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Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces

Yoshihito Kazashi, Yuya Suzuki, Takashi Goda

Abstract

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.

Optimality of quasi-Monte Carlo methods and suboptimality of the sparse-grid Gauss--Hermite rule in Gaussian Sobolev spaces

Abstract

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 , 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 with function evaluations, although the optimal rate is known to be . 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 . 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.

Paper Structure

This paper contains 13 sections, 19 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. The Sobolev space
  4. Tensorised operators
  5. Sobolev embedding for spaces with dominating mixed smoothness
  6. Gauss--Hermite sparse-grid quadrature
  7. Lower bounds
  8. Application: lower bounds for sparse-grid interpolation
  9. Upper bound
  10. Quasi-Monte Carlo methods
  11. Affinely-mapped QMC
  12. QMC with Möbius transformation
  13. Concluding remarks

Key Result

Theorem 2.1

For $N\geq2$, let $A_{N}\colon H_{\rho}^{\alpha}(\mathbb{R}^{d})\to\mathbb{R}$ be a mapping (linear or nonlinear) that uses only $N$ values as information about the argument, i.e., it is of the form $A_{N}(f)=\mathcal{I}_{N}(f(\boldsymbol{x}_{1}),\dots,f(\boldsymbol{x}_{N}))$ for a mapping $\mathcal

Theorems & Definitions (36)

  • Theorem 2.1: Dick.J_Irrgeher_Leobacher_Pillichshammer_2018_SINUM_Hermite
  • Proposition 2.2
  • Proof 1
  • Proposition 2.3
  • Proof 2
  • Theorem 3.1
  • Proof 3
  • Theorem 3.2
  • Proof 4
  • Theorem 3.3
  • ...and 26 more