Stable high-order randomized cubature formulae in arbitrary dimension

We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure $\mu$ defined on a domain $\Gamma \subseteq \mathbb{R}^d$, in any dimension $d$. Each cubature formula is exact on a given finite-dimensional subspace $V_n\subset L^2(\Gamma,\mu)$ of dimension $n$, and uses pointwise evaluations of the integrand function $\phi : \Gamma \to \mathbb{R}$ at $m>n$ independent random points. These points are drawn from a suitable auxiliary probability measure that depends on $V_n$. We show that, up to a logarithmic factor, a linear proportionality between $m$ and $n$ with dimension-independent constant ensures stability of the cubature formula with high probability. We also prove error estimates in probability and in expectation for any $n\geq 1$ and $m>n$, thus covering both preasymptotic and asymptotic regimes. Our analysis shows that the expected cubature error decays as $\sqrt{n/m}$ times the $L(\Gamma, \mu)$-best approximation error of $\phi$ in $V_n$. On the one hand, for fixed $n$ and $m\to \infty$ our cubature formula can be seen as a variance reduction technique for a Monte Carlo estimator, and can lead to enormous variance reduction for smooth integrand functions and subspaces $V_n$ with spectral approximation properties. On the other hand, when we let $n,m\to\infty$, our cubature becomes of high order with spectral convergence. As a further contribution, we analyse also another cubature formula whose expected error decays as $\sqrt{1/m}$ times the $L^2(\Gamma,\mu)$-best approximation error of $\phi$ in $V_n$, which is asymptotically optimal but with constants that can be larger in the preasymptotic regime. Finally we show that, under a more demanding (at least quadratic) proportionality betweeen $m$ and $n$, the weights of the cubature are positive with high probability.