Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866915933802463232 |
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| author | Addy, Elliot J. Teckentrup, Aretha L. |
| author_facet | Addy, Elliot J. Teckentrup, Aretha L. |
| contents | Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Matérn kernels $Φ_{\boldsymbolν,\boldsymbolλ}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^dϕ_{ν_j,λ_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $ν_j$ and the lengthscale $λ_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $ν_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic $λ_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10872 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation Addy, Elliot J. Teckentrup, Aretha L. Numerical Analysis 65D40 65D05 65D15 (Primary), 46E22 (Secondary) Sparse grids are popular tools for high-dimensional function approximation. In this work, we study the use of sparse grids for interpolation using separable Matérn kernels $Φ_{\boldsymbolν,\boldsymbolλ}(\mathbf{x},\mathbf{x}')=\prod_{j=1}^dϕ_{ν_j,λ_j}(x_j,x_j')$, with a particular focus on the anisotropic setting where the regularity $ν_j$ and the lengthscale $λ_j$ vary with dimension $j$. We combine the construction of anisotropic sparse grids, which exploit anisotropic $ν_j$ to improve convergence rates in smooth dimensions, with the construction of lengthscale-informed sparse grids, which diminish the error contribution of less varying dimensions using anisotropic $λ_j$. We provide theory and numerical experiments to showcase the benefits on asymptotic and pre-asymptotic error behaviour of sparse grid kernel interpolation. |
| title | Asymptotic and pre-asymptotic convergence of sparse grids for anisotropic kernel interpolation |
| topic | Numerical Analysis 65D40 65D05 65D15 (Primary), 46E22 (Secondary) |
| url | https://arxiv.org/abs/2604.10872 |