On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs
For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the...
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| Veröffentlicht in: | Engineering analysis with boundary elements 2010-09, Vol.34 (9), p.802-809 |
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| creator | Huang, C.-S. Yen, H.-D. Cheng, A.H.-D. |
| description | For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value. |
| doi_str_mv | 10.1016/j.enganabound.2010.03.002 |
| format | Article |
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In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.</description><identifier>ISSN: 0955-7997</identifier><identifier>EISSN: 1873-197X</identifier><identifier>DOI: 10.1016/j.enganabound.2010.03.002</identifier><language>eng</language><publisher>Kidlington: Elsevier Ltd</publisher><subject>Arbitrary precision computation ; Boundary element method ; Constraining ; Error estimate ; Exact sciences and technology ; Flattening ; Increasingly flat radial basis function ; Interpolation ; Mathematical analysis ; Mathematical models ; Mathematics ; Meshless method ; Methods of scientific computing (including symbolic computation, algebraic computation) ; Multiquadric collocation method ; Numerical analysis. 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In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.</description><subject>Arbitrary precision computation</subject><subject>Boundary element method</subject><subject>Constraining</subject><subject>Error estimate</subject><subject>Exact sciences and technology</subject><subject>Flattening</subject><subject>Increasingly flat radial basis function</subject><subject>Interpolation</subject><subject>Mathematical analysis</subject><subject>Mathematical models</subject><subject>Mathematics</subject><subject>Meshless method</subject><subject>Methods of scientific computing (including symbolic computation, algebraic computation)</subject><subject>Multiquadric collocation method</subject><subject>Numerical analysis. Scientific computation</subject><subject>Optimization</subject><subject>Partial differential equations</subject><subject>Radial basis function</subject><subject>Sciences and techniques of general use</subject><issn>0955-7997</issn><issn>1873-197X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2010</creationdate><recordtype>article</recordtype><recordid>eNqNkE1LJDEQhsOisOO4_yF7WPDSYz76Izkus7orCHpwYW8hnVQ0QyZpk-4F_73REfEoFBRUPfVW1YvQd0o2lND-fLeBeK-jHtMS7YaRWid8Qwj7glZUDLyhcvh3hFZEdl0zSDl8RSel7AihnJB-hcJNxPMDYB9NBl18vA9P2AU946yt1wGPtViwW6KZfYpYR4vTNPt9bZUHPQGedNZ7mCFjl_KrVklheYWTwxCCr7jBt78uyik6djoU-PaW1-jv5cXd9k9zffP7avvzujFctHMDQFvZW7ByJNZqJ8RguWzN2HXOuNaNwkhmrZSWMir63omOkU4MTIzgRqn5Gp0ddKecHhcos9r7YuopOkJaiqL9QDvR8xprJA-oyamUDE5NuT6XnxQl6sVhtVMfHFYvDivCVXW4zv54W6OL0cFlHY0v7wKMSc5aISq3PXBQf_7vIatiPEQD1mcws7LJf2LbM88cmbo</recordid><startdate>20100901</startdate><enddate>20100901</enddate><creator>Huang, C.-S.</creator><creator>Yen, H.-D.</creator><creator>Cheng, A.H.-D.</creator><general>Elsevier Ltd</general><general>Elsevier</general><scope>IQODW</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>7TB</scope><scope>8FD</scope><scope>FR3</scope><scope>JQ2</scope><scope>KR7</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20100901</creationdate><title>On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs</title><author>Huang, C.-S. ; Yen, H.-D. ; Cheng, A.H.-D.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c384t-ee1496ded9b0ddaf887d394cb55fcf4fb8c92dd99d121866f852058728befb9a3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2010</creationdate><topic>Arbitrary precision computation</topic><topic>Boundary element method</topic><topic>Constraining</topic><topic>Error estimate</topic><topic>Exact sciences and technology</topic><topic>Flattening</topic><topic>Increasingly flat radial basis function</topic><topic>Interpolation</topic><topic>Mathematical analysis</topic><topic>Mathematical models</topic><topic>Mathematics</topic><topic>Meshless method</topic><topic>Methods of scientific computing (including symbolic computation, algebraic computation)</topic><topic>Multiquadric collocation method</topic><topic>Numerical analysis. Scientific computation</topic><topic>Optimization</topic><topic>Partial differential equations</topic><topic>Radial basis function</topic><topic>Sciences and techniques of general use</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Huang, C.-S.</creatorcontrib><creatorcontrib>Yen, H.-D.</creatorcontrib><creatorcontrib>Cheng, A.H.-D.</creatorcontrib><collection>Pascal-Francis</collection><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Mechanical & Transportation Engineering Abstracts</collection><collection>Technology Research Database</collection><collection>Engineering Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Civil Engineering Abstracts</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Engineering analysis with boundary elements</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Huang, C.-S.</au><au>Yen, H.-D.</au><au>Cheng, A.H.-D.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs</atitle><jtitle>Engineering analysis with boundary elements</jtitle><date>2010-09-01</date><risdate>2010</risdate><volume>34</volume><issue>9</issue><spage>802</spage><epage>809</epage><pages>802-809</pages><issn>0955-7997</issn><eissn>1873-197X</eissn><abstract>For the interpolation of continuous functions and the solution of partial differential equation (PDE) by radial basis function (RBF) collocation, it has been observed that solution becomes increasingly more accurate as the shape of the RBF is flattened by the adjustment of a shape parameter. In the case of interpolation of continuous functions, it has been proven that in the limit of increasingly flat RBF, the interpolant reduces to Lagrangian polynomials. Does this limiting behavior implies that RBFs can perform no better than Lagrangian polynomials in the interpolation of a function, as well as in the solution of PDE? In this paper, arbitrary precision computation is used to test these and other conjectures. It is found that RBF in fact performs better than polynomials, as the optimal shape parameter exists not in the limit, but at a finite value.</abstract><cop>Kidlington</cop><pub>Elsevier Ltd</pub><doi>10.1016/j.enganabound.2010.03.002</doi><tpages>8</tpages></addata></record> |
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| source | Elsevier ScienceDirect Journals |
| subjects | Arbitrary precision computation Boundary element method Constraining Error estimate Exact sciences and technology Flattening Increasingly flat radial basis function Interpolation Mathematical analysis Mathematical models Mathematics Meshless method Methods of scientific computing (including symbolic computation, algebraic computation) Multiquadric collocation method Numerical analysis. Scientific computation Optimization Partial differential equations Radial basis function Sciences and techniques of general use |
| title | On the increasingly flat radial basis function and optimal shape parameter for the solution of elliptic PDEs |
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