Regularisation for the approximation of functions by mollified discretisation methods
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arXiv
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| Format: | Preprint |
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2024
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| author | Pouchol, Camille Hoffmann, Marc |
| author_facet | Pouchol, Camille Hoffmann, Marc |
| contents | Some prominent discretisation methods such as finite elements provide a way to approximate a function of $d$ variables from $n$ values it takes on the nodes $x_i$ of the corresponding mesh. The accuracy is $n^{-s_a/d}$ in $L^2$-norm, where $s_a$ is the order of the underlying method. When the data are measured or computed with systematical experimental noise, some statistical regularisation might be desirable, with a smoothing method of order $s_r$ (like the number of vanishing moments of a kernel). This idea is behind the use of some regularised discretisation methods, whose approximation properties are the subject of this paper. We decipher the interplay of $s_a$ and $s_r$ for reconstructing a smooth function on regular bounded domains from $n$ measurements with noise of order $σ$. We establish that for certain regimes with small noise $σ$ depending on $n$, when $s_a > s_r$, statistical smoothing is not necessarily the best option and {\it not regularising} is more beneficial than {\it statistical regularising}. We precisely quantify this phenomenon and show that the gain can achieve a multiplicative order $n^{(s_a-s_r)/(2s_r+d)}$. We illustrate our estimates by numerical experiments conducted in dimension $d=1$ with $\mathbb P_1$ and $\mathbb P_2$ finite elements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_13263 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Regularisation for the approximation of functions by mollified discretisation methods Pouchol, Camille Hoffmann, Marc Numerical Analysis Statistics Theory Some prominent discretisation methods such as finite elements provide a way to approximate a function of $d$ variables from $n$ values it takes on the nodes $x_i$ of the corresponding mesh. The accuracy is $n^{-s_a/d}$ in $L^2$-norm, where $s_a$ is the order of the underlying method. When the data are measured or computed with systematical experimental noise, some statistical regularisation might be desirable, with a smoothing method of order $s_r$ (like the number of vanishing moments of a kernel). This idea is behind the use of some regularised discretisation methods, whose approximation properties are the subject of this paper. We decipher the interplay of $s_a$ and $s_r$ for reconstructing a smooth function on regular bounded domains from $n$ measurements with noise of order $σ$. We establish that for certain regimes with small noise $σ$ depending on $n$, when $s_a > s_r$, statistical smoothing is not necessarily the best option and {\it not regularising} is more beneficial than {\it statistical regularising}. We precisely quantify this phenomenon and show that the gain can achieve a multiplicative order $n^{(s_a-s_r)/(2s_r+d)}$. We illustrate our estimates by numerical experiments conducted in dimension $d=1$ with $\mathbb P_1$ and $\mathbb P_2$ finite elements. |
| title | Regularisation for the approximation of functions by mollified discretisation methods |
| topic | Numerical Analysis Statistics Theory |
| url | https://arxiv.org/abs/2407.13263 |