Title: Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss-Hermite rule
Abstract: Computing integrals with respect to the Gaussian measure is a common task across many disciplines. In this talk, I will discuss optimal numerical integration rules for this task. To formalise optimality, we fix a target function class and consider the L2-Sobolev space of dominating mixed smoothness.
In one dimension, the trapezoidal rule is asymptotically optimal in this space, up to a logarithmic factor. By contrast, Gauss–Hermite quadrature converges at only half this rate. A similar pattern holds for analytic functions. We show that these results extend to higher dimensions: several quasi-Monte Carlo methods achieve the optimal rate, whereas the sparse-grid Gauss–Hermite rule based on attains only half the optimal rate.
This talk is based on the following joint works:[1] Kazashi, Suzuki, and Goda (arXiv:2509.18712, 2025)
[2] Goda, Kazashi, and Tanaka. How Sharp Are Error Bounds? –Lower Bounds on Quadrature Worst-Case Errors for Analytic Functions–. SIAM Journal on Numerical Analysis (2024)
[3] Kazashi, Suzuki, and Goda. Suboptimality of Gauss–Hermite quadrature and optimality of the trapezoidal rule for functions with finite smoothness. SIAM Journal on Numerical Analysis (2023)