{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T03:31:15Z","timestamp":1771471875579,"version":"3.50.1"},"reference-count":36,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T00:00:00Z","timestamp":1593993600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T00:00:00Z","timestamp":1593993600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2020,8]]},"DOI":"10.1007\/s10444-020-09797-9","type":"journal-article","created":{"date-parts":[[2020,7,6]],"date-time":"2020-07-06T14:03:53Z","timestamp":1594044233000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["HT-AWGM: a hierarchical Tucker\u2013adaptive wavelet Galerkin method for high-dimensional elliptic problems"],"prefix":"10.1007","volume":"46","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1664-7098","authenticated-orcid":false,"given":"Mazen","family":"Ali","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karsten","family":"Urban","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,7,6]]},"reference":[{"issue":"4","key":"9797_CR1","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1007\/s10208-013-9187-3","volume":"15","author":"M Bachmayr","year":"2015","unstructured":"Bachmayr, M., Dahmen, W.: Adaptive near-optimal rank tensor approximation for high-dimensional operator equations. Found Comput. Math. 15(4), 839\u2013898 (2015)","journal-title":"Found Comput. Math."},{"issue":"4","key":"9797_CR2","doi-asserted-by":"crossref","first-page":"1107","DOI":"10.1051\/m2an\/2015071","volume":"50","author":"M Bachmayr","year":"2016","unstructured":"Bachmayr, M., Dahmen, W.: Adaptive low-rank methods for problems on Sobolev spaces with error control in L2. ESAIM Math. Model. Numer Anal. 50(4), 1107\u20131136 (2016)","journal-title":"ESAIM Math. Model. Numer Anal."},{"issue":"2","key":"9797_CR3","doi-asserted-by":"crossref","first-page":"744","DOI":"10.1137\/140978223","volume":"54","author":"M Bachmayr","year":"2016","unstructured":"Bachmayr, M., Dahmen, W.: Adaptive low-rank methods: problems on Sobolev spaces. SIAM J. Numer. Anal. 54(2), 744\u2013796 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"4","key":"9797_CR4","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1007\/s10208-016-9314-z","volume":"17","author":"M Bachmayr","year":"2017","unstructured":"Bachmayr, M., Schneider, R.: Iterative methods based on soft thresholding of hierarchical tensors. Found Comput. Math. 17(4), 1037\u20131083 (2017)","journal-title":"Found Comput. Math."},{"key":"9797_CR5","doi-asserted-by":"crossref","unstructured":"Bachmayr, M., Schneider, R., Uschmajew, A.: Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations. Found. Comput. Math., pp 1\u201350 (2016)","DOI":"10.1007\/s10208-016-9317-9"},{"issue":"1","key":"9797_CR6","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1002\/nla.1818","volume":"20","author":"J Ballani","year":"2013","unstructured":"Ballani, J., Grasedyck, L.: A projection method to solve linear systems in tensor format. Numer Linear Algebra Appl. 20(1), 27\u201343 (2013)","journal-title":"Numer Linear Algebra Appl."},{"issue":"6","key":"9797_CR7","doi-asserted-by":"crossref","first-page":"1777","DOI":"10.1051\/m2an\/2014019","volume":"48","author":"M Billaud-Friess","year":"2014","unstructured":"Billaud-Friess, M., Nouy, A., Zahm, O.: A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems. ESAIM: Mathematical Modelling and Numerical Analysis 48 (6), 1777\u20131806 (2014)","journal-title":"ESAIM: Mathematical Modelling and Numerical Analysis"},{"issue":"233","key":"9797_CR8","doi-asserted-by":"crossref","first-page":"27","DOI":"10.1090\/S0025-5718-00-01252-7","volume":"70","author":"A Cohen","year":"2001","unstructured":"Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods for elliptic operator equations: convergence rates. Math. Comp. 70(233), 27\u201375 (2001)","journal-title":"Math. Comp."},{"issue":"3","key":"9797_CR9","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1007\/s102080010027","volume":"2","author":"A Cohen","year":"2002","unstructured":"Cohen, A., Dahmen, W., DeVore, R.: Adaptive wavelet methods. II. Beyond the elliptic case. Found. Comput. Math. 2(3), 203\u2013245 (2002)","journal-title":"Found. Comput. Math."},{"issue":"4","key":"9797_CR10","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1007\/s10208-015-9265-9","volume":"16","author":"W Dahmen","year":"2016","unstructured":"Dahmen, W., DeVore, R., Grasedyck, L., S\u00fcli, E.: Tensor-sparsity of solutions to high-dimensional elliptic partial differential equations. Found Comput. Math. 16(4), 813\u2013874 (2016)","journal-title":"Found Comput. Math."},{"key":"9797_CR11","doi-asserted-by":"crossref","first-page":"1171","DOI":"10.1137\/060676386","volume":"19","author":"A D\u2019Aspremont","year":"2008","unstructured":"D\u2019Aspremont, A.: Smooth optimization with approximate gradient. SIAM Journal on Optimization 19, 1171\u20131183 (2008)","journal-title":"SIAM Journal on Optimization"},{"issue":"1-2","key":"9797_CR12","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s10107-013-0677-5","volume":"146","author":"O Devolder","year":"2013","unstructured":"Devolder, O., Glineur, F., Nesterov, Y.: First-order methods of smooth convex optimization with inexact oracle. Mathematical Programming 146(1-2), 37\u201375 (2013)","journal-title":"Mathematical Programming"},{"key":"9797_CR13","doi-asserted-by":"crossref","unstructured":"DeVore, R.A.: Nonlinear approximation. In: Acta Numerica. Cambridge Univ. Press, Cambridge, 1998, vol. 7, pp 51\u2013150 (1998)","DOI":"10.1017\/S0962492900002816"},{"issue":"2","key":"9797_CR14","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1002\/nla.1942","volume":"22","author":"S Dolgov","year":"2015","unstructured":"Dolgov, S., Khoromskij, B.: Simultaneous state-time approximation of the chemical master equation using tensor product formats. Numer. Linear Algebra Appl. 22(2), 197\u2013219 (2015)","journal-title":"Numer. Linear Algebra Appl."},{"issue":"5","key":"9797_CR15","doi-asserted-by":"crossref","first-page":"A2248","DOI":"10.1137\/140953289","volume":"36","author":"SV Dolgov","year":"2014","unstructured":"Dolgov, S.V., Savostyanov, D.V.: Alternating minimal energy methods for linear systems in higher dimensions. SIAM Journal on Scientific Computing 36(5), A2248\u2013A2271 (2014)","journal-title":"SIAM Journal on Scientific Computing"},{"key":"9797_CR16","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-663-11108-5","volume-title":"Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley-Teubner Series Advances in Numerical Mathematics","author":"B Fischer","year":"1996","unstructured":"Fischer, B.: Polynomial Based Iteration Methods for Symmetric Linear Systems. Wiley-Teubner Series Advances in Numerical Mathematics. Wiley, Chichester (1996). B.G. Teubner, Stuttgart"},{"issue":"258","key":"9797_CR17","doi-asserted-by":"crossref","first-page":"615","DOI":"10.1090\/S0025-5718-06-01917-X","volume":"76","author":"T Gantumur","year":"2007","unstructured":"Gantumur, T., Harbrecht, H., Stevenson, R. : An optimal adaptive wavelet method without coarsening of the iterands. Math Comp. 76 (258), 615\u2013629 (2007)","journal-title":"Math Comp."},{"key":"9797_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-663-01354-9","volume-title":"Iterative L\u00f6sung Gro\u00df Er Schwachbesetzter Gleichungssysteme, vol. 69","author":"W Hackbusch","year":"1991","unstructured":"Hackbusch, W.: Iterative L\u00f6sung Gro\u00df Er Schwachbesetzter Gleichungssysteme, vol. 69. B. G. Teubner, Stuttgart (1991)"},{"key":"9797_CR19","volume-title":"Tensor Spaces Vol. 42 of Springer Series in Computational Mathematics","author":"W Hackbusch","year":"2012","unstructured":"Hackbusch, W., calculus, numerical tensor: Tensor Spaces Vol. 42 of Springer Series in Computational Mathematics. Springer, Heidelberg (2012)"},{"issue":"5","key":"9797_CR20","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1007\/s00041-009-9094-9","volume":"15","author":"W Hackbusch","year":"2009","unstructured":"Hackbusch, W., K\u00fchn, S.: A new scheme for the tensor representation. J Fourier Anal. Appl. 15(5), 706\u2013722 (2009)","journal-title":"J Fourier Anal. Appl."},{"issue":"2","key":"9797_CR21","doi-asserted-by":"crossref","first-page":"A683","DOI":"10.1137\/100818893","volume":"34","author":"S Holtz","year":"2012","unstructured":"Holtz, S., Rohwedder, T., Schneider, R.: The alternating linear scheme for tensor optimization in the tensor train format. SIAM J. Sci. Comput. 34(2), A683\u2013A713 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"9797_CR22","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-18785-8","volume-title":"Optimierung","author":"F Jarre","year":"2004","unstructured":"Jarre, F., Stoer, J.: Optimierung. Springer, Berlin (2004)"},{"key":"9797_CR23","unstructured":"Kestler, S.: On the Adaptive Tensor Product Wavelet Galerkin Method with Applications in Finance. PhD thesis, Ulm University (2013)"},{"issue":"3","key":"9797_CR24","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1007\/s00365-009-9068-9","volume":"30","author":"BN Khoromskij","year":"2009","unstructured":"Khoromskij, B.N.: Tensor-structured preconditioners and approximate inverse of elliptic operators in \u211dd. Constr. Approx. 30(3), 599\u2013620 (2009)","journal-title":"Constr. Approx."},{"issue":"4","key":"9797_CR25","doi-asserted-by":"crossref","first-page":"376","DOI":"10.2478\/cmam-2010-0023","volume":"10","author":"BN Khoromskij","year":"2010","unstructured":"Khoromskij, B.N., Oseledets, I.: Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs. Comput. Methods Appl Math. 10(4), 376\u2013394 (2010)","journal-title":"Comput. Methods Appl Math."},{"issue":"1","key":"9797_CR26","doi-asserted-by":"crossref","first-page":"364","DOI":"10.1137\/100785715","volume":"33","author":"BN Khoromskij","year":"2011","unstructured":"Khoromskij, B.N., Schwab, C.: Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33(1), 364\u2013385 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"4","key":"9797_CR27","doi-asserted-by":"crossref","first-page":"1288","DOI":"10.1137\/100799010","volume":"32","author":"D Kressner","year":"2011","unstructured":"Kressner, D., Tobler, C.: Low-rank tensor Krylov subspace methods for parametrized linear systems. SIAM J. Matrix Anal. Appl. 32(4), 1288\u20131316 (2011)","journal-title":"SIAM J. Matrix Anal. Appl."},{"key":"9797_CR28","doi-asserted-by":"crossref","unstructured":"Nochetto, R.H., Siebert, K.G., Veeser, A.: Theory of adaptive finite element methods: an introduction. In: Multiscale, Nonlinear and Adaptive Approximation, pp 409\u2013542. Springer, Berlin (2009)","DOI":"10.1007\/978-3-642-03413-8_12"},{"issue":"5","key":"9797_CR29","doi-asserted-by":"crossref","first-page":"2295","DOI":"10.1137\/090752286","volume":"33","author":"IV Oseledets","year":"2011","unstructured":"Oseledets, I.V.: Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295\u20132317 (2011)","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"9797_CR30","doi-asserted-by":"crossref","first-page":"A2718","DOI":"10.1137\/110833142","volume":"34","author":"IV Oseledets","year":"2012","unstructured":"Oseledets, I.V., Dolgov, S.V.: Solution of linear systems and matrix inversion in the TT-format. SIAM J. Sci. Comput. 34(5), A2718\u2013A2739 (2012)","journal-title":"SIAM J. Sci. Comput."},{"key":"9797_CR31","unstructured":"Rupp, A.: High Dimensional Wavelet Methods for Structured Financial Products. PhD thesis, Ulm University (2013)"},{"key":"9797_CR32","unstructured":"Schmidt, M., Roux, N.L., Bach, F.R.: Convergence rates of inexact proximal-gradient methods for convex optimization. In: Shawe-Taylor, J, Zemel, R.S., Bartlett, P.L., Pereira, F., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems 24, pp 1458\u20131466. Curran Associates, Inc. (2011)"},{"issue":"2","key":"9797_CR33","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.jco.2013.10.001","volume":"30","author":"R Schneider","year":"2014","unstructured":"Schneider, R., Uschmajew, A.: Approximation rates for the hierarchical tensor format in periodic Sobolev spaces. J. Complexity 30(2), 56\u201371 (2014)","journal-title":"J. Complexity"},{"issue":"2","key":"9797_CR34","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/s10208-005-0183-0","volume":"7","author":"R Stevenson","year":"2007","unstructured":"Stevenson, R.: Optimality of a standard adaptive finite element method. Found Comput. Math. 7(2), 245\u2013269 (2007)","journal-title":"Found Comput. Math."},{"key":"9797_CR35","doi-asserted-by":"crossref","unstructured":"Stevenson, R.: Adaptive wavelet methods for solving operator equations: an overview. In: Multiscale, Nonlinear and Adaptive Approximation, pp 543\u2013597. Springer, Berlin (2009)","DOI":"10.1007\/978-3-642-03413-8_13"},{"key":"9797_CR36","volume-title":"Wavelet Methods for Elliptic Partial Differential Equations. Numerical Mathematics and Scientific Computation","author":"K Urban","year":"2009","unstructured":"Urban, K.: Wavelet Methods for Elliptic Partial Differential Equations. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2009)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09797-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-020-09797-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-020-09797-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,6]],"date-time":"2021-07-06T00:46:19Z","timestamp":1625532379000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-020-09797-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,7,6]]},"references-count":36,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2020,8]]}},"alternative-id":["9797"],"URL":"https:\/\/doi.org\/10.1007\/s10444-020-09797-9","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,7,6]]},"assertion":[{"value":"8 August 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 May 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 July 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"59"}}