{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T07:05:17Z","timestamp":1787382317258,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001711","name":"Swiss National Science Foundation","doi-asserted-by":"crossref","award":["174987"],"award-info":[{"award-number":["174987"]}],"id":[{"id":"10.13039\/501100001711","id-type":"DOI","asserted-by":"crossref"}]},{"name":"Excellence Initiative of the German Federal and State Governments"},{"DOI":"10.13039\/501100005714","name":"Technische Universit\u00e4t Darmstadt","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100005714","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>The multiplication of matrices is an important arithmetic operation in computational mathematics. In the context of hierarchical matrices, this operation can be realized by the multiplication of structured blockwise low-rank matrices, resulting in an almost linear cost. However, the computational efficiency of the algorithm is based on a recursive scheme which makes the error analysis quite involved. In this article, we propose a new algorithmic framework for the multiplication of hierarchical matrices. It improves currently known implementations by reducing the multiplication of hierarchical matrices to suitable low-rank approximations of sums of matrix products. We propose several compression schemes to address this task. As a consequence, we are able to compute the best approximation of hierarchical matrix products. A cost analysis shows that, under reasonable assumptions on the low-rank approximation method, the cost of the framework is almost linear with respect to the size of the matrix. Numerical experiments show that the new approach produces indeed the best approximation of the product of hierarchical matrices for a given tolerance. They also show that the new multiplication can accomplish this task in less computation time than the established multiplication algorithm without error control.<\/jats:p>","DOI":"10.1137\/18m1189373","type":"journal-article","created":{"date-parts":[[2019,1,30]],"date-time":"2019-01-30T15:30:46Z","timestamp":1548862246000},"page":"147-174","source":"Crossref","is-referenced-by-count":6,"title":["On the Best Approximation of the Hierarchical Matrix Product"],"prefix":"10.1137","volume":"40","author":[{"given":"J\u00fcrgen","family":"D\u00f6lz","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Helmut","family":"Harbrecht","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Michael D.","family":"Multerer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,1,30]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-013-9714-z"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2015.10.012"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-006-0178-y"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/PL00005410"},{"key":"atypb5","unstructured":"M. Bebendorf,\n                      Hierarchical Matrices. A Means to Efficiently Solve Elliptic Boundary Value problems\n                      , Lect. Notes Comput. Sci. Eng. 63, Springer, Berlin, 2008."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-002-0445-6"},{"key":"atypb7","doi-asserted-by":"crossref","unstructured":"S. B\u00f6rm,\n                      Efficient Numerical Methods for Non-local Operators. ${\\mathcal{H}}^2$-Matrix Compression, Algorithms and Analysis\n                      , EMS Tracts Math. 14, European Mathematical Society, Z\u00fcrich, 2010.","DOI":"10.4171\/091"},{"key":"atypb8","unstructured":"S. B\u00f6rm,\n                      Hierarchical Matrix Arithmetic with Accumulated Updates\n                      , preprint, arXiv:1703.09085, 2017."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803436652"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-014-9965-3"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1137\/16M1074813"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-016-0825-y"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100360"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/0702016"},{"key":"atypb15","unstructured":"G. H. Golub and C. F. Van Loan,\n                      Matrix Computations\n                      , 4th ed., Johns Hopkins University Press, Baltimore, MD, 2012."},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/S0024-3795(96)00301-1"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-003-0019-1"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-002-1470-0"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-008-0098-9"},{"key":"atypb20","unstructured":"G. Guennebaud and B. Jacob,\n                      Eigen v\n                      3,http:\/\/eigen.tuxfamily.org(2010)."},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s006070050015"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"W. Hackbusch,\n                      Hierarchical Matrices: Algorithms and Analysis\n                      , Springer, Berlin, 2015.","DOI":"10.1007\/978-3-662-47324-5"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1007\/PL00021408"},{"key":"atypb24","first-page":"9","author":"Hackbusch W.","year":"2000","journal-title":"Berlin"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1137\/090771806"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"H. Harbrecht and M. Peters,\n                      Comparison of fast boundary element methods on parametric surfaces\n                      , Comput. Methods Appl. Mech. Eng., 261-262 (2013), pp. 39-55.","DOI":"10.1016\/j.cma.2013.03.022"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1007\/s00791-015-0254-y"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/100786617"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1137\/15M1016679"},{"key":"atypb30","first-page":"42","author":"Rouet F.-H.","year":"2016","journal-title":"ACM Trans. Math. Software"},{"key":"atypb31","doi-asserted-by":"crossref","unstructured":"O. Steinbach,\n                      Numerical Approximation Methods for Elliptic Boundary Value Problems\n                      , Springer, Berlin, 2008.","DOI":"10.1007\/978-0-387-68805-3"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1002\/nla.691"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1189373","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:26Z","timestamp":1787327246000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1189373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1189373"],"URL":"https:\/\/doi.org\/10.1137\/18m1189373","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}