{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T10:29:35Z","timestamp":1768732175261,"version":"3.49.0"},"reference-count":47,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T00:00:00Z","timestamp":1539388800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2019,4]]},"DOI":"10.1007\/s10915-018-0851-2","type":"journal-article","created":{"date-parts":[[2018,10,13]],"date-time":"2018-10-13T06:45:25Z","timestamp":1539413125000},"page":"321-344","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels"],"prefix":"10.1007","volume":"79","author":[{"given":"S.","family":"De Marchi","sequence":"first","affiliation":[]},{"given":"A.","family":"Mart\u00ednez","sequence":"additional","affiliation":[]},{"given":"E.","family":"Perracchione","sequence":"additional","affiliation":[]},{"given":"M.","family":"Rossini","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,10,13]]},"reference":[{"key":"851_CR1","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.amc.2010.01.065","volume":"216","author":"M Bozzini","year":"2010","unstructured":"Bozzini, M., Lenarduzzi, L., Rossini, M.: Polyharmonic splines: an approximation method for noisy scattered data of extra-large size. Appl. Math. Comput. 216, 317\u2013331 (2010)","journal-title":"Appl. Math. Comput."},{"key":"851_CR2","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1093\/imanum\/drt071","volume":"35","author":"M Bozzini","year":"2015","unstructured":"Bozzini, M., Lenarduzzi, L., Rossini, M., Schaback, R.: Interpolation with variably scaled kernels. IMA J. Numer. Anal. 35, 199\u2013219 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"851_CR3","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1016\/j.amc.2004.07.001","volume":"165","author":"M Caliari","year":"2005","unstructured":"Caliari, M., De Marchi, S., Vianello, M.: Bivariate polynomial interpolation on the square at new nodal sets. Appl. Math. Comput. 165, 261\u2013274 (2005)","journal-title":"Appl. Math. Comput."},{"key":"851_CR4","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.neunet.2015.07.015","volume":"71","author":"R Cancelliere","year":"2015","unstructured":"Cancelliere, R., Gai, M., Gallinari, P., Rubini, L.: OCReP: an optimally conditioned regularization for pseudoinversion based neural training. Neural Netw. 71, 76\u201387 (2015)","journal-title":"Neural Netw."},{"key":"851_CR5","doi-asserted-by":"publisher","unstructured":"Cavoretto, R., De Rossi, A., Dell\u2019Accio, F., Di Tommaso, F.: Fast computation of triangular Shepard interpolants. J. Comput. Appl. Math. (2018). https:\/\/doi.org\/10.1016\/j.cam.2018.03.012","DOI":"10.1016\/j.cam.2018.03.012"},{"key":"851_CR6","doi-asserted-by":"publisher","first-page":"2568","DOI":"10.1016\/j.camwa.2016.04.021","volume":"71","author":"R Cavoretto","year":"2016","unstructured":"Cavoretto, R., De Rossi, A., Perracchione, E.: Efficient computation of partition of unity interpolants through a block-based searching technique. Comput. Math. Appl. 71, 2568\u20132584 (2016)","journal-title":"Comput. Math. Appl."},{"key":"851_CR7","doi-asserted-by":"publisher","first-page":"1750008","DOI":"10.1142\/S0219876217500086","volume":"14","author":"R Cavoretto","year":"2017","unstructured":"Cavoretto, R., De Rossi, A., Perracchione, E., Venturino, E.: Graphical representation of separatrices of attraction basins in two and three-dimensional dynamical systems. Int. J. Comput. Methods 14, 1750008 (2017)","journal-title":"Int. J. Comput. Methods"},{"key":"851_CR8","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1007\/s11075-014-9850-z","volume":"68","author":"R Cavoretto","year":"2015","unstructured":"Cavoretto, R., Fasshauer, G.E., McCourt, M.: An introduction to the Hilbert\u2013Schmidt SVD using iterated Brownian bridge kernels. Numer. Algorithms 68, 393\u2013422 (2015)","journal-title":"Numer. Algorithms"},{"key":"851_CR9","first-page":"230","volume":"304","author":"O Davydov","year":"2011","unstructured":"Davydov, O., Oanh, D.T.: Adaptive meshless centres and RBF stencils for Poisson equation. J. Comput. Phys. 304, 230\u2013287 (2011)","journal-title":"J. Comput. Phys."},{"key":"851_CR10","first-page":"343","volume":"61","author":"S Marchi De","year":"2003","unstructured":"De Marchi, S.: On optimal center locations for radial basis interpolation: computational aspects. Rend. Sem. Mat. Torino 61, 343\u2013358 (2003)","journal-title":"Rend. Sem. Mat. Torino"},{"key":"851_CR11","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/978-3-319-59912-0_3","volume-title":"Approximation Theory XV: San Antonio 2016","author":"S Marchi De","year":"2017","unstructured":"De Marchi, S., Idda, A., Santin, G.: A rescaled method for RBF approximation. In: Fasshauer, G.E., et al. (eds.) Approximation Theory XV: San Antonio 2016, vol. 201, pp. 39\u201359. Springer, New York (2017)"},{"key":"851_CR12","doi-asserted-by":"publisher","unstructured":"De Marchi, S., Mart\u00ednez, A., Perracchione, E.: Fast and stable rational RBF-based partition of unity interpolation. J. Comput. App. Math. (2018). https:\/\/doi.org\/10.1016\/j.cam.2018.07.020","DOI":"10.1016\/j.cam.2018.07.020"},{"key":"851_CR13","doi-asserted-by":"publisher","first-page":"949","DOI":"10.1007\/s10543-014-0537-6","volume":"55","author":"S Marchi De","year":"2015","unstructured":"De Marchi, S., Santin, G.: Fast computation of orthonormal basis for RBF spaces through Krylov space methods. BIT 55, 949\u2013966 (2015)","journal-title":"BIT"},{"key":"851_CR14","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1186\/s13104-015-1830-4","volume":"9","author":"A Rossi De","year":"2016","unstructured":"De Rossi, A., Perracchione, E., Venturino, E.: Fast strategy for PU interpolation: an application for the reconstruction of separatrix manifolds. Dolom. Res. Notes Approx. 9, 3\u201312 (2016)","journal-title":"Dolom. Res. Notes Approx."},{"key":"851_CR15","doi-asserted-by":"publisher","first-page":"927","DOI":"10.1016\/j.camwa.2006.06.005","volume":"53","author":"TA Driscoll","year":"2007","unstructured":"Driscoll, T.A., Heryudono, A.R.H.: Adaptive residual subsampling methods for radial basis function interpolation and collocation problems. Comput. Math. Appl. 53, 927\u2013939 (2007)","journal-title":"Comput. Math. Appl."},{"key":"851_CR16","unstructured":"Fasshauer, G.E.: Dealing with Ill-conditioned RBF systems. Dolomites Res. Notes Approx. 1 (2008). https:\/\/www.emis.de\/journals\/DRNA\/3-12.html"},{"key":"851_CR17","doi-asserted-by":"publisher","DOI":"10.1142\/6437","volume-title":"Meshfree Approximations Methods with Matlab","author":"GE Fasshauer","year":"2007","unstructured":"Fasshauer, G.E.: Meshfree Approximations Methods with Matlab. World Scientific, Singapore (2007)"},{"key":"851_CR18","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1007\/s11075-007-9072-8","volume":"45","author":"GE Fasshauer","year":"2007","unstructured":"Fasshauer, G.E., Zhang, J.G.: On choosing \u201coptimal\u201d shape parameters for RBF approximation. Numer. Algorithms 45, 345\u2013368 (2007)","journal-title":"Numer. Algorithms"},{"key":"851_CR19","doi-asserted-by":"publisher","first-page":"2403","DOI":"10.1137\/120898383","volume":"51","author":"P Farrell","year":"2013","unstructured":"Farrell, P., Wendland, H.: RBF multiscale collocation for second order elliptic boundary value problems. J. Numer. Anal. 51, 2403\u20132425 (2013)","journal-title":"J. Numer. Anal."},{"key":"851_CR20","doi-asserted-by":"publisher","first-page":"869","DOI":"10.1137\/09076756X","volume":"33","author":"B Fornberg","year":"2011","unstructured":"Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33, 869\u2013892 (2011)","journal-title":"SIAM J. Sci. Comput."},{"key":"851_CR21","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1186\/s13104-016-2323-9","volume":"10","author":"E Francomano","year":"2017","unstructured":"Francomano, E., Hilker, F.M., Paliaga, M., Venturino, E.: An efficient method to reconstruct invariant manifolds of saddle points. Dolom. Res. Notes Approx. 10, 25\u201330 (2017)","journal-title":"Dolom. Res. Notes Approx."},{"key":"851_CR22","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1007\/s11075-011-9498-x","volume":"59","author":"M Fuhry","year":"2012","unstructured":"Fuhry, M., Reichel, L.: A new Tikhonov regularization method. Numer. Algorithms 59, 433\u2013445 (2012)","journal-title":"Numer. Algorithms"},{"key":"851_CR23","doi-asserted-by":"publisher","first-page":"1089","DOI":"10.1007\/s10915-015-0120-6","volume":"67","author":"A Heryudono","year":"2016","unstructured":"Heryudono, A., Larsson, E., Ramage, A., Von Sydow, L.: Preconditioning for radial basis function partition of unity methods. J. Sci. Comput. 67, 1089\u20131109 (2016)","journal-title":"J. Sci. Comput."},{"key":"851_CR24","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/S0096-3003(99)00255-6","volume":"119","author":"YC Hon","year":"2001","unstructured":"Hon, Y.C., Schaback, R.: On unsymmetric collocation by radial basis functions. Appl. Math. Comput. 119, 177\u2013186 (2001)","journal-title":"Appl. Math. Comput."},{"key":"851_CR25","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1023\/A:1022253303343","volume":"32","author":"YC Hon","year":"2003","unstructured":"Hon, Y.C., Schaback, R., Zhou, X.: An adaptive greedy algorithm for solving large RBF collocation problems. Numer. Algorithms 32, 13\u201325 (2003)","journal-title":"Numer. Algorithms"},{"key":"851_CR26","unstructured":"Kansa, E.J.: Application of Hardy\u2019s multiquadric interpolation to hydrodynamics. In: Proceedings of 1986 Simulation Conference, vol. l4, pp. 111\u2013117 (1986)"},{"key":"851_CR27","doi-asserted-by":"publisher","first-page":"84","DOI":"10.1063\/1.4961148","volume":"145","author":"M Kowalewski","year":"2016","unstructured":"Kowalewski, M., Larsson, E., Heryudono, A.: An adaptive interpolation scheme for molecular potential energy surfaces. J. Chem. Phys. 145, 84\u2013104 (2016)","journal-title":"J. Chem. Phys."},{"key":"851_CR28","doi-asserted-by":"publisher","first-page":"891","DOI":"10.1016\/S0898-1221(03)90151-9","volume":"46","author":"E Larsson","year":"2003","unstructured":"Larsson, E., Fornberg, B.: A numerical study of some radial basis function based solution methods for elliptic PDEs. Comput. Math. Appl. 46, 891\u2013902 (2003)","journal-title":"Comput. Math. Appl."},{"key":"851_CR29","doi-asserted-by":"publisher","first-page":"A2096","DOI":"10.1137\/120899108","volume":"35","author":"E Larsson","year":"2013","unstructured":"Larsson, E., Lehto, E., Heryudono, A., Fornberg, B.: Stable computation of differentiation matrices and scattered node stencils based on Gaussian radial basis functions. SIAM J. Sci. Comput. 35, A2096\u2013A2119 (2013)","journal-title":"SIAM J. Sci. Comput."},{"key":"851_CR30","doi-asserted-by":"publisher","first-page":"A2538","DOI":"10.1137\/17M1118087","volume":"39","author":"E Larsson","year":"2017","unstructured":"Larsson, E., Shcherbakov, V., Heryudono, A.: A least squares radial basis function partition of unity method for solving PDEs. SIAM J. Sci. Comput. 39, A2538\u2013A2563 (2017)","journal-title":"SIAM J. Sci. Comput."},{"key":"851_CR31","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10444-004-1809-5","volume":"23","author":"L Ling","year":"2005","unstructured":"Ling, L., Kansa, E.J.: A least-squares preconditioner for radial basis functions collocation methods. Adv. Comput. Math. 23, 31\u201354 (2005)","journal-title":"Adv. Comput. Math."},{"key":"851_CR32","doi-asserted-by":"publisher","first-page":"247","DOI":"10.1016\/j.enganabound.2005.08.008","volume":"30","author":"L Ling","year":"2006","unstructured":"Ling, L., Opfer, R., Schaback, R.: Results on meshless collocation techniques. Eng. Anal. Bound. Elem. 30, 247\u2013253 (2006)","journal-title":"Eng. Anal. Bound. Elem."},{"key":"851_CR33","unstructured":"Melenk, J.M., Babu $$\\check{\\text{s}}$$ s \u02c7 ka, I., Basic theory and applications: The partition of unity finite element method. Comput. Methods Appl. Mech. Eng. 139, 289\u2013314 (1996)"},{"key":"851_CR34","doi-asserted-by":"crossref","unstructured":"Oanh, D.T., Davydov, O., Phu, H.X: Adaptive RBF-FD method for elliptic problems with point singularities in 2D. Preprint (2016)","DOI":"10.1016\/j.amc.2017.06.006"},{"key":"851_CR35","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1016\/j.cam.2011.05.021","volume":"236","author":"M Pazouki","year":"2011","unstructured":"Pazouki, M., Schaback, R.: Bases for kernel-based spaces. J. Comput. Appl. Math. 236, 575\u2013588 (2011)","journal-title":"J. Comput. Appl. Math."},{"key":"851_CR36","doi-asserted-by":"publisher","unstructured":"Romani, L., Rossini, M., Schenone, D.: Edge detection methods based on RBF interpolation. J. Comput. Appl. Math. (2018). https:\/\/doi.org\/10.1016\/j.cam.2018.08.006","DOI":"10.1016\/j.cam.2018.08.006"},{"key":"851_CR37","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1186\/s13104-017-3107-6","volume":"11","author":"M Rossini","year":"2018","unstructured":"Rossini, M.: Interpolating functions with gradient discontinuities via variably scaled kernels. Dolom. Res. Notes Approx. 11, 3\u201314 (2018)","journal-title":"Dolom. Res. Notes Approx."},{"key":"851_CR38","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/s10915-014-9935-9","volume":"64","author":"A Safdari-Vaighani","year":"2015","unstructured":"Safdari-Vaighani, A., Heryudono, A., Larsson, E.: A radial basis function partition of unity collocation method for convection-diffusion equations arising in financial applications. J. Sci. Comput. 64, 341\u2013367 (2015)","journal-title":"J. Sci. Comput."},{"key":"851_CR39","unstructured":"Santin, G., Haasdonk, B.: Convergence rate of the data-independent $$P$$ P -greedy algorithm in kernel-based approximation. In: Dolomites Research Notes on Approximation, vol. 10, pp. 68\u201378. Special issue (2017)"},{"key":"851_CR40","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5334\/jors.131","volume":"5","author":"SA Sarra","year":"2017","unstructured":"Sarra, S.A.: The Matlab radial basis function toolbox. J. Open Res. Softw. 5, 1\u201310 (2017)","journal-title":"J. Open Res. Softw."},{"key":"851_CR41","doi-asserted-by":"crossref","unstructured":"Sarra, S.A., Bay, Y.: A rational radial basis function method for accurately resolving discontinuities and steep gradients. Preprint (2017)","DOI":"10.1016\/j.apnum.2018.04.001"},{"key":"851_CR42","doi-asserted-by":"crossref","unstructured":"Shepard, D.: A two-dimensional interpolation function for irregularly spaced data. In: Proceedings of 23-rd National Conference, Brandon\/Systems Press, Princeton, pp. 517\u2013524 (1968)","DOI":"10.1145\/800186.810616"},{"key":"851_CR43","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1137\/050633366","volume":"45","author":"R Schaback","year":"2007","unstructured":"Schaback, R.: Convergence of unsymmetric kernel-based meshless collocation methods. SIAM J. Numer. Anal. 45, 333\u2013351 (2007)","journal-title":"SIAM J. Numer. Anal."},{"key":"851_CR44","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1016\/j.camwa.2015.11.007","volume":"71","author":"V Shcherbakov","year":"2016","unstructured":"Shcherbakov, V., Larsson, E.: Radial basis function partition of unity methods for pricing vanilla basket options. Comput. Math. Appl. 71, 185\u2013200 (2016)","journal-title":"Comput. Math. Appl."},{"key":"851_CR45","first-page":"1035","volume":"4","author":"AN Tikhonov","year":"1963","unstructured":"Tikhonov, A.N.: Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035\u20131038 (1963)","journal-title":"Sov. Math. Dokl."},{"key":"851_CR46","first-page":"473","volume-title":"Approximation Theory X: Wavelets, Splines, and Applications","author":"H Wendland","year":"2002","unstructured":"Wendland, H.: Fast evaluation of radial basis functions: methods based on partition of unity. In: Chui, C.K. (ed.) Approximation Theory X: Wavelets, Splines, and Applications, pp. 473\u2013483. Vanderbilt University Press, Nashville (2002)"},{"key":"851_CR47","volume-title":"Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics","author":"H Wendland","year":"2005","unstructured":"Wendland, H.: Scattered Data Approximation, Cambridge Monographs on Applied and Computational Mathematics, vol. 17. Cambridge University Press, Cambridge (2005)"}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0851-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10915-018-0851-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-018-0851-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,9,3]],"date-time":"2022-09-03T23:11:26Z","timestamp":1662246686000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10915-018-0851-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,10,13]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,4]]}},"alternative-id":["851"],"URL":"https:\/\/doi.org\/10.1007\/s10915-018-0851-2","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2018,10,13]]},"assertion":[{"value":"16 July 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 October 2018","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 October 2018","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 October 2018","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}