Compactly supported radial basis functions: How and why?
The use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an practical method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods fo...
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| Format: | Report |
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SIAM
2012
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| _version_ | 1826294147801153536 |
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| author | Zhu, S |
| author_facet | Zhu, S |
| author_sort | Zhu, S |
| collection | OXFORD |
| description | The use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an practical method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial dierential equations (PDEs), an emerging method for solving PDEs on surfaces, a novel method for mesh repair and so on. All these applications share one mathematical foundation: high dimensional approximation/interpolation. This paper explains why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space; and gives a brief description on how to construct the most commonly used compactly supported radial basis functions. Without sophisticated mathematics, one can construct a compactly supported (radial) basis function with required smoothness according to procedures described here. Short programs and tables for compactly supported radial basis functions are supplied. |
| first_indexed | 2024-03-07T03:41:09Z |
| format | Report |
| id | oxford-uuid:bdef99c8-71b0-43e0-b8bc-511588612cbd |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:41:09Z |
| publishDate | 2012 |
| publisher | SIAM |
| record_format | dspace |
| spelling | oxford-uuid:bdef99c8-71b0-43e0-b8bc-511588612cbd2022-03-27T05:35:36ZCompactly supported radial basis functions: How and why?Reporthttp://purl.org/coar/resource_type/c_93fcuuid:bdef99c8-71b0-43e0-b8bc-511588612cbdMathematical Institute - ePrintsSIAM2012Zhu, SThe use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an practical method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial dierential equations (PDEs), an emerging method for solving PDEs on surfaces, a novel method for mesh repair and so on. All these applications share one mathematical foundation: high dimensional approximation/interpolation. This paper explains why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space; and gives a brief description on how to construct the most commonly used compactly supported radial basis functions. Without sophisticated mathematics, one can construct a compactly supported (radial) basis function with required smoothness according to procedures described here. Short programs and tables for compactly supported radial basis functions are supplied. |
| spellingShingle | Zhu, S Compactly supported radial basis functions: How and why? |
| title | Compactly supported radial basis functions: How and why? |
| title_full | Compactly supported radial basis functions: How and why? |
| title_fullStr | Compactly supported radial basis functions: How and why? |
| title_full_unstemmed | Compactly supported radial basis functions: How and why? |
| title_short | Compactly supported radial basis functions: How and why? |
| title_sort | compactly supported radial basis functions how and why |
| work_keys_str_mv | AT zhus compactlysupportedradialbasisfunctionshowandwhy |