{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T01:51:55Z","timestamp":1780365115065,"version":"3.54.1"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"1-2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computing"],"published-print":{"date-parts":[[2009,4]]},"DOI":"10.1007\/s00607-008-0018-3","type":"journal-article","created":{"date-parts":[[2008,10,30]],"date-time":"2008-10-30T08:22:05Z","timestamp":1225354925000},"page":"49-67","source":"Crossref","is-referenced-by-count":59,"title":["Application of hierarchical matrices for computing the Karhunen\u2013Lo\u00e8ve expansion"],"prefix":"10.1007","volume":"84","author":[{"given":"B. N.","family":"Khoromskij","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Litvinenko","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"H. G.","family":"Matthies","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2008,10,31]]},"reference":[{"key":"18_CR1","doi-asserted-by":"crossref","unstructured":"Atkinson KE (1997) The numerical solution of integral equations of the second kind. Cambridge monographs on applied and computational mathematics, vol 4. Cambridge University Press, Cambridge","DOI":"10.1017\/CBO9780511626340"},{"key":"18_CR2","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka I, Tempone R, Zouraris GE (2004) Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J Numer Anal 42(2):800\u2013825 (electronic)","DOI":"10.1137\/S0036142902418680"},{"key":"18_CR3","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka I, Nobile F, Tempone R (2007) A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J Numer Anal 45(3):1005\u20131034 (electronic)","DOI":"10.1137\/050645142"},{"key":"18_CR4","doi-asserted-by":"crossref","unstructured":"Bai Z, Demmel J, Dongarra J, Ruhe A, van der Vorst H (eds) (2000) Templates for the solution of algebraic eigenvalue problems\u2014 practical guide, vol 11 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA","DOI":"10.1137\/1.9780898719581"},{"issue":"4","key":"18_CR5","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1007\/PL00005410","volume":"86","author":"M Bebendorf","year":"2000","unstructured":"Bebendorf M (2000) Approximation of boundary element matrices. Numer Math 86(4): 565\u2013589","journal-title":"Numer Math"},{"issue":"5","key":"18_CR6","doi-asserted-by":"crossref","first-page":"407","DOI":"10.1002\/nla.525","volume":"14","author":"M Bebendorf","year":"2007","unstructured":"Bebendorf M, Hackbusch W (2007) Stabilized rounded addition of hierarchical matrices. Numer Linear Algebra Appl 14(5): 407\u2013423","journal-title":"Numer Linear Algebra Appl"},{"issue":"1","key":"18_CR7","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s00607-002-1469-6","volume":"70","author":"M Bebendorf","year":"2003","unstructured":"Bebendorf M, Rjasanow S (2003) Adaptive low-rank approximation of collocation matrices. Computing 70(1): 1\u201324","journal-title":"Computing"},{"issue":"2","key":"18_CR8","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/s00211-005-0618-1","volume":"101","author":"S B\u00f6rm","year":"2005","unstructured":"B\u00f6rm S, Grasedyck L (2005) Hybrid cross approximation of integral operators. Numer Math 101(2): 221\u2013249","journal-title":"Numer Math"},{"key":"18_CR9","volume-title":"Hierarchical Matrices, Lecture Note","author":"S B\u00f6rm","year":"2003","unstructured":"B\u00f6rm S, Grasedyck L, Hackbusch W (2003) Hierarchical Matrices, Lecture Note. Max-Planck Institute for Mathematics, Leipzig"},{"issue":"1","key":"18_CR10","doi-asserted-by":"crossref","first-page":"3","DOI":"10.1007\/s00791-006-0047-4","volume":"10","author":"M Eiermann","year":"2007","unstructured":"Eiermann M, Ernst OG, Ullmann E (2007) Computational aspects of the stochastic finite element method. Comput Vis Sci 10(1): 3\u201315","journal-title":"Comput Vis Sci"},{"issue":"9\u201311","key":"18_CR11","doi-asserted-by":"crossref","first-page":"1037","DOI":"10.1016\/j.cma.2004.06.028","volume":"194","author":"HC Elman","year":"2005","unstructured":"Elman HC, Ernst OG, O\u2019Leary DP, Stewart M (2005) Efficient iterative algorithms for the stochastic finite element method with application to acoustic scattering. Comput Methods Appl Mech Eng 194(9\u201311): 1037\u20131055","journal-title":"Comput Methods Appl Mech Eng"},{"key":"18_CR12","doi-asserted-by":"crossref","unstructured":"Frigo M, Johnson S (1998) FFTW: An adaptive software architecture for the FFT. In: Proc. ICASSP, IEEE, Seattle, WA, (3):1381\u20131384. http:\/\/www.fftw.org","DOI":"10.1109\/ICASSP.1998.681704"},{"issue":"10","key":"18_CR13","doi-asserted-by":"crossref","first-page":"2351","DOI":"10.1061\/(ASCE)0733-9399(1991)117:10(2351)","volume":"117","author":"R Ghanem","year":"1991","unstructured":"Ghanem R, Spanos D (1991) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech 117(10): 2351\u20132370","journal-title":"J Eng Mech"},{"key":"18_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-3094-6","volume-title":"Stochastic finite elements: a spectral approach","author":"R Ghanem","year":"1991","unstructured":"Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer, New York"},{"key":"18_CR15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0024-3795(96)00301-1","volume":"261","author":"SA Goreinov","year":"1997","unstructured":"Goreinov SA, Tyrtyshnikov EE, Zamarashkin NL (1997) A theory of pseudoskeleton approximations. Linear Algebra Appl 261: 1\u201321","journal-title":"Linear Algebra Appl"},{"key":"18_CR16","unstructured":"Grasedyck L, B\u00f6rm S: $${\\mathcal{H}}$$ -matrix library. http:\/\/www.hlib.org"},{"issue":"4","key":"18_CR17","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s00607-003-0019-1","volume":"70","author":"L Grasedyck","year":"2003","unstructured":"Grasedyck L, Hackbusch W (2003) Construction arithmetics of $${\\mathcal{H}}$$ -matrices. Computing 70(4): 295\u2013334","journal-title":"Computing"},{"issue":"1","key":"18_CR18","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1007\/PL00021408","volume":"64","author":"W Hackbusch","year":"2000","unstructured":"Hackbusch W, Khoromskij BN (2000) A sparse $${\\mathcal{H}}$$ -matrix arithmetic. II. Application to multi-dimensional problems. Computing 64(1): 21\u201347","journal-title":"Computing"},{"key":"18_CR19","unstructured":"Hackbusch W (1995) Integral equations, vol 120 of International Series of Numerical Mathematics. Birkh\u00e4user Verlag, Basel, 1995. Theory and numerical treatment, Translated and revised by the author from the 1989 German original"},{"issue":"2","key":"18_CR20","doi-asserted-by":"crossref","first-page":"89","DOI":"10.1007\/s006070050015","volume":"62","author":"W Hackbusch","year":"1999","unstructured":"Hackbusch W (1999) A sparse matrix arithmetic based on $${\\mathcal{H}}$$ -matrices. I. Introduction to $${\\mathcal{H}}$$ -matrices. Computing 62(2): 89\u2013108","journal-title":"Computing"},{"key":"18_CR21","volume-title":"Hierarchische Matrizen\u2014Algorithmen und Analysis","author":"W Hackbusch","year":"2004","unstructured":"Hackbusch W (2004) Hierarchische Matrizen\u2014Algorithmen und Analysis. Max-Planck-Institut f\u00fcr Mathematik, Leipzig"},{"issue":"3","key":"18_CR22","doi-asserted-by":"crossref","first-page":"207","DOI":"10.1007\/s00607-004-0080-4","volume":"73","author":"W Hackbusch","year":"2004","unstructured":"Hackbusch W, Khoromskij BN, Kriemann R (2004) Hierarchical matrices based on a weak admissibility criterion. Computing 73(3): 207\u2013243","journal-title":"Computing"},{"issue":"2","key":"18_CR23","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1515\/1569395054012767","volume":"13","author":"W Hackbusch","year":"2005","unstructured":"Hackbusch W, Khoromskij BN, Tyrtyshnikov EE (2005) Hierarchical Kronecker tensor-product approximations. J Numer Math 13(2): 119\u2013156","journal-title":"J Numer Math"},{"key":"18_CR24","doi-asserted-by":"crossref","unstructured":"Hida T, Kuo H-H, Potthoff J, Streit L (1993) White noise\u2014an infinite-dimensional calculus, vol 253 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1993)","DOI":"10.1007\/978-94-017-3680-0"},{"key":"18_CR25","unstructured":"Keese A (2004) Numerical solution of systems with stochastic uncertainties. A general purpose framework for stochastic finite elements. Ph.D. Thesis, TU Braunschweig, Germany"},{"key":"18_CR26","doi-asserted-by":"crossref","first-page":"255","DOI":"10.6028\/jres.045.026","volume":"45","author":"C Lanczos","year":"1995","unstructured":"Lanczos C (1995) An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. J Res Nat Bur Standards 45: 255\u2013282","journal-title":"J Res Nat Bur Standards"},{"key":"18_CR27","doi-asserted-by":"crossref","unstructured":"Lehoucq RB, Sorensen DC, Yang C (1998) ARPACK users\u2019 guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods, vol 6 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA","DOI":"10.1137\/1.9780898719628"},{"key":"18_CR28","doi-asserted-by":"crossref","unstructured":"Lo\u00e8ve, M (1977) Probability theory I. Graduate Texts in Mathematics, vol 45, 6, 4th edn. Springer, New York","DOI":"10.1007\/978-1-4684-9464-8"},{"key":"18_CR29","doi-asserted-by":"crossref","unstructured":"Matthies HG (2007) Uncertainty quantification with stochastic finite elements. Part 1. Fundamentals. Encyclopedia of Computational Mechanics. Wiley, New York","DOI":"10.1002\/0470091355.ecm071"},{"issue":"12\u201316","key":"18_CR30","doi-asserted-by":"crossref","first-page":"1295","DOI":"10.1016\/j.cma.2004.05.027","volume":"194","author":"HG Matthies","year":"2005","unstructured":"Matthies HG, Keese A (2005) Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput Methods Appl Mech Eng 194(12\u201316): 1295\u20131331","journal-title":"Comput Methods Appl Mech Eng"},{"issue":"8\u20139","key":"18_CR31","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1016\/S0965-9978(00)00034-X","volume":"31","author":"MF Pellissetti","year":"2000","unstructured":"Pellissetti MF, Ghanem R (2000) Iterative solution of systems of linear equations arising in the context of stochastic finite elements. Adv Eng Softw 31(8\u20139): 607\u2013616","journal-title":"Adv Eng Softw"},{"key":"18_CR32","unstructured":"Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical recipes in C\u2014the art of scientific computing, 2nd edn. Cambridge University Press, Cambridge"},{"key":"18_CR33","unstructured":"Riesz F, Sz.-Nagy B (1990) Functional analysis. Dover Books on Advanced Mathematics. Dover Publications Inc., New York, 1990. Translated from the second French edition by Leo F. Boron, Reprint of the 1955 original"},{"key":"18_CR34","volume-title":"Numerical methods for large eigenvalue problems","author":"Y Saad","year":"1992","unstructured":"Saad Y (1992) Numerical methods for large eigenvalue problems. Manchester University Press, Manchester"},{"issue":"4","key":"18_CR35","doi-asserted-by":"crossref","first-page":"707","DOI":"10.1007\/s00211-003-0455-z","volume":"95","author":"C Schwab","year":"2003","unstructured":"Schwab C, Todor RA (2003) Sparse finite elements for elliptic problems with stochastic loading. Numer Math 95(4): 707\u2013734","journal-title":"Numer Math"},{"key":"18_CR36","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1007\/s00607-003-0024-4","volume":"71","author":"C Schwab","year":"2003","unstructured":"Schwab C, Todor RA (2003) Sparse finite elements for stochastic elliptic problems\u2014higher order moments. Computing 71: 43\u201363","journal-title":"Computing"},{"issue":"1","key":"18_CR37","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1016\/j.jcp.2006.01.048","volume":"217","author":"C Schwab","year":"2006","unstructured":"Schwab C, Todor RA (2006) Karhunen\u2013Lo\u00e8ve approximation of random fields by generalized fast multipole methods. J Comput Phys 217(1): 100\u2013122","journal-title":"J Comput Phys"},{"issue":"1","key":"18_CR38","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.jcp.2006.12.011","volume":"224","author":"B Seynaeve","year":"2007","unstructured":"Seynaeve B, Rosseel E, Nicola\u00ef B, Vandewalle S (2007) Fourier mode analysis of multigrid methods for partial differential equations with random coefficients. J Comput Phys 224(1): 132\u2013","journal-title":"J Comput Phys"},{"issue":"2","key":"18_CR39","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1093\/imanum\/drl025","volume":"27","author":"RA Todor","year":"2007","unstructured":"Todor RA, Schwab C (2007) Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J Numer Anal 27(2): 232\u2013261","journal-title":"IMA J Numer Anal"},{"key":"18_CR40","unstructured":"Werner D (2000) Funktionalanalysis, extended edn. Springer, Berlin"},{"key":"18_CR41","doi-asserted-by":"crossref","first-page":"897","DOI":"10.2307\/2371268","volume":"60","author":"N Wiener","year":"1938","unstructured":"Wiener N (1938) The homogeneous chaos. Am J Math 60: 897\u2013936","journal-title":"Am J Math"},{"key":"18_CR42","doi-asserted-by":"crossref","unstructured":"Wu K, Simon H (2000) Thick-restart Lanczos method for large symmetric eigenvalue problems. SIAM J Matrix Anal Appl 22(2):602\u2013616 (electronic)","DOI":"10.1137\/S0895479898334605"}],"container-title":["Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00607-008-0018-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,20]],"date-time":"2023-05-20T20:45:10Z","timestamp":1684615510000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00607-008-0018-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,10,31]]},"references-count":42,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2009,4]]}},"alternative-id":["18"],"URL":"https:\/\/doi.org\/10.1007\/s00607-008-0018-3","relation":{},"ISSN":["0010-485X","1436-5057"],"issn-type":[{"value":"0010-485X","type":"print"},{"value":"1436-5057","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,10,31]]}}}