Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration

1996 ◽  
Vol 121 (3) ◽  
pp. 231-253 ◽  
Author(s):  
Gerhard Larcher ◽  
Harald Niederreiter ◽  
Wolfgang Ch. Schmid
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Integration ◽  
Finite Rings ◽  
Quasi Monte Carlo ◽  
Digital Nets

scholarly journals The Bb-adic Symmetrization of Digital Nets for Quasi-Monte Carlo Integration

2017 ◽  
Vol 12 (1) ◽  
pp. 1-25
Author(s):  
Takashi Goda
Keyword(s):  
Monte Carlo ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Unit Cube ◽  
Monte Carlo Integration ◽  
Point Sets ◽  
Quasi Monte Carlo ◽  
Integration Error ◽  
Geometric Technique ◽  
Digital Nets

Abstract The notion of symmetrization, also known as Davenport’s reflection principle, is well known in the area of the discrepancy theory and quasi- Monte Carlo (QMC) integration. In this paper we consider applying a symmetrization technique to a certain class of QMC point sets called digital nets over ℤb. Although symmetrization has been recognized as a geometric technique in the multi-dimensional unit cube, we give another look at symmetrization as a geometric technique in a compact totally disconnected abelian group with dyadic arithmetic operations. Based on this observation we generalize the notion of symmetrization from base 2 to an arbitrary base b ∈ ℕ, b ≥ 2. Subsequently, we study the QMC integration error of symmetrized digital nets over ℤb in a reproducing kernel Hilbert space. The result can be applied to component-by-component construction or Korobov construction for finding good symmetrized (higher order) polynomial lattice rules which achieve high order convergence of the integration error for smooth integrands at the expense of an exponential growth of the number of points with the dimension. Moreover, we consider two-dimensional symmetrized Hammersley point sets in prime base b, and prove that the minimum Dick weight is large enough to achieve the best possible order of Lp discrepancy for all 1 ≤ p < ∞.

Higher order Quasi-Monte Carlo integration for Bayesian PDE Inversion

2019 ◽  
Vol 77 (1) ◽  
pp. 144-172 ◽  
Author(s):  
Josef Dick ◽  
Robert N. Gantner ◽  
Quoc T. Le Gia ◽  
Christoph Schwab
Keyword(s):  
Monte Carlo ◽  
Higher Order ◽  
Monte Carlo Integration ◽  
Quasi Monte Carlo
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