Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature

Abstract

Abstract. Integration lattices are one of the main types of low discrepancy sets used in quasiMonte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which forms a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.

Current Issues in Sampling-Based Motion Planning

“…Secondarily, these sequences should be easy to generate and allow easy neighbor-finding, to fully exploit the advantages of regular structures. Examples of infinite sequences based on regular structures include the extensible grid sequence of the authors [48] and the extensible low-discrepancy lattices of Hickernell et al [31]. The derandomized version of the Lazy PRM (see [11]) also progressively increases the resolution of the grid used to build the roadmap [10].…”

Quasi-Monte Carlo Progressive Photon Mapping

“…The images in Figure 2 have been computed using the Halton sequence. We also verified the theoretical results using fast implementations of (t, s)-sequences in base b, especially the Sobol' sequence [17,19], and rank-1 lattices sequences [4].…”

Parallel Quasi-Monte Carlo Integration by Partitioning Low Discrepancy Sequences

“…For a suitable generator vector h ∈ N s+1 , the points of a rank-1 lattice sequence [7] are computed by…”

“…6.4] and reported in [2,12,18,9], transferring the approach to (t, s)-sequences in a straightforward way has defects and rank-1 lattice sequences [7] have not yet been considered.…”