{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:24:10Z","timestamp":1787329450637,"version":"build-2736575974"},"reference-count":47,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, we discuss the numerical solution of certain matrix-valued PDEs. Such PDEs arise, for example, when constructing a Riemannian contraction metric for a dynamical system given by an autonomous ODE. We develop and analyze a new meshfree discretization scheme using kernel-based approximation spaces. However, since these approximation spaces have now to be matrix-valued, the kernels we need to use are fourth-order tensors. We will review and extend recent results on even more general reproducing kernel Hilbert spaces. We will then apply this general theory to solve a matrix-valued PDE and derive error estimates for the approximate solution. The paper ends with applications to typical examples from dynamical systems.<\/jats:p>","DOI":"10.1137\/16m1092842","type":"journal-article","created":{"date-parts":[[2018,11,29]],"date-time":"2018-11-29T11:18:02Z","timestamp":1543490282000},"page":"3386-3406","source":"Crossref","is-referenced-by-count":12,"title":["Kernel-Based Discretization for Solving Matrix-Valued PDEs"],"prefix":"10.1137","volume":"56","author":[{"given":"Peter","family":"Giesl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9614-5321","authenticated-orcid":true,"given":"Holger","family":"Wendland","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,11,29]]},"reference":[{"key":"atypb1","first-page":"17","volume-title":"Surface Fitting and Multiresolution Methods, A. L. M\u00e9haut'e","author":"Amodei L.","year":"1997"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9947-1950-0051437-7"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1016\/j.automatica.2007.12.012"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1137\/080743743"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4419-9096-9"},{"key":"atypb6","volume-title":"Elanders","author":"Borg G.","year":"1960"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511543241"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1007\/s00607-007-0248-9"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-47969-4_17"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511801389"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1090\/S0273-0979-01-00923-5"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492913000044"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1142\/6437"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2010.08.005"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492914000130"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1109\/TAC.2013.2285771"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1023\/A:1018916902176"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-02096-0"},{"key":"atypb19","volume-title":"Lect. Notes Math.","author":"Giesl P.","year":"1904"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2014.12.010"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.na.2013.03.012"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/140988802"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.3934\/dcdsb.2015.20.2291"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1137\/060658813"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-50085-5"},{"key":"atypb27","volume-title":"Ordinary Differential Equations","author":"Hartman P.","year":"1964"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(90)90270-T"},{"key":"atypb29","volume-title":"Nonlinear systems","author":"Khalil H.","year":"1992"},{"key":"atypb30","volume-title":"Mir","author":"Krasovskii N. N.","year":"1959"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-0193-3"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1016\/S0005-1098(98)00019-3"},{"key":"atypb33","unstructured":"S. Lowitzsch,\n                      Approximation and Interpolation Employing Divergence-Free Radial Basis Functions with Applications\n                      , PhD thesis, Texas A&M University, College Station, TX, 2002."},{"key":"atypb34","doi-asserted-by":"crossref","first-page":"301","DOI":"10.57262\/ade\/1355867466","volume":"12","author":"Mardare S.","year":"2007","journal-title":"Adv. Differential Equations"},{"key":"atypb35","first-page":"1099","volume":"6","author":"Micchelli C. A.","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1254147-6"},{"key":"atypb37","first-page":"643","volume":"74","author":"Narcowich F. J.","year":"2005","journal-title":"Math. Comput."},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-009-9142-7"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492906270016"},{"key":"atypb40","volume-title":"Regularization, Optimization, and Beyond","author":"Sch\u00f6lkopf B.","year":"2002"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-77242-4"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10661"},{"key":"atypb43","volume-title":"Statistical Learning Theory","author":"Vapnik V.","year":"1998"},{"key":"atypb44","first-page":"251","volume-title":"Inverse Problems, Image Analysis","author":"Weickert J.","year":"2001"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1007\/BF02123482"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01102-3"},{"key":"atypb47","volume-title":"Scattered Data Approximation","author":"Wendland H.","year":"2005"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/080730299"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1092842","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:18:07Z","timestamp":1787325487000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1092842"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1]]},"references-count":47,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2018,1]]}},"alternative-id":["10.1137\/16M1092842"],"URL":"https:\/\/doi.org\/10.1137\/16m1092842","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,1]]}}}