K. Entacher, P. Hellekalek, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....# min 0#=k#L # #k# r # # . 3. 12) For r = 2 this discrepancy is equivalent to the spectral test, commonly employed to measure the quality of linear congruential pseudo random number generators [31, 34] The spectral test has been used to select lattices for quasi Monte Carlo quadrature in [7, 35, 36, 37]. The case r = 1, which one might call an # 1 spectral test, is also interesting. We will return to these two cases in the next section. 4. Good Generating Vectors for Lattice Sequences. As mentioned at the beginning of the previous section, finding good generating vectors for lattices typically ....

K. Entacher, P. Hellekalek, and P. L' Ecuyer, Quasi-Monte Carlo node sets from linear congruential generators, in Niederreiter and Spanier [47].

.... kkk 2 oe Gammaff : This discrepancy is equivalent to the spectral test, which is used to measure the quality of linear congruential pseudo random number generators. The spectral test has been used to generate lattices for quasi Monte Carlo quadrature by L Ecuyer and his collaborators [EHL99,L E99,LL98,LL99] 5.2 Periodic Integrands Another family of error bounds for periodic integrands is derived in [Hic98a] This is a generalization of (26) and is derived without reference to Fourier series. For ease of presentation we extend the notation for the p and L p norms. Let (f u ....

K. Entacher, P. Hellekalek, and P. L' ' Ecuyer, Quasi-monte carlo node sets from linear congruential generators, In Niederreiter and Spanier [NS99].

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K. Entacher, P. Hellekalek, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998.

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K. Entacher, P. Hellekalek, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. In H. Niederreiter and J. Spanier, editors, Monte Carlo and Quasi-Monte Carlo Methods 1998.

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K. Entacher, P. Hellekalek, P. L'Ecuyer, Quasi-Monte Carlo node sets from linear congruential generators, in: H. Niederreiter, J. Spanier (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 1998.

....t . If P n has n points, it turns out that each coordinate of each vector of L t is a multiple of 1 n. One simple and convenient way to construct an integration lattice is by taking P n as the set of all t dimensional vectors of successive output values from a linear congruential generator (LCG) [11, 24, 33]; that is, take P n as the set of all vectors (u 0 , u t 1 ) where x 0 # Z n = 0, n 1 and the u i obey the recurrence x i = ax i 1 ) mod n, u i = x i n, 1.5) for some positive integer a in Z n . The corresponding integration rule Q n given by (1.2) was proposed by ....

....measures of the form D(P n ) # 0#=h#L # t w(h) or D(P n ) sup 0#=h#L # t w(h) 3. 4) with weights w(h) that decrease with #h# in a way that corresponds to how we think the squared Fourier coe#cients f(h) 2 decrease with #h#, and where # # is an arbitrary norm (see, e.g. [11, 15, 18, 38]) For a given choice of weights w(h) each of the two definitions of D(P n ) in (3.4) can be used as a criterion (to be minimized) for selecting a rule over a given set of lattice rules having n points. If the rule thus selected is used to integrate f and if the weights w(h) behave similarly to ....

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K. Entacher, P. Hellekalek, and P. L'Ecuyer, Quasi-Monte Carlo node sets from linear congruential generators, in Monte Carlo and Quasi-Monte Carlo Methods

....t . If P n has n points, it turns out that each coordinate of each vector of L t is a multiple of 1 n. One simple and convenient way to construct an integration lattice is by taking P n as the set of all t dimensional vectors of successive output values from a linear congruential generator (LCG) [10, 23, 32]; that is, take P n as the set of all vectors (u 0 , u t 1 ) where x 0 # Z n = 0, n 1 and the u i obey the recurrence x i = ax i 1 ) mod n, u i = x i n, 1.5) for some positive integer a in Z n . The corresponding integration rule Q n given by (1.2) was proposed by ....

....# t w(h) or D(P n ) sup 0#=h#L # t w(h) 3. 4) with weights w(h) that decrease with #h# in a way that corresponds to how we think the squared Fourier coe#cients f(h) 2 decrease with #h#, and where # # is an 6 CHRISTIANE LEMIEUX AND PIERRE L ECUYER arbitrary norm (see, e.g. [10, 14, 17, 37]) For a given choice of weights w(h) each of the two definitions of D(P n ) in (3.4) can be used as a criterion (to be minimized) for selecting a rule over a given set of lattice rules having n points. Most selection criteria in the literature are of the form (3.4) Examples are P# and the ....

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K. Entacher, P. Hellekalek, and P. L'Ecuyer, Quasi-Monte Carlo node sets from linear congruential generators, in Monte Carlo and Quasi-Monte Carlo Methods

.... integration errors of the different terms of this expansion, assuming that the Fourier series of f converges absolutely, one obtains E n = 1 n X 06=h2Z s f(h) n Gamma1 X i=0 exp(2 p Gamma1 h Delta u i ) 4) Examples of discrepancies yielding error bounds (3) via (4) can be found in [3,5,7,8,16,20]. We now suppose that P n is the intersection of a lattice L s with the unit 3 hypercube [0; 1) s . Then, Q n is called a lattice rule with node set P n , and (4) becomes [6,20] E n = X 06=h2L s f(h) 5) where L s = fh 2 Z s : k Delta h 2 Z for all k 2 L s g is the dual lattice to L s ....

....If w(h) 1=khk 2 , the inverse Euclidean norm of h, then minimizing the second D(P n ) with the sup) is equivalent to maximizing the Euclidean distance between the origin and the nearest nonzero point in L s . This criterion is identical to the spectral test which is commonly used to rate LCGs [3 5,9]. Lattice rules are typically chosen for a fixed s (i.e. different rules for different s) based on the minimization of some figure of merit of the form (6) 20] Tables of rules that are good uniformly for a range of values of s are given in [13] for several values of n that are either powers ....

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K. Entacher, P. Hellekalek, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. In Niederreiter and Spanier [18]. To appear.

....06=k2L kkk r oe Gammaff : 3. 12) For r = 2 this discrepancy is equivalent to the spectral test, commonly employed to measure the quality of linear congruential pseudo random number generators [30, 33] The spectral test has been used to select lattices for quasi Monte Carlo quadrature in [7, 34, 35, 36]. The case r = 1, which one might call an 1 spectral test is also interesting. We will return to these two cases in the next section. 4. Good Generating Vectors for Lattice Sequences. As mentioned at the beginning of the previous section, finding good generating vectors for lattices typically ....

K. Entacher, P. Hellekalek, and P. L' ' Ecuyer, Quasi-Monte Carlo node sets from linear congruential generators, in Niederreiter and Spanier [46].

....needed to generate all the points in L. The lattice rules that we use in the numerical examples of Sect. 6 all have rank 1. When the vector h defining a rank 1 lattice is of the form h = 1; a; a s Gamma1 ) mod N and N is prime, the rule is called a Korobov type rule. As mentioned in [9,22], this type of rule is closely related to linear congruential generators (LCGs) because, for these rules, P is equal to the set of all successive overlapping s tuples generated by an LCG with modulo N and multiplier a, from all possible initial seeds. When a is a primitive element modulo N , the ....

....select lattice rules, but seems harder to compute than P ff . For this reason, searches based on ae rarely extend to dimensions greater than 8 or 10. If we use the euclidean norm khk e = i P s j=1 jh j j 2 j 1=2 instead of the product norm k Delta k in the definition of ae (as suggested in [9]) we get d Gamma1 s = min h2L nf0g khk e ; where d s corresponds to the distance between the farthest family of hyperplanes that cover L, and is the quantity measured by the spectral test, a figure of merit commonly used for LCGs ( 18,19,16] See [9] for the connections between ae ....

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K. Entacher, P. Hellekalek, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. Submitted, 1998.

.... vectors needed to generate all of its points) When the generating vector is of the form h = 1; a; a s Gamma1 ) P is equal to the set of all successive overlapping s tuples produced by a linear congruential generator (LCG) with modulo N and multiplier a, from all possible initial seeds [7, 13, 6, 14]. Thus, if the LCG has maximal period, the points in P may be generated quickly by just running the LCG over its full period, taking the successive s tuples, and adding the origin. How should one choose h As mentioned in the introduction, the usual way to do this is to use a particular ....

....often used as a figure of merit to choose lattice rules [16] The properties of the lattice allow the derivation of a convenient formula for P ff , that can be calculated in O(Ns) time. One can also use the connection with LCGs to derive a selection criterion for P . For example, as suggested in [7], one might think of using the spectral test, a well known selection criterion for LCGs, that can be calculated efficiently. In practice, the spectral test value can be computed more quickly than P ff and one can also construct a more global figure of merit that takes account of the quality of the ....

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P. Hellekalek, K. Entacher, and P. L'Ecuyer. Quasi-Monte Carlo node sets from linear congruential generators. In Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, 1998. Submitted.

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