{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:32:28Z","timestamp":1787329948536,"version":"build-2736575974"},"reference-count":10,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>The generalized singular value expansion (GSVE) simultaneously diagonalizes a pair of operators on Hilbert space. From a theoretical point of view, the GSVE enables a straightforward analysis of, for example, weighted least-squares problems and the method of Tikhonov regularization with seminorms. When the operators are discretized, an approximate GSVE can be computed from the generalized singular value decomposition of a pair of Galerkin matrices. Unless the discretization is carefully chosen, spurious modes can appear, but a natural condition on the discretization guarantees convergence of the approximate GSVE to the exact one. Numerical examples illustrate the pitfalls of a poor discretization and efficacy of the convergence conditions.<\/jats:p>","DOI":"10.1137\/18m1163713","type":"journal-article","created":{"date-parts":[[2018,9,11]],"date-time":"2018-09-11T16:33:48Z","timestamp":1536683628000},"page":"2776-2795","source":"Crossref","is-referenced-by-count":2,"title":["Approximating the Generalized Singular Value Expansion"],"prefix":"10.1137","volume":"56","author":[{"given":"Mark S.","family":"Gockenbach","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Matthew J.","family":"Roberts","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,9,11]]},"reference":[{"key":"atypb1","volume-title":"Collectively Compact Operator Approximation Theory and Applications to Integral Equations","author":"Anselone P. M.","year":"1971"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1016\/S1570-8659(05)80042-0"},{"key":"atypb3","volume-title":"Handbook of Numerical Analysis","volume":"1","author":"Bj\u00f6rck AA.","year":"1990"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492910000012"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-1740-8"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/15M1019453"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1090\/car\/032"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/BF02790238"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1016\/0022-247X(80)90145-6"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5280-1"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1163713","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:40:36Z","timestamp":1787326836000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1163713"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":10,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/18M1163713"],"URL":"https:\/\/doi.org\/10.1137\/18m1163713","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}