G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatshefte fur Mathematik, 121(3):231--253, 1996.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....rank k. Digital nets can in fact be defined over an arbitrary commutative ring R of cardinality b, with an identity element. One simply define bijections between R and Z b to map the digits of the b ary expansion of i to elements of R and to recover the b ary digits of u i,j from elements of R [14, 18, 24, 31]. A similar generalization applies to polynomial lattices as well, where the bijections from R to Z b can be incorporated into #. To simplify the exposition in this paper, we will assume that R = Z b and that all the bijections are the identity. Quasi Monte Carlo point sets are justified by a ....

....lattice rules. This is helpful for comparing them with their polynomial versions. The reader can consult [17, 28] for more details. PLRs with coe#cients in the ring Z b for an arbitrary b are defined and studied in section 3. These PLRs generalize the PLRs of rank 1 introduced and studied in [14, 24, 29]. We extend definitions and results of [18, 21] to an arbitrary b and gives new ones (e.g. the development of section 3.7) In section 4, we introduce a resolutionwise version of PLRs, based on the notion of resolutionwise lattice of Tezuka [30, 31] The links between PLRs and digital nets [14, ....

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G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatshefte fur Mathematik, 121(3):231--253, 1996.

....fact that f(0) the second equality is obtained by changing the order of summation, which is allowed by Fubini s theorem [55] since # h#N t f(h) #; the third equality follows from Lemma 4.20. Error bounds for functions having su#ciently fast decaying Walsh coe#cients are given in [29, 28] for di#erent types of digital nets. This is in analogy with existing results for ordinary lattice rules that can be found in [56] for example. We do not explore this subject here, as we are rather interested in studying randomizations of P n and their corresponding variance expressions. This is ....

....every non negative integer vector (q 1 , q t ) such that q 1 . q t = k q. In the same way that the result of [7] can be expressed in terms of a shortest non zero vector h in L # t using the norm #h## , a point set P n having the (q, k, t) net property amounts to say that [28, 47, 61] min 0 #=h#L # t #h# # # 2 k q t 1 , 5.3) where #h# # = # t j=1 h j l , and h j l = h j p if h j 0 and h j l = 2 1 if h j = 0; the equality holds for the smallest value of q for which the (q, k, t) net property holds [47] This result is valid for ....

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G. Larcher, H. Niederreiter, and W. C. Schmid, Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration, Monatshefte fur Mathematik, 121 (1996), pp. 231--253.

....with Lemma 4.19, we obtain the next proposition. Proposition 4.20. If f is such that # h#N t f(h) #, then the integration error with P n can be written as E n = # 0#=h#L # t f(h) 4. 18) Error bounds for functions having su#ciently fast decaying Walsh coe#cients are given in [28, 27] for di#erent types of digital nets. This is in analogy with existing results for ordinary lattice rules that can be found in [54] for example. We do not explore this subject here, as we are rather interested in studying randomizations of P n and their corresponding variance expressions. This is ....

....every non negative integer vector (q 1 , q t ) such that q 1 . q t = k q. In the same way that the result of [6] can be expressed in terms of a shortest non zero vector h in L # t using the norm #h## , a point set P n having the (q, k, t) net property amounts to say that [27, 46, 59] min 0 #=h#L # t #h# # # 2 k q t 1 , 5.3) where #h# # = # t j=1 h j l , and h j l = h j p if h j 0 and h j l = 2 1 if h j = 0; the equality holds for the smallest value of q for which the (q, k, t) net property holds POLYNOMIAL LATTICE RULES 15 [46] ....

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G. Larcher, H. Niederreiter, and W. C. Schmid, Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration, Monatshefte fur Mathematik, 121 (1996), pp. 231--253. 24 CHRISTIANE LEMIEUX AND PIERRE L'ECUYER

....1) b Gamma 1) see Mullen and Whittle [60] For k = 2 equality holds if b is a prime power. Lawrence [40] provides improvements for k 2 using a theory of generalized orthogonal arrays. Similar questions arise concerning bounds for so called digital nets, see Larcher, Niederreiter and Schmid [39] for details as well as for the definition of a digital net. We close this section with a problem concerning digital nets first raised by Schmid [72] Problem 36. Let q be a prime power and t 0; k 2 and s 1 be integers. Then there exists a (t; t k; s) net in base q if and only if there exists ....

....[72] Problem 36. Let q be a prime power and t 0; k 2 and s 1 be integers. Then there exists a (t; t k; s) net in base q if and only if there exists a digital (t; t k; s) net constructed over F q . This is known to be false for (0; k; s) nets in base b where b is not a prime power, see [39]. 7 Character Sums Let (z) be a non trivial additive character of F q . We let S(n; q) max a2F q fi fi fi fi fi fi X x2Fq (ax n ) fi fi fi fi fi fi : The general Weil bound for character sums with an arbitrary polynomial provides the bound S(n; q) nq 1=2 . It is a simple ....

G. Larcher, H. Niederreiter and W. C. Schmid, `Digital nets and sequences constructed over finite rings and their application to quasiMonte Carlo integration', Monatsh. Math., to appear.

....and Traunfellner[12] to present an analogue to Korobov s results for generalized Walsh systems. In the latter case a similar relation between certain function classes b E ff s (C) and smoothness with respect to the b adic derivatives exists (see [8] Further references for this approach are [9, 11, 10, 7] . Larcher et al. derived the following integration error estimates for their function classes b E ff s (C) ff 1=2: If P is a (t; m; s) net to base b 2, then RN (f; P) C 0 (s; b; ff; C) Delta b t(ff Gamma 1 2 ) Delta (log N) s Gamma1 N ff Gamma 1 2 : If P is a special kind ....

G. Larcher, H. Niederreiter, and W.Ch. Schmid. Digital nets and sequences constructed over finite rings and their applications to quasi-Monte-Carlo integration. Preprint, Institut fur Mathematik, Universitat Salzburg, 1994.

....and ffl any finite set of indices I ae N s 0 for the construction of the partial sum. If we restrict our considerations to the functions in the classes b E ff s (C) see section 2) we can apply the the number theoretical numerical integration methods introduced by Larcher et. al ([9, 5, 7, 6, 4]) for the computation of the corresponding Walsh coefficients with optimal error bounds. The special nature of the functions in b E ff s (C) also suggests the construction of the s dimensional partial sums by the use of hyperbolic index sets instead of the cube like sets of the classical DWT. ....

G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math., 1995. (to appear).

.... Numerical integration of multivariate Walsh series 92 4 A comparison of results on classical numerical integration methods with the results for our integration method 99 1 Introduction In a series of papers (see [Lar93a] Lar93b] LT94] LS94] LSW94] LSW95] LS95b] LLNS95] LS95a] [LNS95], Sch93] and [Sch94] by the authors of this paper and several co authors a theory for the numerical integration of multivariate Walsh series by number theoretical methods, that is with the help of extremely well distributed point sets and especially with the help of digital (t; m; s) nets was ....

....was given in a series of papers by Niederreiter. See [Nie87] Nie88] Nie91] Nie92d] Nie92c] Nie92b] Nie92a] Contributions to the theory of (t; m; s) nets and (T; s) sequences given by the authors can be found in [Lar93b] LS94] LSW94] LSW95] LN93] LN95] LLNS95] LS95a] and [LNS95]. We give now the basic definitions for (t; m; s) nets and (T; s) sequences in a base b and we introduce the fundamental construction method for (t; m; s) nets and (T; s) sequences namely the so called digital method. For more details we refer to the excellent monograph [Nie92d] of ....

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G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. To appear in: Monatsh. Math., 1995.

....concerning this system. 1 Introduction In a series of papers by the last two authors and several co authors, a theoretical method for the numerical integration of high dimensional functions, based on number theoretical methods was developed in detail. See for example [7] 4] 5] 6] 3] 1] [2], and [8] This method is called digital lattice rule and essentially is based on the use of a new class of highdimensional, extremely well distributed, number theoretical point sets in an s dimensional unit cube. Recently, we have implemented this theoretical method and at the moment we can ....

....some special values of the base b, parts of these nets already have been implemented in a (sequential) test version. You will find numerical results of the tests in our last deliverable. The theoretical work for generating the nets was done in [9] 10] and [11] galois fields) and above all in [2] (residue class rings) see also our deliverables) The usefulness of these nets for the numerical integration of high dimensional multivariate Walsh series in certain bases (the Walsh series lattice rules) was established by the authors in several publications of the last years (for example see ....

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G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math., 121:231--253, 1996.

....the point set consisting of the xn with kq m n (k 1)q m is a (t; m; s) net in base q. For a more general definition of digital (t; s) sequences and digital (t; m; s) nets (over arbitrary finite commutative rings) see for example Niederreiter [12] or Larcher, Niederreiter, and Schmid [9]. Remark: The construction of the points of digital nets and sequences and therefore the quality of their distribution only depend on the matrices C (1) C (s) So, the crucial point for a concrete implementation is the construction of the matrices. Examples for the construction of ....

G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatsh. Math., 121:231--253, 1996.

....m; s) nets, and so the tables presented here should be of value to researchers and to those who use nets. We also note that in recent years, considerable emphasis has been placed by a number of researchers on the subclass of (t; m; s) nets known as digital (t; m; s) nets in base b; see for example [11, 19, 22, 51, 53]. 2. New (t; m; s) net constructions. In this section we briefly describe the new constructions for (t; m; s) nets which have arisen since the publication of [35] We continue with the numbering which in [35] ended with Construction 16. Construction 17. In [31] the authors show that given a ....

.... yielding the same parameters as that of Niederreiter was discussed in Tezuka [54, Chapter 6] and Tezuka and Tokuyama [55] An investigation of general digital constructions of (t; m; s) nets and (t; s) sequences with the use of finite rings was carried out in Larcher, Niederreiter, and Schmid [22]. A generalization of the concept of a (t; s) sequence was introduced and analyzed in Larcher and Niederreiter [21] This paper contains also a study of Kronecker type sequences, i.e. of nonarchimedean analogs of classical Kronecker sequences. Further work on Kronecker type sequences and their ....

[Article contains additional citation context not shown here]

G. Larcher, H. Niederreiter, and W.C. Schmid, Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration, Monatsh. Math., 121 (1996), pp. 231-253.

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G. Larcher, H. Niederreiter, and W.Ch. Schmid. Digital nets and sequences constructed over finite rings and their applications to quasi-Monte-Carlo integration. To appear in: Monatsh. Math.

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