Randomized Projection Operators onto Piecewise Polynomial Spaces

TL;DR

This work develops randomized projection operators onto piecewise polynomial spaces by Monte Carlo sampling and discrete least-squares, achieving (almost) optimal $L^2$ and $H^{-1}$ approximation properties and enabling computable FE discretizations for PDEs with rough right-hand sides. The authors introduce a lowest-order operator $\hat{\Pi}_0$ that is unbiased and diagonalizes the $H^{-1}$ norm, and higher-order operators $\tilde{\Pi}_k$ with a correction to ensure unbiased first moments via $\hat{\tilde{\Pi}}_k$, with rigorous $L^2$ and $H^{-1}$ error bounds tied to best-approximation and data-oscillation. They show these operators act as smoothers for rough data, yielding expected convergence rates in Poisson-type problems, and provide high-probability bounds for the sampling error. Numerical experiments corroborate the theory, demonstrating robust performance even with few samples per element and highlighting advantages of higher-order randomized smoothing in standard FE settings.

Abstract

We introduce computable projection operators onto piecewise polynomial spaces, defined via sampling and discrete least-squares polynomial approximations. The resulting mappings exhibit (almost) optimal approximation properties in $L^2$ and $H^{-1}$. As smoothers for incomplete or rough data, they yield computable finite element discretizations with optimal convergence rates.

Figures (2)

• Figure 1: Convergence history plot of the relative errors with respect to $\lVert \nabla \bigcdot \rVert_{L^2(\mathcal{D})}$ (solid line) and $\lVert \bigcdot \rVert_{L^2(\mathcal{D})}$ (dotted line) for various approximations of the right-hand side in the experiment of Section \ref{['subsec:Oscillation']}. The dash-dotted line illustrates the slope $\textup{ndof}^{-1/2}$. On the left we evaluated $f$ using the standard NGSolve settings, $\Pi_0 f$ using midpoint quadrature, and $\hat{\Pi}_0 f$ using $N=1$ samples per element. On the right, we increased the standard NGSolve polynomial order of accuracy by 10, approximated $\Pi_0 f$ by a routine at least exact for polynomials of degree 10, and used $N= 20$ samples per element for the evaluation of $\hat{\Pi}_0 f$.
• Figure 2: Convergence history of the relative errors with respect to $\lVert \nabla \bigcdot \rVert_{L^2(\mathcal{D})}$ (left) and $\lVert \bigcdot \rVert_{L^2(\mathcal{D})}$ (right) for various approximations of the right-hand side in the experiment of Section \ref{['subsec:Waterfall']}. We display three different realizations of $\hat{\tilde{\Pi}}_1 f$. The dash-dotted line illustrates the rate $\mathcal{O}(\textup{ndof}^{-1/2})$, the dotted line illustrates $\mathcal{O}(\textup{ndof}^{-1})$.

Theorems & Definitions (36)

• lemma 1: Unbiased approximation of $\Pi_0$
• proof
• lemma 2: Approximation properties in $L^2(K)$
• proof
• lemma 3: $H^{-1}(\mathcal{D})$ norm of locally supported functions
• proof
• lemma 4: Expected error for cell averages
• proof
• remark 1: Diagonalization
• theorem 1: Approximation properties of $\hat{\Pi}_0$