Regularisation for the approximation of functions by mollified discretisation methods
<|reference_start|>Regularisation for the approximation of functions by mollified discretisation methods: 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 $\sigma$. We establish that for certain regimes with small noise $\sigma$ 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.<|reference_end|>
@article{pouchol2024regularisation,
    title={Regularisation for the approximation of functions by mollified
  discretisation methods},
    author={Camille Pouchol (MAP5), Marc Hoffmann (CEREMADE, IUF)},
    journal={arXiv preprint arXiv:2407.13263},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.13263},
    primaryClass={math.NA cs.NA math.ST stat.TH}
}