{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:29:28Z","timestamp":1787336968874,"version":"build-2736575974"},"reference-count":52,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"6","funder":[{"DOI":"10.13039\/501100006221","name":"Cordis","doi-asserted-by":"publisher","award":["235730"],"award-info":[{"award-number":["235730"]}],"id":[{"id":"10.13039\/501100006221","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1552238"],"award-info":[{"award-number":["1552238"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>Recently, collocation-based radial basis function (RBF) partition of unity methods (PUMs) for solving partial differential equations have been formulated and investigated numerically and theoretically. When combined with stable evaluation methods such as the RBF-QR method, high order convergence rates can be achieved and sustained under refinement. However, some numerical issues remain. The method is sensitive to the node layout, and condition numbers increase with the refinement level. Here, we propose a modified formulation based on least squares approximation. We show that the sensitivity to node layout is removed and that conditioning can be controlled through oversampling. We derive theoretical error estimates both for the collocation and least squares RBF-PUMs. Numerical experiments are performed for the Poisson equation in two and three space dimensions for regular and irregular geometries. The convergence experiments confirm the theoretical estimates, and the least squares formulation is shown to be 5--10 times faster than the collocation formulation for the same accuracy.<\/jats:p>","DOI":"10.1137\/17m1118087","type":"journal-article","created":{"date-parts":[[2017,11,9]],"date-time":"2017-11-09T10:51:35Z","timestamp":1510224695000},"page":"A2538-A2563","source":"Crossref","is-referenced-by-count":99,"title":["A Least Squares Radial Basis Function Partition of Unity Method for Solving PDEs"],"prefix":"10.1137","volume":"39","author":[{"given":"Elisabeth","family":"Larsson","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Victor","family":"Shcherbakov","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alfa","family":"Heryudono","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,11,9]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.10.045"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/s10898-011-9667-4"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.12.008"},{"key":"atypb5","first-page":"431","volume":"3","author":"Cavoretto R.","year":"2012","journal-title":"Commun. Appl. Ind. Math."},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2014.01.007"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/140989157"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2016.04.021"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1063\/1.4965349"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2011.06.037"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0722-9"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(01)00295-4"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1142\/6437"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1137\/110824784"},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.05.026"},{"key":"atypb16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2012.01.028"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.2009.0033"},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611974041"},{"key":"atypb19","doi-asserted-by":"publisher","DOI":"10.1137\/09076756X"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2010.12.014"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2012.11.006"},{"key":"atypb22","doi-asserted-by":"publisher","DOI":"10.1137\/060671991"},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2003.08.010"},{"key":"atypb24","volume-title":"Efficient Numerical Solution Technique for Partial Differential Equations with Mixed Regularity, tech. rep., Marie Curie FP7","author":"Heryudono A.","year":"2012"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-015-0120-6"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/120893975"},{"key":"atypb27","first-page":"367","volume":"1","author":"Kwok T.-O.","year":"2009","journal-title":"Adv. Appl. Math. Mech."},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1016\/S0898-1221(03)90151-9"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2005.01.010"},{"key":"atypb30","volume-title":"A Partition of Unity Radial Basis Function Collocation Method for Partial Differential Equations, manuscript in preparation","author":"Larsson E.","year":"2017"},{"key":"atypb31","doi-asserted-by":"publisher","DOI":"10.1137\/120899108"},{"key":"atypb32","doi-asserted-by":"publisher","DOI":"10.1007\/BF01893414"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2003.08.007"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-008-9089-0"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-013-9211-5"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-014-9935-9"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1007\/s00365-004-0585-2"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1137\/050633366"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-015-0731-8"},{"key":"atypb40","doi-asserted-by":"publisher","DOI":"10.2307\/1968466"},{"key":"atypb41","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-016-0616-y"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2015.11.007"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1145\/800186.810616"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00618-7"},{"key":"atypb45","volume-title":"Proceedings of the 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation","author":"Tolstykh A. I.","year":"2000"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1080\/00207160.2015.1072172"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1007\/BF02123482"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01102-3"},{"key":"atypb49","first-page":"473","volume-title":"Approximation Theory X (St.","author":"Wendland H.","year":"2001"},{"key":"atypb50","volume-title":"Scattered Data Approximation","author":"Wendland H.","year":"2005"},{"key":"atypb51","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2005.05.030"},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2016.11.030"}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/17M1118087","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:38:39Z","timestamp":1787333919000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/17M1118087"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":52,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/17M1118087"],"URL":"https:\/\/doi.org\/10.1137\/17m1118087","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}