{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:20Z","timestamp":1747198040323,"version":"3.40.5"},"reference-count":27,"publisher":"Walter de Gruyter GmbH","issue":"1","funder":[{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"publisher","award":["DE-AC52-07NA27344"],"award-info":[{"award-number":["DE-AC52-07NA27344"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1619640"],"award-info":[{"award-number":["DMS-1619640"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,1,1]]},"abstract":"Abstract<\/jats:title>\n We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate \n \n \n \n \n |<\/m:mo>\n \n \n \u03bb<\/m:mi>\n k<\/m:mi>\n <\/m:msub>\n -<\/m:mo>\n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:mrow>\n |<\/m:mo>\n <\/m:mrow>\n \u2264<\/m:mo>\n \n \n C<\/m:mi>\n k<\/m:mi>\n <\/m:msub>\n \u2062<\/m:mo>\n \n h<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {\\lvert\\lambda_{k}-\\lambda_{k,h}\\rvert\\leq C_{k}h^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in the case of lowest order approximation for both Galerkin and Nystr\u00f6m methods, where h<\/jats:italic> is the mesh size, \n \n \n \n \u03bb<\/m:mi>\n k<\/m:mi>\n <\/m:msub>\n <\/m:math>\n \n {\\lambda_{k}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n <\/m:msub>\n <\/m:math>\n \n {\\lambda_{k,h}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> are the exact and approximate k<\/jats:italic>th largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that \n \n \n \n \n |<\/m:mo>\n \n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n \n (<\/m:mo>\n G<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:msubsup>\n -<\/m:mo>\n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n \n (<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:msubsup>\n <\/m:mrow>\n |<\/m:mo>\n <\/m:mrow>\n \u2264<\/m:mo>\n \n C<\/m:mi>\n \u2062<\/m:mo>\n \n h<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {|\\lambda^{(G)}_{k,h}-\\lambda^{(N)}_{k,h}|\\leq Ch^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, where \n \n \n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n \n (<\/m:mo>\n G<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:msubsup>\n <\/m:math>\n \n {\\lambda^{(G)}_{k,h}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and \n \n \n \n \u03bb<\/m:mi>\n \n k<\/m:mi>\n ,<\/m:mo>\n h<\/m:mi>\n <\/m:mrow>\n \n (<\/m:mo>\n N<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:msubsup>\n <\/m:math>\n \n {\\lambda^{(N)}_{k,h}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> denote the k<\/jats:italic>th largest eigenvalues computed by Galerkin and Nystr\u00f6m methods, respectively, and C<\/jats:italic> is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2018-0186","type":"journal-article","created":{"date-parts":[[2019,4,7]],"date-time":"2019-04-07T02:48:35Z","timestamp":1554605315000},"page":"61-78","source":"Crossref","is-referenced-by-count":5,"title":["Eigenvalue Problems for Exponential-Type Kernels"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9482-6425","authenticated-orcid":false,"given":"Difeng","family":"Cai","sequence":"first","affiliation":[{"name":"Department of Mathematics , Purdue University , West Lafayette , IN 47907-2067 , USA"}]},{"given":"Panayot S.","family":"Vassilevski","sequence":"additional","affiliation":[{"name":"Fariborz Maseeh Department of Mathematics and Statistics , Portland State University , Portland , OR 97207; and Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, Livermore, CA 94550 , USA"}]}],"member":"374","published-online":{"date-parts":[[2019,4,6]]},"reference":[{"key":"2023033110163494066_j_cmam-2018-0186_ref_001","doi-asserted-by":"crossref","unstructured":"K. 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Keller,\nOn the accuracy of finite difference approximations to the eigenvalues of differential and integral operators,\nNumer. Math. 7 (1965), 412\u2013419.","DOI":"10.1007\/BF01436255"},{"key":"2023033110163494066_j_cmam-2018-0186_ref_013","unstructured":"M. A. Krasnosel\u2019skii, G. M. Vainikko, R. P. Zabreyko, Y. B. Ruticki and V. Y. Stetsenko,\nApproximate Solution of Operator Equations,\nSpringer, Cham, 2012."},{"key":"2023033110163494066_j_cmam-2018-0186_ref_014","doi-asserted-by":"crossref","unstructured":"R. Kress,\nLinear Integral Equations, 3rd ed.,\nAppl. Math. Sci. 82,\nSpringer, New York, 2014.","DOI":"10.1007\/978-1-4614-9593-2"},{"key":"2023033110163494066_j_cmam-2018-0186_ref_015","unstructured":"P. D. Lax,\nFunctional Analysis,\nPure Appl. Math. (New York),\nWiley-Interscience, New York, 2002."},{"key":"2023033110163494066_j_cmam-2018-0186_ref_016","doi-asserted-by":"crossref","unstructured":"N. M. Nasrabadi,\nPattern recognition and machine learning,\nJ. Electron. 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Eigensystems,\nSociety for Industrial and Applied Mathematics (SIAM), Philadelphia, 2001.","DOI":"10.1137\/1.9780898718058"},{"key":"2023033110163494066_j_cmam-2018-0186_ref_024","doi-asserted-by":"crossref","unstructured":"R. A. Todor,\nRobust eigenvalue computation for smoothing operators,\nSIAM J. Numer. Anal. 44 (2006), no. 2, 865\u2013878.","DOI":"10.1137\/040616449"},{"key":"2023033110163494066_j_cmam-2018-0186_ref_025","doi-asserted-by":"crossref","unstructured":"G. M. Vainikko,\nOn the speed of convergence of approximate methods in the eigenvalue problem,\nZh. Vychisl. Mat. Mat. Fiz. 7 (1967), 977\u2013987.","DOI":"10.1016\/0041-5553(67)90091-2"},{"key":"2023033110163494066_j_cmam-2018-0186_ref_026","unstructured":"H. Wendland,\nScattered Data Approximation,\nCambridge Monogr. Appl. Comput. Math. 17,\nCambridge University Press, Cambridge, 2005."},{"key":"2023033110163494066_j_cmam-2018-0186_ref_027","doi-asserted-by":"crossref","unstructured":"H. Weyl,\nDas asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie\nder Hohlraumstrahlung),\nMath. 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