Regularisation for the approximation of functions by mollified discretisation methods

TL;DR

This paper analyzes how regularisation affects the accuracy of discretisation methods like finite elements when approximating functions from noisy data, revealing regimes where regularisation may be unnecessary or even detrimental.

Contribution

It provides a detailed theoretical analysis of the interplay between approximation order and regularisation, quantifying when regularisation improves or worsens reconstruction accuracy.

Findings

Regularisation can be unnecessary or harmful when the approximation order exceeds the regularisation order.

In certain regimes, non-regularised methods outperform regularised ones, especially with small noise.

Theoretical results are supported by numerical experiments in one dimension using finite element methods.

Abstract

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.