Regularisation for the approximation of functions by mollified discretisation methods

TL;DR

The paper analyzes mollified discretisation methods for reconstructing a smooth function on a bounded domain from $n$ noisy samples, introducing discretisation order $s_a$ and regularisation order $s_r$, and showing a regime where skipping regularisation can outperform standard regularisation with potential gains in the error rate up to $n^{(s_a-s_r)/(2s_r+d)}$. It derives upper and lower bounds on reconstruction error, derives the optimal regularisation parameter $\beta^*(\sigma,n) \sim \sigma^{2/(2s_r+d)} n^{-1/(2s_r+d)}$, and identifies a threshold $\lambda_M$ in the noise scaling that delineates regimes where regularisation helps or hurts, particularly when $s_a > s_r$. The methodology relies on a linear estimator $P_n y_\sigma$ and a mollified estimator $K_\beta * P_n y_\sigma$ with a moment-reproducing kernel, and the results hold in $L^2$-norm with Sobolev-smoothness classes as the function space. Numerical experiments in $d=1$ using $\mathbb{P}_1$ and $\mathbb{P}_2$ finite elements corroborate the predicted rates and illustrate the trade-offs across regularisation and discretisation orders. These findings provide practical guidance on when to apply standard regularisation versus forgoing it in mollified discretisation schemes under various noise regimes.

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 $σ$. 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.

Figures (4)

• Figure 1: For $0 \leqslant \lambda \leqslant \lambda_M$, plot of the order of convergence of $e_{\mathrm{noreg}}(n)$ and $e_{\mathrm{reg}}(n)$ towards $0$, as given by Theorem \ref{['interesting_regime']}. In red, the function is $\lambda \mapsto \frac{1}{d}\min(\lambda, s_a)$, and in blue $\lambda \mapsto \frac{2\lambda+d}{2 s_r +d} \frac{s_r}{d}$. Parameters for this figure are chosen to be $d=2$, $s_a=3$, $s_r =2$, for which $\lambda_M = 3.5$.
• Figure 2: No regularisation: plot of $\lambda \mapsto \gamma_{\text{noreg}}(\lambda)$ defined by \ref{['error_noreg']}. The left panel shows the case of $\mathbb{P}_1$ finite elements, the panel right that of $\mathbb{P}_2$ finite elements. In both cases, the theoretical curve $\lambda \mapsto \min(\lambda, s_a)$ is plotted in orange against the numerically obtained curve, in magenta.
• Figure 3: Regularisation with kernel $K$: plot of $\lambda \mapsto \gamma_{\text{noreg}}(\lambda)$ defined by \ref{['error_noreg']}, with regularisation through the kernel $K$. The left panel shows the case of $\mathbb{P}_1$ finite elements ($\lambda_M = 2.5$), the panel right that of $\mathbb{P}_2$ finite elements ($\lambda_M = 4$). In both cases, the theoretical curves without regularisation $\lambda \mapsto \min(\lambda, s_a)$ and with regularisation $\lambda \mapsto \frac{1}{3}(2\lambda +1)$ are plotted in orange and blue, respectively. The numerically obtained curve (with regularisation through $K$) is plotted in green.
• Figure 4: Regularisation with kernel $H$: plot of $\lambda \mapsto \gamma_{\text{reg}}(\lambda)$ defined by \ref{['error_noreg']}, with regularisation through the kernel $H$. The left panel shows the case of $\mathbb{P}_1$ finite elements ($\lambda_M = 2$), the panel right that of $\mathbb{P}_2$ finite elements ($\lambda_M = 3.25$). In both cases, the theoretical curves without regularisation $\lambda \mapsto \min(\lambda, s_a)$ and with regularisation $\lambda \mapsto \frac{2}{5} (2\lambda+1)$ are plotted in orange and blue, respectively. The numerically obtained curve (with regularisation through $H$) is plotted in green.

Theorems & Definitions (24)

• Theorem 1
• Theorem 2
• Definition 1
• Remark 1
• Proposition 1
• proof
• Proposition 2
• proof
• Lemma 1
• proof