Making conditionally negative definite radial basis function interpolation well-conditioned by adding cardinal basis functions
A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1+(∊r)2 (MQ) and log(1+(∊r)2) (LOG). The condition number of the...
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| Published in | Ain Shams Engineering Journal Vol. 9; no. 4; pp. 2587 - 2598 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Elsevier B.V
01.12.2018
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| ISSN | 2090-4479 |
| DOI | 10.1016/j.asej.2017.03.013 |
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| Abstract | A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1+(∊r)2 (MQ) and log(1+(∊r)2) (LOG). The condition number of the interpolation matrix arising from this basis is of O(N), where N is the number of center nodes. This order is independent of shape parameter and therefore applying this basis functions would recover the ill–posed linear system associated with the order 1 conditionally negative definite RBFs interpolation. |
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| AbstractList | A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1+(∊r)2 (MQ) and log(1+(∊r)2) (LOG). The condition number of the interpolation matrix arising from this basis is of O(N), where N is the number of center nodes. This order is independent of shape parameter and therefore applying this basis functions would recover the ill–posed linear system associated with the order 1 conditionally negative definite RBFs interpolation. Keywords: Radial basis functions, Cardinal functions, Multiquadrics (MQ), AMS subject classifications: 65M20, 65M06 A class of basis functions so called well-conditioned RBF (WRBFs) has been introduced. This basis has been manipulated by adding cardinal functions to the conditionally negative definite RBFs of order 1, such as Multiquadric functions 1+(∊r)2 (MQ) and log(1+(∊r)2) (LOG). The condition number of the interpolation matrix arising from this basis is of O(N), where N is the number of center nodes. This order is independent of shape parameter and therefore applying this basis functions would recover the ill–posed linear system associated with the order 1 conditionally negative definite RBFs interpolation. |
| Author | Hatam, Ali Kazem, Saeed Chadwick, Edmund A. |
| Author_xml | – sequence: 1 givenname: Saeed surname: Kazem fullname: Kazem, Saeed email: saeedkazem@aut.ac.ir organization: Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914 Tehran, Iran – sequence: 2 givenname: Edmund A. surname: Chadwick fullname: Chadwick, Edmund A. email: e.a.chadwick@salford.ac.uk organization: School of Computing, Science and Engineering, University of Salford, Salford M5 4WT, UK – sequence: 3 givenname: Ali surname: Hatam fullname: Hatam, Ali email: alihatam@aut.ac.ir organization: Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914 Tehran, Iran |
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| Cites_doi | 10.1080/00207160.2012.741227 10.1016/0898-1221(90)90270-T 10.1016/j.jcp.2015.05.032 10.1007/s00211-005-0637-y 10.1007/s00365-004-0585-2 10.1515/zna-2015-0100 10.1016/0898-1221(90)90271-K 10.1007/s10915-015-0120-6 10.1137/120899108 10.1137/0912070 10.1016/j.enganabound.2014.04.019 10.2307/2002065 10.1080/00207160.2015.1127359 10.1016/j.cam.2011.05.021 10.1140/epjp/i2015-15047-y 10.1137/S1064827599361771 10.1137/060671991 10.1137/09076756X 10.1016/0021-9045(91)90087-Q 10.1016/j.camwa.2014.12.003 10.1016/j.camwa.2015.11.013 10.1016/j.enganabound.2016.03.003 10.1016/j.jcp.2016.02.078 10.1016/j.camwa.2007.01.028 10.1016/j.camwa.2012.10.014 10.1016/j.jcp.2016.02.030 10.1016/j.camwa.2003.08.010 10.1016/j.enganabound.2010.05.011 10.1016/0021-9045(92)90058-V 10.1016/j.compfluid.2016.01.009 |
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| Title | Making conditionally negative definite radial basis function interpolation well-conditioned by adding cardinal basis functions |
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