Stable high-order randomized cubature formulae in arbitrary dimension

TL;DR

This paper introduces randomized high-order cubature formulas for numerical integration in arbitrary dimensions, providing stability, error estimates, and spectral convergence properties, with potential variance reduction benefits.

Contribution

It develops and analyzes new randomized cubature formulas that are exact on finite-dimensional subspaces, with proven stability, error bounds, and positivity of weights under certain conditions.

Findings

Error decays as √(n/m) times best approximation error

Expected error scales as √(1/m) times best approximation error

Weights are positive with high probability under quadratic m/n proportionality

Abstract

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.