Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces
This addresses the efficiency of numerical integration methods for high-dimensional problems in computational mathematics, identifying limitations in a common approach and offering superior alternatives.
The paper proves that sparse-grid Gauss--Hermite quadrature achieves a suboptimal convergence rate of N^{-α/2} in Gaussian Sobolev spaces, while several quasi-Monte Carlo methods with variable changes achieve the optimal rate of N^{-α}(ln N)^{(d-1)/2}.
Optimality of several quasi-Monte Carlo methods and suboptimality of the sparse-grid quadrature based on the univariate Gauss--Hermite rule is proved in the Sobolev spaces of mixed dominating smoothness of order $α$, where the optimality is in the sense of worst-case convergence rate. For sparse-grid Gauss--Hermite quadrature, lower and upper bounds are established, with rates coinciding up to a logarithmic factor. The dominant rate is found to be only $N^{-α/2}$ with $N$ function evaluations, although the optimal rate is known to be $N^{-α}(\ln N)^{(d-1)/2}$. The lower bound is obtained by exploiting the structure of the Gauss--Hermite nodes and is independent of the quadrature weights; consequently, no modification of the weights can improve the rate $N^{-α/2}$. In contrast, several quasi-Monte Carlo methods with a change of variables are shown to achieve the optimal rate, some up to, and one including, the logarithmic factor.