The tent transformation can improve the convergence rate of quasi-Monte Carlo algorithms using digital nets

Abstract

In this paper we investigate multivariate integration in reproducing kernel Sobolev spaces for which the second partial derivatives are square integrable. As quadrature points for our quasi-Monte Carlo algorithm we use digital (t, m, s)-nets over Z 2 which are randomly digitally shifted and then folded using the tent transformation. For this QMC algorithm we show that the root mean square worst-case error converges with order 2 m(−2+ε) for any ε > 0, where 2 m is the number of points. A similar result for lattice rules has previously been shown by Hickernell.

Quasi-Monte Carlo methods with applications in finance

“…He also provides a simplified expression for the case of a randomly shifted lattice, and uses it to show the existence of lattice rules for which this discrepancy is O(n −2+δ ) for fixed s. In other words, adding the baker's transformation to the random shift reduces the variance from O(n −2+δ ) to O(n −4+δ ) for non-periodic smooth functions. A similar result applies to a digital net with a random digital shift [14]. Empirical results showing significant variance reductions provided by the baker's transformation can be found in [57,62], for example.…”

“…Most of the properties of ordinary lattice rules have counterparts for the polynomial rules [56]. In particular, figures of merit similar to (3.4), (3.5), and (3.6) can be defined in terms of shortest vectors in the dual lattices, and CBC constructions can provide good parameters for discrepancies based on the Walsh expansion, of the general form (2.16), with product weights [14][15][16].…”

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

“…The obtained cost for the fast CBC construction becomes O(sα|P | α/2 log |P |) arithmetic operations using O(|P | α/2 ) memory. This is a generalization of the study in [3]. However, this result not only restricts the base b to 2, but also needs a randomization by a random digital shift.…”

Theb-adic tent transformation for quasi-Monte Carlo integration using digital nets

“…The tent transformation was later analyzed in the context of QMC rules using digital nets by Cristea et al [4], where the tent transformation was successfully applied to randomly digitally shifted digital nets over Z 2 to achieve almost the optimal convergence rate for integrands in unanchored Sobolev spaces of smoothness of second order. Their result has been generalized very recently by one of the authors [10] to unanchored Sobolev spaces of smoothness of arbitrary high order for the purpose of constructing good higher order polynomial lattice rules over Z 2 with modulus of reduced degree as compared to [3,8].…”

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In this paper we investigate quasi-Monte Carlo (QMC) integration using digital nets over Z b in reproducing kernel Hilbert spaces. The tent transformation, or the baker's transformation, was originally used for lattice rules by Hickernell (2002) to achieve higher order convergence of the integration error for smooth non-periodic integrands, and later, has been successfully applied to digital nets over Z2 by Cristea et al (2007) and Goda (2014). The aim of this paper is to generalize the latter two results to digital nets over Z b for an arbitrary prime b.

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