On the spectral distribution of kernel matrices related to radial basis functions
This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an ort...
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| Format: | Journal article |
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2012
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| _version_ | 1797085170806816768 |
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| author | Wathen, A Zhu, S |
| author_facet | Wathen, A Zhu, S |
| author_sort | Wathen, A |
| collection | OXFORD |
| description | This paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants. |
| first_indexed | 2024-03-07T02:05:11Z |
| format | Journal article |
| id | oxford-uuid:9ec03b0f-02d7-4af5-98f7-21c145f78670 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T02:05:11Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:9ec03b0f-02d7-4af5-98f7-21c145f786702022-03-27T00:52:20ZOn the spectral distribution of kernel matrices related to radial basis functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:9ec03b0f-02d7-4af5-98f7-21c145f78670Mathematical Institute - ePrints2012Wathen, AZhu, SThis paper focuses on the spectral distribution of kernel matrices related to radial basis functions. The asymptotic behaviour of eigenvalues of kernel matrices related to radial basis functions with different smoothness are studied. These results are obtained by estimated the coefficients of an orthogonal expansion of the underlying kernel function. Beside many other results, we prove that there are exactly (k+d−1/d-1) eigenvalues in the same order for analytic separable kernel functions like the Gaussian in Rd. This gives theoretical support for how to choose the diagonal scaling matrix in the RBF-QR method (Fornberg et al, SIAM J. Sci. Comput. (33), 2011) which can stably compute Gaussian radial basis function interpolants. |
| spellingShingle | Wathen, A Zhu, S On the spectral distribution of kernel matrices related to radial basis functions |
| title | On the spectral distribution of kernel matrices related to
radial basis functions |
| title_full | On the spectral distribution of kernel matrices related to
radial basis functions |
| title_fullStr | On the spectral distribution of kernel matrices related to
radial basis functions |
| title_full_unstemmed | On the spectral distribution of kernel matrices related to
radial basis functions |
| title_short | On the spectral distribution of kernel matrices related to
radial basis functions |
| title_sort | on the spectral distribution of kernel matrices related to radial basis functions |
| work_keys_str_mv | AT wathena onthespectraldistributionofkernelmatricesrelatedtoradialbasisfunctions AT zhus onthespectraldistributionofkernelmatricesrelatedtoradialbasisfunctions |