Construction of Interlaced Scrambled Polynomial Lattice Rules of Arbitrary High Order

Abstract

Higher order scrambled digital nets are randomized quasi-Monte Carlo rules which have recently been introduced in [J. Dick, Ann. Statist., 39 (2011), 1372-1398 and shown to achieve the optimal rate of convergence of the root mean square error for numerical integration of smooth functions defined on the s-dimensional unit cube. The key ingredient there is a digit interlacing function applied to the components of a randomly scrambled digital net whose number of components is ds, where the integer d is the so-called interlacing factor. In this paper, we replace the randomly scrambled digital nets by randomly scrambled polynomial lattice point sets, which allows us to obtain a better dependence on the dimension while still achieving the optimal rate of convergence. Our results apply to Owen's full scrambling scheme as well as the simplifications studied by Hickernell, Matoušek and Owen. We consider weighted function spaces with general weights, whose elements have square integrable partial mixed derivatives of order up to α ≥ 1, and derive an upper bound on the variance of the estimator for higher order scrambled polynomial lattice rules. Employing our obtained bound as a quality criterion, we prove that the component-by-component construction can be used to obtain explicit constructions of good polynomial lattice point sets. By first constructing classical polynomial lattice point sets in base b and dimension ds, to which we then apply the interlacing scheme of order d, we obtain a construction cost of the algorithm of order O(dsmb m ) operations 1 using O(b m ) memory in case of product weights, where b m is the number of points in the polynomial lattice point set.

Computational Higher Order Quasi-Monte Carlo Integration

“…A key component for obtaining higher-order convergence rates is the interlacing of lattice point sets, as introduced in [6]. To this end, we define the digit interlacing function, which maps α points in [0, 1) to one point in [0, 1).…”

“…In the present paper, we consider the realization of novel higher-order interlaced polynomial lattice rules introduced in [4,6], which allow an integrand-adapted construction of a quasi-Monte Carlo quadrature rule that exploits sparsity of the parameter-to-solution map. We consider in what follows the problem of integrating a function f : [0, 1) s → R of s variables y 1 , .…”

4. Recent advances in higher order quasi-Monte Carlo methods

“…, q αj should be searched for simultaneously. The papers [35,33] originally gave a justification for employing the former approach, i.e., component-by-component construction.…”

Hot New Directions for Quasi-Monte Carlo Research in Step with Applications

“…We denote the ν-th partial derivative of f by ∂ ν f = (∂ |ν| f )/(∂ ν 1 y 1 ∂ ν 2 y 2 · · · ∂ νs ys ). This function space setting can be paired with interlaced polynomial lattice rules [30,31] to achieve higher order convergence rates in the unit cube. A polynomial lattice rule [64] is similar to a lattice rule (see (3) without the random shift ∆), but instead of a generating vector of integers we have a generating vector of polynomials, and thus the regular multiplication and division are replaced by their polynomial equivalents.…”