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.
Abstract. An L 2 -type discrepancy arises in the average-and worst-case error analyses for multidimensional quadrature rules. This discrepancy is uniquely defined by K(x, y), which serves as the covariance kernel for the space of random functions in the average-case analysis and a reproducing kernel for the space of functions in the worst-case analysis. This article investigates the asymptotic order of the root mean square discrepancy for randomized (0, m, s)-nets in base b. For moderately smooth K(x, y) the discrepancy is O(N −1 [log(N )] (s−1)/2 ), and for K(x, y) with greater smoothness the discrepancy is O(N −3/2 [log(N )] (s−1)/2 ), where N = b m is the number of points in the net. Numerical experiments indicate that the (t, m, s)-nets of Faure, Niederreiter and Sobol do not necessarily attain the higher order of decay for sufficiently smooth kernels. However, Niederreiter nets may attain the higher order for kernels corresponding to spaces of periodic functions.
Random scrambling of deterministic (t, m, s)-nets and (t, s)-sequences eliminates their inherent bias while retaining their low-discrepancy properties. This article describes an implementation of two types of random scrambling, one proposed by Owen and another proposed by Faure and Tezuka. The four different constructions of digital sequences implemented are those proposed by Sobol', Faure, Niederreiter, and Niederreiter and Xing. Because the random scrambling involves manipulating all digits of each point, the code must be written carefully to minimize the execution time. Computed root mean square discrepancies of the scrambled sequences are compared to known theoretical results. Furthermore, the performances of these sequences on various test problems are discussed.
Quasi-Monte Carlo methods are a way of improving the efficiency of Monte Carlo methods. Digital nets and sequences are one of the low discrepancy point sets used in quasi-Monte Carlo methods. This thesis presents the three new results pertaining to digital nets and sequences: implementing randomized digital nets, finding the distribution of the discrepancy of scrambled digital nets, and obtaining better quality of digital nets through evolutionary computation. Finally, applications of scrambled and non-scrambled digital nets are provided. i
List of Figures1.1 Dimensions 7 and 8 of the a) glp and b) randomly shifted glp for N = 256 1.2 Dimensions 2 and 5 of the a) Sobol' and b) Scrambled Sobol' sequences
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