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On the use of low discrepancy sequences in Monte Carlo methods
 MONTE CARLO METHODS AND APPLICATIONS
, 1996
"... Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 ..."
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Cited by 40 (2 self)
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Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 (ln N) s ), where s is the mathematical dimension of the problem considered. The disadvantage of these methods is that error bounds, even if they exist theoretically, are inefficient in practice. Nevertheless, to take advantage of these methods for what concerns their convergence speed, we use them as a variance reduction technique, which lead to great improvements with respect to standard Monte Carlo methods. We consider in this paper two different approaches which combine Monte Carlo and quasiMonte Carlo methods. The first one can use every low discrepancy sequence and the second one, called Owen's method, uses only Niederreiter sequences. We prove that the first approach has the same...
Extensible Lattice Sequences For QuasiMonte Carlo Quadrature
 SIAM Journal on Scientific Computing
, 1999
"... 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 form a lattice for any nonnegative ..."
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Cited by 35 (10 self)
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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 form 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.
Construction algorithms for polynomial lattice rules for multivariate integration
 Math. Comp
"... Abstract. We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worstcase error for integration using digital nets ..."
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Cited by 23 (8 self)
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Abstract. We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worstcase error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worstcase error by computer search. We prove an upper bound on the worstcase error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights. We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worstcase error is slightly different. Again digital nets obtained from our search algorithm yield a worstcase error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules. We conclude the article with numerical results comparing the expected worstcase error for randomly digitally shifted digital nets with those for randomly shifted lattice rules. 1.
Randomized Halton Sequences
 Mathematical and Computer Modelling
, 2000
"... The Halton sequence is a wellknown multidimensional low discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence: we randomize the start point of each component of the sequence. This method combines the potential accuracy advantage of Halton sequence in mult ..."
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Cited by 23 (1 self)
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The Halton sequence is a wellknown multidimensional low discrepancy sequence. In this paper, we propose a new method for randomizing the Halton sequence: we randomize the start point of each component of the sequence. This method combines the potential accuracy advantage of Halton sequence in multidimensional integration with the practical error estimation advantage of Monte Carlo methods. Theoretically, using multiple randomized Halton sequences as a variance reduction technique we can obtain an efficiency improvement over standard Monte Carlo under rather general conditions. Numerical results show that randomized Halton sequences have better performance than not only Monte Carlo, but also randomly shifted Halton sequences and (single long) purely deterministic skipped Halton sequence. Key Words: QuasiMonte Carlo methods, low discrepancy sequences, Monte Carlo methods, numerical integration, variance reduction. AMS 1991 Subject Classification: 65C05, 65D30. This work was suppor...
Acceleration of the multipletry Metropolis algorithm using antithetic and stratified sampling. Statistics and Computing 17:109
, 2007
"... The MultipleTry Metropolis is a recent extension of the Metropolis algorithm in which the next state of the chain is selected among a set of proposals. We propose a modification of the MultipleTry Metropolis algorithm which allows the use of correlated proposals, particularly antithetic and strat ..."
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Cited by 20 (5 self)
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The MultipleTry Metropolis is a recent extension of the Metropolis algorithm in which the next state of the chain is selected among a set of proposals. We propose a modification of the MultipleTry Metropolis algorithm which allows the use of correlated proposals, particularly antithetic and stratified proposals. The method is particularly useful for random walk Metropolis in high dimensional spaces and can be used easily when the proposal distribution is Gaussian. We explore the use of quasiMonte Carlo (QMC) methods to generate highly stratified samples. A series of examples is presented to evaluate the potential of the method.
Monte Carlo Extension Of QuasiMonte Carlo
 Proceedings of the 1998 Winter Simulation Conference
, 1998
"... This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin super ..."
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Cited by 20 (0 self)
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This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin supercube sampling it is possible to use Monte Carlo methods to extend quasiMonte Carlo methods to higher dimensional problems. 1 INTRODUCTION The problem we consider is the estimation of an integral I = Z [0;1] d f(x)dx: (1) Standard manipulations can be applied to express integrals over domains other than the unit cube or with respect to nonuniform measures in the form (1). Similarly, the integrand f in (1) subsumes weighting functions from importance sampling or periodization. We are especially interested in cases where the dimension d is large, and some of the methods considered here apply to the case d = 1. The focus of this article is on ways of combining Monte Carlo and quasiMo...
SphericalRadial Integration Rules for Bayesian Computation
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1997
"... The common numerical problem in Bayesian analysis is the numerical integration of the posterior. In high dimensions, this problem becomes too formidable for fixed quadrature methods, and Monte Carlo integration is the usual approach. Through the use of modal standardization and a sphericalradial tr ..."
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Cited by 18 (6 self)
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The common numerical problem in Bayesian analysis is the numerical integration of the posterior. In high dimensions, this problem becomes too formidable for fixed quadrature methods, and Monte Carlo integration is the usual approach. Through the use of modal standardization and a sphericalradial transformation, we reparameterize in terms of a radius r and point z on the surface of the sphere in d dimensions. We propose two types of methods for sphericalradial integration. A completely randomized method uses randomly placed abscissas for the radial integration and for the sphere surface. A mixed method uses fixed quadrature (Simpson's rule) on the radius and randomized spherical integration. The mixed methods show superior accuracy in comparisons, require little or no assumptions, and provide diagnostics to detect difficult problems. Moreover, if the posterior is close to the multivariate normal, the mixed methods can give remarkable accuracy.
Fast Generation of Randomized LowDiscrepancy Point Sets
, 2001
"... We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html. ..."
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Cited by 16 (2 self)
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We introduce two novel techniques for speeding up the generation of digital (t,s)sequences. Based on these results a new algorithm for the construction of Owen's randomly permuted (t,s)sequences is developed and analyzed. An implementation is available at http://www.mcqmc.org/Software.html.
Randomized Polynomial Lattice Rules For Multivariate Integration And Simulation
 SIAM JOURNAL ON SCIENTIFIC COMPUTING
, 2001
"... Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical fra ..."
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Cited by 13 (3 self)
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Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasiMonte Carlo set of tools. A new class of lattice rules, defined in a space of polynomials with coefficients in a finite field, is introduced in this paper, and a theoretical framework for these polynomial lattice rules is developed. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, turn out to be special cases of this method.
QuasiMonte Carlo Via Linear ShiftRegister Sequences
, 1999
"... Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period leng ..."
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Cited by 13 (3 self)
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Linear recurrences modulo 2 with long periods have been widely used for contructing (pseudo)random number generators. Here, we use them for quasiMonte Carlo integration over the unit hypercube. Any stochastic simulation fits this framework. The idea is to choose a recurrence with a short period length and to estimate the integral by the average value of the integrand over all vectors of successive output values produced by the small generator. We examine randomizations of this scheme, discuss criteria for selecting the parameters, and provide examples. This approach can be viewed as a polynomial version of lattice rules.