744 research outputs found
From van der Corput to modern constructions of sequences for quasi-Monte Carlo rules
In 1935 J.G. van der Corput introduced a sequence which has excellent uniform
distribution properties modulo 1. This sequence is based on a very simple
digital construction scheme with respect to the binary digit expansion.
Nowadays the van der Corput sequence, as it was named later, is the prototype
of many uniformly distributed sequences, also in the multi-dimensional case.
Such sequences are required as sample nodes in quasi-Monte Carlo algorithms,
which are deterministic variants of Monte Carlo rules for numerical
integration. Since its introduction many people have studied the van der Corput
sequence and generalizations thereof. This led to a huge number of results.
On the occasion of the 125th birthday of J.G. van der Corput we survey many
interesting results on van der Corput sequences and their generalizations. In
this way we move from van der Corput's ideas to the most modern constructions
of sequences for quasi-Monte Carlo rules, such as, e.g., generalized Halton
sequences or Niederreiter's -sequences
Discrepancy bounds for low-dimensional point sets
The class of -nets and -sequences, introduced in their most
general form by Niederreiter, are important examples of point sets and
sequences that are commonly used in quasi-Monte Carlo algorithms for
integration and approximation. Low-dimensional versions of -nets and
-sequences, such as Hammersley point sets and van der Corput sequences,
form important sub-classes, as they are interesting mathematical objects from a
theoretical point of view, and simultaneously serve as examples that make it
easier to understand the structural properties of -nets and
-sequences in arbitrary dimension. For these reasons, a considerable
number of papers have been written on the properties of low-dimensional nets
and sequences
Quasi-random numbers for copula models
The present work addresses the question how sampling algorithms for commonly
applied copula models can be adapted to account for quasi-random numbers.
Besides sampling methods such as the conditional distribution method (based on
a one-to-one transformation), it is also shown that typically faster sampling
methods (based on stochastic representations) can be used to improve upon
classical Monte Carlo methods when pseudo-random number generators are replaced
by quasi-random number generators. This opens the door to quasi-random numbers
for models well beyond independent margins or the multivariate normal
distribution. Detailed examples (in the context of finance and insurance),
illustrations and simulations are given and software has been developed and
provided in the R packages copula and qrng
Weak multipliers for generalized van der Corput sequences
Generalized van der Corput sequences are onedimensional, infinite sequences in the unit interval. They are generated from permutations in integer base b and are the building blocks of the multi-dimensional Halton sequences. Motivated by recent progress of Atanassov on the uniform distribution behavior of Halton sequences, we study, among others, permutations of the form P(i) = ai (mod b) for coprime integers a and b. We show that multipliers a that either divide b - 1 or b + 1 generate van der Corput sequences with weak distribution properties. We give explicit lower bounds for the asymptotic distribution behavior of these sequences and relate them to sequences generated from the identity permutation in smaller bases, which are, due to Faure, the weakest distributed generalized van der Corput sequences.Les suites de Van der Corput gĂ©nĂ©ralisĂ©es sont dessuites unidimensionnelles et infinies dans lâintervalle de lâunitĂ©.Elles sont gĂ©nĂ©rĂ©es par permutations des entiers de la basebetsont les Ă©lĂ©ments constitutifs des suites multi-dimensionnelles deHalton. Suites aux progrĂšs rĂ©cents dâAtanassov concernant le com-portement de distribution uniforme des suites de Halton nous nousintĂ©ressons aux permutations de la formuleP(i) =ai(modb)pour les entiers premiers entre euxaetb. Dans cet article nousidentifions des multiplicateursagĂ©nĂ©rant des suites de Van derCorput ayant une mauvaise distribution. Nous donnons les bornesinfĂ©rieures explicites pour cette distribution asymptotique asso-ciĂ©e Ă ces suites et relions ces derniĂšres aux suites gĂ©nĂ©rĂ©es parpermutation dâidentitĂ©, qui sont, selon Faure, les moins bien dis-tribuĂ©es des suites gĂ©nĂ©ralisĂ©es de Van der Corput dans une basedonnĂ©e
Generalized von NeumannâKakutani transformation and random-start scrambled Halton sequences
AbstractIt is a well-known fact that the Halton sequence exhibits poor uniformity in high dimensions. Starting with Braaten and Weller in 1979, several researchers introduced permutations to scramble the digits of the van der Corput sequences that make up the Halton sequence, in order to improve the uniformity of the Halton sequence. These sequences are called scrambled Halton, or generalized Halton sequences. Another significant result on the Halton sequence was the fact that it could be represented as the orbit of the von NeumannâKakutani transformation, as observed by Lambert in 1982. In this paper, I will show that a scrambled Halton sequence can be represented as the orbit of an appropriately generalized von NeumannâKakutani transformation. A practical implication of this result is that it gives a new family of randomized quasi-Monte Carlo sequences: random-start scrambled Halton sequences. This work generalizes random-start Halton sequences of Wang and Hickernell. Numerical results show that random-start scrambled Halton sequences can improve on the sample variance of random-start Halton sequences by factors as high as 7000
On the Optimal Spatial Design for Groundwater Level Monitoring Networks
Effective groundwater monitoring networks are important, as systematic data collected at observation wells provide a crucial understanding of the dynamics of hydrogeological systems as well as the basis for many other applications. This study investigates the influence of six groundwater level monitoring network (GLMN) sampling designs (random, grid, spatial coverage, and geostatistical) with varying densities on the accuracy of spatially interpolated groundwater surfaces. To obtain spatially continuous prediction errors (in contrast to point crossâvalidation errors), we used nine potentiometric groundwater surfaces from three regional MODFLOW groundwater flow models with different resolutions as a priori references. To assess the suitability of frequentlyâused crossâvalidation error statistics (MAE, RMSE, RMSSE, ASE, and NSE), we compared them with the actual prediction errors (APE). Additionally, we defined upper and lower thresholds for an appropriate spatial density of monitoring wells. Below the lower threshold, the observation density appears insufficient, and additional wells lead to a significant improvement of the results. Above the upper threshold, additional wells lead to only minor and inefficient improvements. According to the APE, systematic sampling lead to the best results but is often not suited for GLMN due to its nonprogressive characteristic. Geostatistical and spatial coverage sampling are considerable alternatives, which are in contrast progressive and allow evenly spaced and, in the case of spatial coverage sampling, yet reproducible coverage with accurate results. We found that the global crossâvalidation error statistics are not suitable to compare the performance of different sampling designs, although they allow rough conclusions about the quality of the GLMN
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