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 xᵢ of the corresponding mesh. The accuracy is n-sₐ/d in L²-norm, where sₐ 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 sᵣ (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 sₐ and sᵣ 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 sₐ > sᵣ, statistical smoothing is not necessarily the best option and { not regularising} is more beneficial than { statistical regularising}. We precisely quantify this phenomenon and show that the gain can achieve a multiplicative order n(sₐ-sᵣ)/(2sᵣ+d). We illustrate our estimates by numerical experiments conducted in dimension $d=1$ with P₁ and P₂ finite elements.