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 accepted method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for...
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| Format: | Journal article |
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2012
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| _version_ | 1826277073364189184 |
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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 accepted method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial differential 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 | 2024-03-06T23:23:25Z |
| format | Journal article |
| id | oxford-uuid:698c1230-9719-455f-a486-80eb9ed80163 |
| institution | University of Oxford |
| last_indexed | 2024-03-06T23:23:25Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:698c1230-9719-455f-a486-80eb9ed801632022-03-26T18:51:44ZCompactly supported radial basis functions: how and why?Journal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:698c1230-9719-455f-a486-80eb9ed80163Mathematical Institute - ePrints2012Zhu, SThe use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an accepted method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial differential 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 |