Digital Point Sets: Analysis and Application

Summary. In analogy to the recent paper [11] which studied lattice rule algorithms for approximation in weighted Korobov spaces, we consider the approximation problem in a weighted Hilbert space of Walsh series. Our approximation uses a truncated Walsh series with Walsh coefficients approximated by numerical integration using digital nets. We show that digital nets (or more precisely, polynomial lattices) tailored specially for the approximation problem lead to better error bounds. The error bounds can be independent of the dimension s, or depend only polynomially on s, under certain conditions on the weights defining the function space.

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“…, b − 1} with b prime. For a more general definition, see, e.g., [13,14,19]. Throughout the paper, means the transpose of a vector or matrix.…”

Section: Integration Using Digital Netsmentioning

confidence: 99%

Approximation of Functions Using Digital Nets

Dick

Kritzer

Kuo

2008

Monte Carlo and Quasi-Monte Carlo Methods 2006

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“…To avoid too many technical notions, we restrict ourselves in the following to digital nets defined over the finite field F b and hence b is restricted to prime powers. For a more general definition (over arbitrary finite commutative rings) see for example Niederreiter [21], Larcher [13], or Larcher, Niederreiter and Schmid [15].…”

Section: Definition 21 ((T M S)-netsmentioning

confidence: 99%

Construction algorithms for polynomial lattice rules for multivariate integration

et al. 2005

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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 worst-case 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 worst-case error by computer search. We prove an upper bound on the worst-case 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 worst-case 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 worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

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“…Practical construction methods for (t, m, s)-nets are based upon the concept of a digital (t, m, s)-net. We refer the reader to the surveys Niederreiter [12] and Larcher [7] for further reading.In this paper we will study the following question. Suppose that the dimension s is given.…”

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confidence: 99%

“…We assume that the reader is familiar with the notion of arbitrary and digital (t, m, s)-nets. The monograph [12] and the survey papers [7,15] contain comprehensive discussions of this topic. We refer to [2,4] …”

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confidence: 99%

Digital (t,m,s)-nets and the spectral test

Hellekalek

2002

Acta Arith.

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Digital (t, m, s)-nets and the spectral test byPeter Hellekalek (Salzburg) 1. Introduction. The notion of a (t, m, s)-net to a given base b is a central concept of the modern theory of uniform distribution of sequences modulo one. It has been introduced by Niederreiter [11] in a highly successful attempt to unify and extend existing construction methods for lowdiscrepancy point sets. Such nets are of fundamental importance in the theory and practice of quasi-Monte Carlo methods.The optimal choice for the quality parameter t is t = 0. Practical construction methods for (t, m, s)-nets are based upon the concept of a digital (t, m, s)-net. We refer the reader to the surveys Niederreiter [12] and Larcher [7] for further reading.In this paper we will study the following question. Suppose that the dimension s is given. What will be the best possible uniform distribution of a (t, m, s)-net in base b on the s-dimensional torus [0, 1[ s ? More precisely, for an appropriately chosen measure of uniform distribution, is it possible to give exact upper and lower bounds for (digital) (t, m, s)-nets?We will employ the concept of the generalized spectral test introduced in Hellekalek [4] to find an answer for this question. Our results include an upper bound for the general case and lower bounds for digital (t, m, s)-nets in prime base b. All bounds are best possible.Our method is based upon the exact computation of Weyl sums with respect to an appropriate Walsh function system and the application of elementary concepts of linear algebra.In our proofs for strict digital (t, m, s)-nets in base b we use results for associated linear codes established in Niederreiter and Pirsic [14] and Skriganov [16]. Interestingly, duality comes into play with all applications of the spectral test: see the survey Hellekalek [5].

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Digital Point Sets: Analysis and Application

Cited by 48 publications

References 33 publications

Approximation of Functions Using Digital Nets

Approximation of Functions Using Digital Nets

Construction algorithms for polynomial lattice rules for multivariate integration

Digital (t,m,s)-nets and the spectral test

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