3322258"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s11222-019-09910-z"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1137\/120864210"},{"key":"atypb21","unstructured":"P. Gel\u00df,\n                      The Tensor-Train Format and Its Applications: Modeling and Analysis of Chemical Reaction Networks, Catalytic Processes, Fluid Flows, and Brownian Dynamics\n                      , Ph.D. thesis, Fachbereich Mathematik und Informatikder Freien Universit\u00e4t Berlin, Germany, 2017."},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201310004"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1145\/2512329"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/100818893"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-69802-1_9"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.04.047"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1003359"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2013.01.009"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-011-9131-1"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/07070111X"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1137\/100801482"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.02.033"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-23099-8"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/18M1209465"},{"key":"atypb36","volume-title":"The Art of Differentiating Computer Programs: An Introduction to Algorithmic Differentiation","author":"Naumann U.","year":"2012"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1134\/S1064562409050056"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/090757861"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1137\/090752286"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2009.07.024"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.2478\/cmam-2011-0022"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.26421\/QIC7.5-6-1"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1137\/130934805"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1109\/nDS.2011.6076873"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/140977308"},{"key":"atypb46","first-page":"3891","volume-title":"Proceedings of the 34th International Conference on Machine Learning","volume":"70","author":"Yang Y.","year":"2017"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1190\/geo2015-0641.1"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/20M131936X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:15:19Z","timestamp":1787332519000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/20M131936X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,1]]},"references-count":47,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2020,1]]}},"alternative-id":["10.1137\/20M131936X"],"URL":"https:\/\/doi.org\/10.1137\/20m131936x","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,1]]}}}