2018/12/19 by Giovanni Migliorati, Fabio Nobile, Migliorati, Giovanni +1 · 1 citation
Decision Sciences · Mathematics · #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Probabilistic and Robust Engineering Design #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1812.07761
openalex publication_date 2018/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose and analyse randomized cubature formulae for the numerical integration of functions with respect to a given probability measure μ defined on a domain Γ⊆ ℝd, in any dimension d. Each cubature formula is exact on a given finite-dimensional subspace Vn⊂ L2(Γ,μ) of dimension n, and uses pointwise evaluations of the integrand function ϕ: Γ→ ℝ at m>n independent random points. These points are drawn from a suitable auxiliary probability measure that depends on Vn. 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≥ 1 and m>n, thus covering both preasymptotic and asymptotic regimes. Our analysis shows that the expected cubature error decays as √(n/m) times the L(Γ, μ)-best approximation error of ϕ in Vn. On the one hand, for fixed n and m→ ∞ 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 Vn with spectral approximation properties. On the other hand, when we let n,m→∞, our cubature becomes of high order with spectral convergence. As a further contribution, we analyse also another cubature formula whose expected error decays as √(1/m) times the L2(Γ,μ)-best approximation error of ϕ in Vn, 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.