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 μ defined on domain Γ⊆Rd, in any dimension d. Each formula is exact finite-dimensional subspace Vn⊂L2(Γ,μ) n, uses pointwise evaluations integrand function ϕ:Γ→R at m>n independent random points. These points are drawn from suitable auxiliary that depends Vn. show that, up logarithmic factor, linear proportionality between m n dimension-independent constant ensures stability high probability. also prove error estimates expectation n≥1 m>n, thus covering both preasymptotic asymptotic regimes. Our analysis shows expected decays as n/m times L2(Γ,μ)-best approximation ϕ On one hand, fixed m→∞ our can be seen variance reduction technique Monte Carlo estimator, lead enormous smooth subspaces Vn spectral properties. other when n,m→∞, becomes order convergence. As further contribution, we another whose 1/m Vn, but constants larger regime. Finally under more demanding (at least quadratic) all weights strictly positive an example application, discuss case where Γ has structure Cartesian product, product contains algebraic multivariate polynomials.