{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T12:40:28Z","timestamp":1777552828386,"version":"3.51.4"},"reference-count":29,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2017,7,1]]},"abstract":"Abstract<\/jats:title>\n We consider an iteration method for solving an elliptic type boundary value problem\n\n \n \n \n \n \ud835\udc9c<\/m:mi>\n \u2062<\/m:mo>\n u<\/m:mi>\n <\/m:mrow>\n =<\/m:mo>\n f<\/m:mi>\n <\/m:mrow>\n <\/m:math>\n {\\mathcal{A}u=f}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where a positive definite operator \n \n \n \ud835\udc9c<\/m:mi>\n <\/m:math>\n {\\mathcal{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is generated by a quasi-periodic structure\nwith rapidly changing coefficients (a typical period is characterized by a small\nparameter \u03f5). The method is based on using a simpler operator \n \n \n \n \ud835\udc9c<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n {\\mathcal{A}_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n(inversion of \n \n \n \n \ud835\udc9c<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n {\\mathcal{A}_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> is much simpler than inversion of \n \n \n \ud835\udc9c<\/m:mi>\n <\/m:math>\n {\\mathcal{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>),\nwhich can be viewed as a preconditioner for \n \n \n \ud835\udc9c<\/m:mi>\n <\/m:math>\n {\\mathcal{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. We prove contraction of the\niteration method and establish explicit estimates of the contraction\nfactor q<\/jats:italic>. Certainly the value of q<\/jats:italic> depends on the difference between \n \n \n \ud835\udc9c<\/m:mi>\n <\/m:math>\n {\\mathcal{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and\n\n \n \n \n \ud835\udc9c<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n {\\mathcal{A}_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nFor typical quasi-periodic structures, we establish simple relations that\nsuggest an optimal \n \n \n \n \ud835\udc9c<\/m:mi>\n 0<\/m:mn>\n <\/m:msub>\n <\/m:math>\n {\\mathcal{A}_{0}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> (in a selected set of \u201csimple\u201d structures) and compute the\ncorresponding contraction factor. Further, this allows us to deduce fully computable two-sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient\nif the coefficients of \n \n \n \ud835\udc9c<\/m:mi>\n <\/m:math>\n {\\mathcal{A}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> admit low-rank representations and if algebraic operations\nare performed in tensor structured formats.\nUnder moderate assumptions the storage and solution complexity\nof our approach depends only weakly (merely linear-logarithmically)\non the frequency parameter \n \n \n \n 1<\/m:mn>\n \u03f5<\/m:mi>\n <\/m:mfrac>\n <\/m:math>\n \\frac{1}{\\epsilon}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/cmam-2017-0014","type":"journal-article","created":{"date-parts":[[2017,6,20]],"date-time":"2017-06-20T10:01:30Z","timestamp":1497952890000},"page":"457-477","source":"Crossref","is-referenced-by-count":8,"title":["Rank Structured Approximation Method for Quasi-Periodic Elliptic Problems"],"prefix":"10.1515","volume":"17","author":[{"given":"Boris","family":"Khoromskij","sequence":"first","affiliation":[{"name":"Max Planck Institute for Mathematics in the Sciences , Inselstr. 22\u201326, 04103 Leipzig , Germany"}]},{"given":"Sergey","family":"Repin","sequence":"additional","affiliation":[{"name":"Russian Academy of Sciences , Saint Petersburg Department of V.\u2009A. Steklov Institute of Mathematics , Fontanka 27, 191 011 Saint Petersburg , Russia ; and University of Jyv\u00e4skyl\u00e4, P.O. Box 35, FI-40014, Jyv\u00e4skyl\u00e4, Finland"}]}],"member":"374","published-online":{"date-parts":[[2017,6,17]]},"reference":[{"key":"2023033114491597718_j_cmam-2017-0014_ref_001_w2aab3b7e2112b1b6b1ab2b1b1Aa","doi-asserted-by":"crossref","unstructured":"N. S. Bakhvalov and G. Panasenko,\nHomogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials,\nSpringer, Berlin, 1989.","DOI":"10.1007\/978-94-009-2247-1"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_002_w2aab3b7e2112b1b6b1ab2b1b2Aa","unstructured":"P. Benner, V. Khoromskaia and B. N. Khoromskij,\nRange-separated tensor formats for numerical modeling of many-particle interaction potentials,\npreprint (2016), http:\/\/arxiv.org\/abs\/1606.09218."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_003_w2aab3b7e2112b1b6b1ab2b1b3Aa","unstructured":"A. Bensoussan, J.-L. Lions and G. Papanicolaou,\nAsymptotic Analysis for Periodic Structures,\nNorth-Holland, Amsterdam, 1978."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_004_w2aab3b7e2112b1b6b1ab2b1b4Aa","unstructured":"S. Dolgov, V. Kazeev and B. N. Khoromskij,\nThe tensor-structured solution of one-dimensional elliptic differential equations with high-dimensional parameters,\nPreprint 51\/2012, MPI MiS, Leipzig, 2012."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_005_w2aab3b7e2112b1b6b1ab2b1b5Aa","doi-asserted-by":"crossref","unstructured":"I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij,\nHierarchical tensor-product approximation to the inverse and related operators in high-dimensional elliptic problems,\nComputing 74 (2005), 131\u2013157.","DOI":"10.1007\/s00607-004-0086-y"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_006_w2aab3b7e2112b1b6b1ab2b1b6Aa","doi-asserted-by":"crossref","unstructured":"A. Gloria and F. Otto,\nQuantitative estimates on the periodic approximation of the corrector in stochastic homogenization,\nESAIM Proc. 48 (2015), 80\u201397.","DOI":"10.1051\/proc\/201448003"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_007_w2aab3b7e2112b1b6b1ab2b1b7Aa","unstructured":"R. Glowinski, J.-L. Lions and R. Tr\u00e9molier\u00e9s,\nAnalyse Num\u00e9rique des In\u00e9quations Variationnelles,\nDunod, Paris, 1976."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_008_w2aab3b7e2112b1b6b1ab2b1b8Aa","doi-asserted-by":"crossref","unstructured":"V. V. Jikov, S. M. Kozlov and O. A. Oleinik,\nHomogenization of Differential Operators and Integral Functionals,\nSpringer, Berlin, 1994.","DOI":"10.1007\/978-3-642-84659-5"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_009_w2aab3b7e2112b1b6b1ab2b1b9Aa","unstructured":"L. V. Kantorovich and V. L. Krylov,\nApproximate Methods of Higher Analysis,\nInterscience, New York, 1958."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_010_w2aab3b7e2112b1b6b1ab2b1c10Aa","doi-asserted-by":"crossref","unstructured":"V. Kazeev, O. Reichmann and C. Schwab,\nLow-rank tensor structure of linear diffusion operators in the TT and QTT formats,\nLinear Algebra Appl. 438 (2013), no. 11, 4204\u20134221.","DOI":"10.1016\/j.laa.2013.01.009"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_011_w2aab3b7e2112b1b6b1ab2b1c11Aa","doi-asserted-by":"crossref","unstructured":"V. Khoromskaia and B. N. Khoromskij,\nGrid-based lattice summation of electrostatic potentials by assembled rank-structured tensor approximation,\nComp. Phys. Commun. 185 (2014), no. 12, 3162\u20133174.","DOI":"10.1016\/j.cpc.2014.08.015"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_012_w2aab3b7e2112b1b6b1ab2b1c12Aa","unstructured":"V. Khoromskaia and B. N. Khoromskij,\nTensor approach to linearized Hartree\u2013Fock equation for lattice-type and periodic systems,\npreprint (2014), https:\/\/arxiv.org\/abs\/1408.3839."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_013_w2aab3b7e2112b1b6b1ab2b1c13Aa","doi-asserted-by":"crossref","unstructured":"V. Khoromskaia and B. N. Khoromskij,\nTensor numerical methods in quantum chemistry: From Hartree\u2013Fock to excitation energies,\nPhys. Chem. Chem. Phys. 17 (2015), 31491\u201331509.","DOI":"10.1039\/C5CP01215E"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_014_w2aab3b7e2112b1b6b1ab2b1c14Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij,\nTensor-structured preconditioners and approximate inverse of elliptic operators in \u211dd{\\mathbb{R}^{d}},\nJ. Constr. Approx. 30 (2009), 599\u2013620.","DOI":"10.1007\/s00365-009-9068-9"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_015_w2aab3b7e2112b1b6b1ab2b1c15Aa","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), 257\u2013280.","DOI":"10.1007\/s00365-011-9131-1"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_016_w2aab3b7e2112b1b6b1ab2b1c16Aa","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":"2023033114491597718_j_cmam-2017-0014_ref_017_w2aab3b7e2112b1b6b1ab2b1c17Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij and S. Repin,\nA fast iteration method for solving elliptic problems with quasiperiodic coefficients,\nRussian J. Numer. Anal. Math. Modelling 30 (2015), no. 6, 329\u2013344.","DOI":"10.1515\/rnam-2015-0030"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_018_w2aab3b7e2112b1b6b1ab2b1c18Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij, S. Sauter and A. Veit,\nFast quadrature techniques for retarded potentials based on TT\/QTT tensor approximation,\nComput. Methods Appl. Math. 11 (2011), no. 3, 342\u2013362.","DOI":"10.2478\/cmam-2011-0019"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_019_w2aab3b7e2112b1b6b1ab2b1c19Aa","doi-asserted-by":"crossref","unstructured":"B. N. Khoromskij and G. Wittum,\nNumerical Solution of Elliptic Differential Equations by Reduction to the Interface,\nLect. Notes Comput. Sci. Eng. 36,\nSpringer, Berlin, 2004.","DOI":"10.1007\/978-3-642-18777-3"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_020_w2aab3b7e2112b1b6b1ab2b1c20Aa","doi-asserted-by":"crossref","unstructured":"J.-L. Lions and G. Stampacchia,\nVariational inequalities,\nComm. Pure Appl. Math. 20 (1967), 493\u2013519.","DOI":"10.1002\/cpa.3160200302"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_021_w2aab3b7e2112b1b6b1ab2b1c21Aa","doi-asserted-by":"crossref","unstructured":"O. Mali, P. Neittaanmaki and S. Repin,\nAccuracy Verification Methods. Theory and Algorithms,\nSpringer, New York, 2014.","DOI":"10.1007\/978-94-007-7581-7"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_022_w2aab3b7e2112b1b6b1ab2b1c22Aa","unstructured":"P. Neittaanmaki and S. Repin,\nReliable Methods for Computer Simulation. Error Control and a Posteriori Estimates,\nElsevier, Amsterdam, 2004."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_023_w2aab3b7e2112b1b6b1ab2b1c23Aa","doi-asserted-by":"crossref","unstructured":"I. V. Oseledets and S. V. Dolgov,\nSolution of linear systems and matrix inversion in the TT-format,\nSIAM J. Sci. Comput. 34 (2012), no. 5, A2718\u2013A2739.","DOI":"10.1137\/110833142"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_024_w2aab3b7e2112b1b6b1ab2b1c24Aa","unstructured":"A. Ostrowski,\nLes estimations des erreurs a posteriori dans les proc\u00e9d\u00e9s it\u00e9ratifs,\nC. R. Acad. Sci Paris S\u00e9r. A\u2013B 275 (1972), A275\u2013A278."},{"key":"2023033114491597718_j_cmam-2017-0014_ref_025_w2aab3b7e2112b1b6b1ab2b1c25Aa","doi-asserted-by":"crossref","unstructured":"S. Repin,\nA posteriori error estimation for variational problems with uniformly convex functionals,\nMath. Comp. 69 (2000), no. 230, 481\u2013500.","DOI":"10.1090\/S0025-5718-99-01190-4"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_026_w2aab3b7e2112b1b6b1ab2b1c26Aa","doi-asserted-by":"crossref","unstructured":"S. Repin,\nA Posteriori Estimates for Partial Differential Equations,\nWalter de Gruyter, Berlin, 2008.","DOI":"10.1515\/9783110203042"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_027_w2aab3b7e2112b1b6b1ab2b1c27Aa","doi-asserted-by":"crossref","unstructured":"S. Repin, T. Samrowski and S. Sauter,\nCombined a posteriori modeling-discretization error estimate for elliptic problems with complicated interfaces,\nESAIM Math. Model. Numer. Anal. 46 (2012), no. 6, 1389\u20131405.","DOI":"10.1051\/m2an\/2012007"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_028_w2aab3b7e2112b1b6b1ab2b1c28Aa","doi-asserted-by":"crossref","unstructured":"S. Repin, S. Sauter and A. Smolianski,\nA posteriori estimation of dimension reduction errors for elliptic problems on thin domains,\nSIAM J. Numer. Anal. 42 (2004), no. 4, 1435\u20131451.","DOI":"10.1137\/030602381"},{"key":"2023033114491597718_j_cmam-2017-0014_ref_029_w2aab3b7e2112b1b6b1ab2b1c29Aa","doi-asserted-by":"crossref","unstructured":"E. Zeidler,\nNonlinear Functional Analysis and Its Applications. I: Fixed-Point Theorems,\nSpringer, New York, 1986.","DOI":"10.1007\/978-1-4612-4838-5"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/17\/3\/article-p457.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0014\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0014\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:18:44Z","timestamp":1680297524000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2017-0014\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,17]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,6,17]]},"published-print":{"date-parts":[[2017,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0014"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0014","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,6,17]]}}}