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Extrapolated Lattice Rule Integration in Computational Uncertainty Quantification

by J. Dick and M. Longo and Ch. Schwab

(Report number 2020-29)

Abstract
We present an extension of the convergence analysis for Richardson-extrapolated polynomial lattice rules from [Josef Dick, Takashi Goda and Takehito Yoshiki: Richardson extrapolation of polynomial lattice rules, SIAM J. Numer. Anal. {\bf 57}(2019) 44-69] for high-dimensional, numerical integration to classes of integrand functions with quantified smoothness and Quasi-Monte Carlo (``QMC'' for short) integration rules with so-called smoothness-driven, product and order dependent (SPOD for short) weights. We establish in particular sufficient conditions for the existence of an asymptotic expansion of the QMC integration error with respect to suitable powers of $N$, the number of QMC integration nodes. We derive a dimension-separated criterion for a fast component-by-component (``CBC'' for short) construction algorithm for the computation of the QMC generating vector with quadratic scaling with respect to the integration dimension $s$. We prove that the proposed QMC integration strategies a) are free from the curse of dimensionality, b) afford higher-order convergence rates subject to suitable summability conditions on the QMC weights, c) allow for certain classes of high-dimensional integrands functions a computable, asymptotically exact numerical estimate of the QMC quadrature error, with reliability and efficiency independent of the dimension of the integration domain and d) accommodate fast, FFT-based matrix-vector multiplication from [Dick, Josef; Kuo, Frances Y.; Le Gia, Quoc T.; Schwab, Christoph: Fast QMC matrix-vector multiplication. SIAM J. Sci. Comput. 37 (2015), no. 3, A1436-A1450] when applied to parametric operator equations. The integration methods are applicable for large classes of many-parametric integrand functions with quantified parametric smoothness. We verify all hypotheses and present numerical examples arising from the Galerkin Finite-Element discretization of a model, linear parametric elliptic PDE illustrating a) - d). We verify computationally the scaling of the fast CBC construction algorithm with SPOD QMC weights, and examine the extrapolation-based a-posteriori numerical estimation of the QMC quadrature error. We find in examples with parameter spaces of dimension $s=10,...,128$ that the extrapolation-based, computable QMC integration error indicator has an efficiency index between $0.9$ and $1.1$, for a moderate number $N$ of QMC points.

Keywords: High-dimensional Quadrature, Quasi-Monte Carlo, Richardson Extrapolation, A-posterior Error Estimation

@Techreport{DLS20_902,
  author = {J. Dick and M. Longo and Ch. Schwab},
  title = {Extrapolated Lattice Rule Integration in Computational Uncertainty Quantification},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-29},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-29.pdf },
  year = {2020}
}

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