A multi-level solution of scalar and vectorial interpolation problems based on iterated elliptic operators
One of the most popular methods to solve scattered data interpolation problems is the method of radial basis functions. However, it leads to a linear system with large, dense and ill‐conditioned matrix, which causes severe numerical difficulties. Here a relatively new approach is presented which com...
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| Published in | Proceedings in applied mathematics and mechanics Vol. 3; no. 1; pp. 535 - 536 |
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| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Berlin
WILEY-VCH Verlag
01.12.2003
WILEY‐VCH Verlag |
| Online Access | Get full text |
| ISSN | 1617-7061 1617-7061 |
| DOI | 10.1002/pamm.200310537 |
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| Abstract | One of the most popular methods to solve scattered data interpolation problems is the method of radial basis functions. However, it leads to a linear system with large, dense and ill‐conditioned matrix, which causes severe numerical difficulties. Here a relatively new approach is presented which completely avoids the large and dense matrices and significantly reduces the computational cost. The interpolation problem is converted to a higher order (typically to an iterated elliptic) partial differential equation supplied with the interpolation condition as special boundary conditions. This new problem is well‐posed in a suitable Sobolev space and can be solved by using robust, multi‐level methods. The approach is generalized also to vectorial interpolation problems, where the interpolation vector field is assumed to satisfy some additional conditions e.g. it is irrotational or divergence‐free. |
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| AbstractList | One of the most popular methods to solve scattered data interpolation problems is the method of radial basis functions. However, it leads to a linear system with large, dense and ill‐conditioned matrix, which causes severe numerical difficulties. Here a relatively new approach is presented which completely avoids the large and dense matrices and significantly reduces the computational cost. The interpolation problem is converted to a higher order (typically to an iterated elliptic) partial differential equation supplied with the interpolation condition as special boundary conditions. This new problem is well‐posed in a suitable Sobolev space and can be solved by using robust, multi‐level methods. The approach is generalized also to vectorial interpolation problems, where the interpolation vector field is assumed to satisfy some additional conditions e.g. it is irrotational or divergence‐free. |
| Author | Gáspár, Csaba |
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| Cites_doi | 10.1016/0898-1221(90)90270-T 10.1016/S0955-7997(00)00036-9 10.4203/ccp.39.1.4 10.1007/978-3-642-56103-0_11 10.2307/2007474 |
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| References | Franke, R.: Scattered Data Interpolation: Test of Some Methods. Mathematics of Computation, 38 (157) (1982), 181-200. Gáspár, C.: Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method. Engineering Analysis with Boundary Elements 24 (7-8) (2000), 559-573. Golberg, M.A.; Chen, C. S.: A bibliography on radial basis function approximation. Boundary Element Communications, 7 (4) (1996), 155-163. Gáspár, C.: Fast multi-level meshless methods based on the implicit use of radial basis functions. Lecture Notes in Computational Science and Engineering 26 (2002), 143-160. Kansa, E.J.: Multiquadrics - a Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics I-II. Computers Math. Applic. 19 (8/9) (1990), 127-161. 2002; 26 2000; 24 1982; 38 1990; 19 1996; 7 e_1_2_1_6_2 e_1_2_1_7_2 e_1_2_1_5_2 e_1_2_1_2_2 e_1_2_1_3_2 Golberg M.A. (e_1_2_1_4_2) 1996; 7 |
| References_xml | – reference: Golberg, M.A.; Chen, C. S.: A bibliography on radial basis function approximation. Boundary Element Communications, 7 (4) (1996), 155-163. – reference: Gáspár, C.: Multi-level biharmonic and bi-Helmholtz interpolation with application to the boundary element method. Engineering Analysis with Boundary Elements 24 (7-8) (2000), 559-573. – reference: Kansa, E.J.: Multiquadrics - a Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics I-II. Computers Math. Applic. 19 (8/9) (1990), 127-161. – reference: Franke, R.: Scattered Data Interpolation: Test of Some Methods. Mathematics of Computation, 38 (157) (1982), 181-200. – reference: Gáspár, C.: Fast multi-level meshless methods based on the implicit use of radial basis functions. Lecture Notes in Computational Science and Engineering 26 (2002), 143-160. – volume: 19 start-page: 127 issue: 8/9 year: 1990 end-page: 161 article-title: Multiquadrics ‐ a Scattered Data Approximation Scheme with Applications to Computational Fluid Dynamics I‐II publication-title: Computers Math. Applic. – volume: 7 start-page: 155 issue: 4 year: 1996 end-page: 163 article-title: A bibliography on radial basis function approximation publication-title: Boundary Element Communications – volume: 24 start-page: 559 issue: 7‐8 year: 2000 end-page: 573 article-title: Multi‐level biharmonic and bi‐Helmholtz interpolation with application to the boundary element method publication-title: Engineering Analysis with Boundary Elements – volume: 38 start-page: 181 issue: 157 year: 1982 end-page: 200 article-title: Scattered Data Interpolation: Test of Some Methods publication-title: Mathematics of Computation – volume: 26 start-page: 143 year: 2002 end-page: 160 article-title: Fast multi‐level meshless methods based on the implicit use of radial basis functions publication-title: Lecture Notes in Computational Science and Engineering – ident: e_1_2_1_3_2 doi: 10.1016/0898-1221(90)90270-T – ident: e_1_2_1_5_2 doi: 10.1016/S0955-7997(00)00036-9 – ident: e_1_2_1_7_2 doi: 10.4203/ccp.39.1.4 – ident: e_1_2_1_6_2 doi: 10.1007/978-3-642-56103-0_11 – ident: e_1_2_1_2_2 doi: 10.2307/2007474 – volume: 7 start-page: 155 issue: 4 year: 1996 ident: e_1_2_1_4_2 article-title: A bibliography on radial basis function approximation publication-title: Boundary Element Communications |
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| Title | A multi-level solution of scalar and vectorial interpolation problems based on iterated elliptic operators |
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