J. Beck and W. W. L. Chen, Irregularities of Distribution, Cambridge University Press, Cambridge, England, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(x i ) in [0, 1) with D # n (x 1 , x n ) O(log(n) n) Such sequences are called low discrepancy sequences, and some of them are described in chapter 1.5. It is suspected but not proven that infinite sequences cannot be constructed with D # n = o(log(n) n) see Beck and Chen [1]. In an infinite sequence, the first m points of x 1 , x n are the same for any m. If we knew in advance the value of n that we wanted then we might use a sequence customized for that value of n, such as x n1 , x nn , without insisting that x ni = x n 1 i . In this setting D ....

J. Beck and W. W. L. Chen. Irregularities of Distribution. Cambridge University Press, New York, 1987.

....theory. The second is in [AIKPS] and is completely elementary. Our construction is a variant of the second. Note that, due to the remark above, the sets A ;N we construct in this paper also satisfy (A ;N ) 2. 3 Higher dimensional discrepancy A central problem in Discrepancy Theory (see [BC] and the references within) is to determine the smallest size of sets in [0; 1) that have small discrepancy 3 with the family B of all aligned boxes (i.e. cartesian product of d intervals in [0,1) De ning the discrepancy DA (B) of a set A in the natural way, the question is how many points ....

J. Beck, W. Chen, Irregularities of Distributions, Cambridge University Press, 1987.

....a One way of doing this is to use variance reducing techniques sample such that 20 of the points from statistical sampling theory [85, 88] Other methods is within the polygon. use techniques from image sampling theory [57, 63] A third measure for the quality of sampling patterns is discrepancy [14]. Suppose that a pixel is partially covered by a polygon, as in Fig. 19. When we sample this pixel we would like the fraction of the pixel area covered by the polygon to be the same as the fraction of the sample points inside the polygon. The concept of discrepancy captures this type of quality ....

J. Beck and W. Chen. Irregularities of distribution. Cambridge University Press, 1987.

....N) 1.6) where the constant C d;q is independent of N . Constructions of distributions DN satisfying (1. 6) for 1 q 2 were given in dimensions d = 2 and d = 3 by Davenport [D] and Roth [Ro2] respectively, and in arbitrary dimensions by Frolov [Fr] and Roth [Ro3] Shortly thereafter, Chen [C1], C2] extended (1.6) to the interval 2 q 1. At a later time, other constructions for arbitrary exponents q were also given by Skriganov [Sk1] Notice that in the cited papers constructions of the distributions DN satisfying (1.6) are based either on lattices in Euclidean spaces with very ....

....also given by Skriganov [Sk1] Notice that in the cited papers constructions of the distributions DN satisfying (1. 6) are based either on lattices in Euclidean spaces with very special diophantine properties (cf. D] F2] Ro2] Sk1] or on specific sequences over finite fields and rings (cf. [C1], C2] Ro3] In all instances such distributions have rich inner structures. Hovewer, in dimensions d 3 all these constructions also involve additional probabilistic arguments and therefore are not explicit. Known explicit constructions in two dimensions can be found in [BC] DT] M] It ....

W.W.L.Chen, On irregularities of distributions, Mathematika 27 (1980), 153--170.

....[19] 4 ) We do not see straightforward extensions of our results to higher dimension. It would be natural to guess that the proper extension of the boundedness of the variance in particle number, in one dimension, might be that the variance is of the order of the surface area. In fact, J. Beck [20] proved that the variance in the particle number in a ball of radius r, averaged over r uniformly distributed in an interval (0; R) must grow at least like R d Gamma1 , and such a rate is realized in some cases of the OCP. However, examples of the OCP in d 2 dimensions show that surface rate ....

J. Beck, "Irregularities of distribution. I," Acta Math, 1, 159, (1987).

....by DN [5, 3] Furthermore observe that the value of DN resp. that of D (2) N will not change if we use closed boxes [0; x] instead of the half open boxes [0; x) Roth [7] proved that there are constants c k 0 such that DN (x n ) D (2) N (x n ) c k (log N) 1 2 (k 1) 4) In [1] these lower bounds are generalized in the following sense. The corresponding cube discrepancies DN (x n ; Q) sup y2[0;1) k ;0 1 j N (x n ; Q( y)j (5) and D (2) N (x n ; Q) Z 1 0 Z [0;1] k ( N (x n ; Q( y) 2 dy d 1 2 ; 6) where Q( 1 2 ; 1 2 ] k ....

....y2[0;1) 2 ;0 1 j (x n ; C y)j N 1 2 1 2k : 3 This lower bound is optimal despite of a logarithmic factor p log N It should be further mentioned that in the 2 dimensional case k = 2 there exist much more precise results. Schmidt [9] showed that DN (x n ) log N and Hal asz (see [1]) improved this bound to DN (x n ; Q) log N: Both bounds are optimal. Moreover, Beck [1] provided a bound of the form DN (x n ; C) log N) 1 2 for arbitrary convex sets C [0; 1) 2 . Therefore Theorem 3 is only new for dimensions k 2. 2 A Variation of Beck s Method First, let us note ....

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J. Beck and W. W. L. Chen, \Irregularities of Distribution," Cambridge University Press, Cambridge, 1987.

....two dimensional domain whereas N rooks samples are equidistributed in all one dimensional sub domains. However it is currently difficult to predict the behavior of quasi random samples. Although mathematicians have studied quasi random samples and related them to equidistribution and its measure [1, 28, 71, 83, 50, 4] the application of these mathematical results in computer graphics is still limited. In this chapter, we also generalize the idea of equidistribution and propose a new multi jittered sampling method [7] whose samples are equidistributed in any sub domain. In a two dimensional square domain, the ....

Jozsef Beck and William W.L. Chen. Irregularities of distribution. Cambridge University Press, Cambridge, 1987.

.... (n) Gamma Z [0;1] s f(u)du fi fi fi fi fi V (f)D N ( 1) Delta Delta Delta ; N) For an infinite sequence P, we can not have better than D N (P) O(N Gamma1 log N) for s = 1, and O(N Gamma1 (log N) ff(s) for any s (see [17, pages 23 25] 2, page 10] or [1] for a demonstration) Coefficient ff(s) verifies s=2 ff(s) s (and it is conjectured that ff(s) s) Sequences with such convergence rate are called low discrepancy sequences. Asymptotically, convergence is then quicker than for standard Monte Carlo method which, let us recall it, is in O(1= ....

J. Beck and W. Chen. -- Irregularities of Distribution. -- Cambridge University Press, 1987.

.... sequences in Monte Carlo methods 5 fi fi fi fi fi 1 N N X n=1 f( n) Gamma Z [0;1) s f(u)du fi fi fi fi fi V (f)D N (P) A sequence P = n) n2IN is said to be a low discrepancy sequence if and only if D N (P) O(N Gamma1 (ln N) s ) It has been proven by Roth [2] that, for infinite sequences, we can not have a better convergence speed than O(N Gamma1 (ln N) ff(s) for the error bound where ff(s) is such that ff(1) 1 and (s Gamma 1) 2 ff(s) s. It is conjectured in [3] that ff(s) s. There exists other error bounds having the same convergence ....

J. Beck and W. Chen. Irregularities of Distribution. Cambridge University Press, 1987.

....the i th coordinate of vector x n . For an infinite sequence P, we can not have better than D N (P) O(N Gamma1 log N) for s = 1, and O(N Gamma1 (log N) ff(s) for any s; Irisa Improvement of Halton sequences distribution 7 with s=2 ff(s) s (see [11, pages 23 25] 2, page 10] or [1] for a demonstration) The same remarks can be applied to T (p) N . Sequences with such convergence rates are called low discrepancy sequences. Asymptotically, the convergence is then faster than for the standard Monte Carlo method which, let us recall it, is in O(1= p N ) We can consider ....

J. Beck and W. Chen. -- Irregularities of Distribution. -- Cambridge University Press, 1987.

....X 2 ; X n with values in f1; 2; mg where Prob[X i = j] p i;j , 1 i n and 1 j m. In [EGLNV] constructions of small sample spaces were given which approximate the joint distribution of X 1 ; X 2 ; X n . To do so, they used the combinatorial notion of discrepancy, cf. [BC]. Let R n be the set of all axis aligned rectangles of the n dimensional cube [0; 1) n . For any finite set S ae [0; 1) n and any rectangle R 2 R n with volume vol(R) the discrepancy of S on R n is defined by disc S (R n ) sup R2Rn jvol(R) Gamma jS Rj=jSjj. A sample space S f1; 2; ....

J. Beck and W. Chen, Irregularities of Distribution, Cambridge University Press, 1987.

....be exceptionally intriguing. The reader interested in the fundamental aspects of the coding theory is refered to the encyclopaedic book MacWilliams and Sloane [17] and references therein. We recall basic facts in theory of uniform distributions. For necessary details we refer to Beck and Chen [4], Kuipers and Niederreiter [12] and a recent book Matousek [18] Let D # U n be a distribution of finitely many points in the n dimensional unit cube U n = 0, 1) n . The L# discrepancy L[D] is defined by L[D] sup Y #U n # D # B(Y ) # D volB(Y ) 1.1) where B(Y ) 0, y ....

J. Beck, W. W. L. Chen, Irregularities of Distributions, Cambridge Univ. Press, Cambridge (1987).

....we improve are lattice approximation, ffl approximations of range spaces of bounded VC exponent, sampling in geometric configuration spaces, and approximation of integer linear programs. 1 Introduction Problem and previous work. Discrepancy is an important concept in combinatorics, see e.g. [1, 5], and theoretical computer science, see e.g. 27, 23, 9] It attempts to capture the idea of a good sample from a set. The simplest example, the discrepancy problem, DP, considers a set system (X; S) where X is a ground set and S 2 X is a family of subsets of X . Then one is interested in a ....

.... independence guarantees that with nonzero probability for each S 2 S: fi fi fijR Sj Gamma jR Sj fi fi fi (jSj=2; 1=m) q 2jSj ln(2m) This DP has been extensively studied in combinatorics and combinatorial geometry (where S is determined from X IR d by specific geometric objects) [5, 1]. Computationally, it has also been object of extensive research [23, 25, 6] Because of its importance, we consider in detail the work and time requirements for its solution in NC. Also, it is shown in [23] that the DP can be used to solve the more general LAP problem. So if we are willing to ....

J. Beck and W. Chen. Irregularities of distribution. Cambridge University Press, 1987.

....with low discrepancy [MWW91] We want to color a set of n points in the plane by red and blue, such that every halfplane contains roughly the same number of red and blue points. How well can we achieve that goal This type of questions are investigated in the field of discrepancy ( Spe87] BC87] For technical reasons we switch to colors Gamma1 and 1. A coloring of a point set S is a mapping : S f Gamma1; 1g. The discrepancy of is defined as max h j (S h )j, where (A) P p2A (p) and the maximum is taken over all halfplanes h . Theorem 3.2 For every set S of n ....

J'ozsef Beck and William Chen. Irregularities of Distributions. Cambridge University Press, 1987.

....of size O(r 2 Gamma2= d 1) log r) 2 Gamma1= d 1) a similar bound is given there in terms of the socalled dual shatter function. Here also a close relation of approximations to the notion of discrepancy is observed, which is a classical object of study in combinatorics, see e. g, AS93] BC87] There are few geometric situations where the existence of (1=r) nets of size O(r) has been established (improving the general O(r log r) bound) most notably for halfspaces in IR 3 [MSW90] This has a nice algorithmic application in a polytope approximation problem; see Bronniman and Goodrich ....

J. Beck and W. Chen. Irregularities of Distribution. Cambridge University Press, 1987.

....1 Introduction Let P ae [0; 1] d be an n point set in the d dimensional unit cube. The discrepancy of P quantifies the irregularity of distribution of P in some sense; the more uniformly distributed P is, the smaller discrepancy. Discrepancy is a classical object of study (see e.g. 2] [1], or [4] for history and background) and it is closely related to error estimates in numerical integration (see e.g. 10] There are various notions of discrepancy; here we consider one of them, the so called L 2 discrepancy (or, more verbosely, the L 2 discrepancy for anchored boxes) Among ....

....is a mathematical gem proving the lower bound D 2 (2; n) c2 n p ln n for some c 2 0. A straightforward generalization of his proof to dimension d yields the asymptotically tight bound D 2 (d; n) c d n (ln n) d Gamma1) 2 . This proof is presented, among others, in the books [8] and [1]. Kuipers and Niederreiter [8] give an explicit estimate for the value of c d , namely c d 2 Gamma4d (log 2 e= d Gamma 1) d Gamma1) 2 (while Beck and Chen [1] do not write out an explicit bound) This already leads to a nontrivial lower bound p 1:00103 (by a calculation analogous to ....

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J. Beck and W. Chen. Irregularities of Distribution. Cambridge University Press, 1987.

....support for believing in the Beck Fiala conjecture itself. On the other hand, situations are known in geometric discrepancy theory where the pth degree average discrepancy is provably smaller than the worst case discrepancy (for instance, for axis parallel rectangles in the plane; see e.g. [3]) In combinatorial discrepancy, a natural case where disc p ( is asymptotically smaller than disc( is noted in [6] Also, the method of our proof of Theorem 1.1 seems to be inadequate for attacking the Beck Fiala conjecture. 2 Summary of the entropy method Let us define a partial coloring to ....

J. Beck and W. Chen. Irregularities of distribution. Cambridge University Press, 1987.

....all fractal sets. 2.2 Optimality for the L 2 discrepancy In this section, we show that the the lower bound for the L 2 discrepancy in Theorem 1 is asymptotically optimal for many fractals. To get a low discrepancy N point sequence, we use a random sampling similar to the one employed by Beck [4] in a slightly different context (let us remark that a similar sampling is used, too, in computer graphics, where it is called a jittered sampling) Let us consider self similar sets F with parameters a and fi 1 as defined in Section 1. Using the mappings 1 ; a , the set F is ....

Beck, J. Irregularities of distribution I. Acta Math., 159, 1--49 (1987).

....more recently, lot of work has also been devoted to investigating the discrepancy for other families, such as the balls or the halfspaces in R d , and so on. The history and overview of discrepancy theory are presented, among others, in the survey Beck and S os [10] in the book Beck and Chen [7] or, more recently, in Drmota and Tichy [14] The above defined notion of discrepancy will be called the Lebesgue measure discrepancy , in order to distinguish it from a seemingly different but closely related notion of combinatorial Research supported by Czech Republic Grant GA CR 0194 and by ....

.... log n) This theorem is proved in Section 4. Note that both part (i) and part (ii) imply Theorem 1.2, and so in order to improve the dependence on n for A or H fixed one would have to improve the bounds for Tusn ady s problem as well. In conclusion, let us remark that Beck [4] see also [7]) used bounds on the discrepancy for convex gons in an investigation of the Lebesgue measure discrepancy D(n; TA ) for an arbitrary convex set A in the plane. He discovered that the behavior of this quantity depends mainly on the smoothness of the boundary of A, and that the smoothness can be ....

J. Beck and W. Chen. Irregularities of Distribution. Cambridge University Press, 1987.

....intervals (i.e. axis parallel boxes in R d ) but more recently, discrepancy has been investigated for other families as well, such as the balls or the halfspaces in R d . The history and overview of discrepancy theory are presented, for example, in the surveys [13, 3] and in the books [12, 19, 26]. The above defined notion of discrepancy will be called the Lebesgue measure discrepancy , in order to distinguish it from combinatorial discrepancy , to be defined next. Let Research supported by Charles University grants No. 193,194. X be a finite set and let S 2 X be a family of ....

....by O(n 1=2 Gamma1=2d p log n) although the combinatorial discrepancy is n 2 and the dual shatter function is exponential. Here the Lebesgue measure discrepancy bound is usually derived by approximation and decomposition arguments using auxiliary families consisting of Tarski cells (see [12], 18] for examples) In the present paper, we do not consider such families. For many families of Tarski cells in R d , there is a lower bound of Omega Gamma n 1=2 Gamma1=2d ) for the discrepancy, and for those families, the upper bound derived from the dual shatter function provides a quite ....

J. Beck and W. L. Chen. Irregularities of Distribution. Cambridge University Press, Cambridge, 1987.

....be placed in the unit cube, with respect to a given collection of simple geometric shapes (such as axis parallel boxes, balls, convex sets etc. Discrepancy theory is a rich discipline today, with lots of difficult open problems, and here we can only touch it (more comprehensive sources are [BC87] BS95] discrepancy from a combinatorial point of view is treated in some chapters of [Spe87] AS93] PA95] The whole discrepancy theory, or theory of irregularities of distribution, starts with the following Van Aardenne Ehrenfest theorem (proving a conjecture of Van der Corput; see [BC87] ....

.... [BC87] BS95] discrepancy from a combinatorial point of view is treated in some chapters of [Spe87] AS93] PA95] The whole discrepancy theory, or theory of irregularities of distribution, starts with the following Van Aardenne Ehrenfest theorem (proving a conjecture of Van der Corput; see [BC87] for references on earlier work in discrepancy theory) For each infinite sequence of real numbers in the interval [0; 1] and for any k 0 there exists an initial segment (x 1 ; x n ) of the considered sequence and a subinterval (ff; fi) 0; 1] such that the number of elements of fx 1 ....

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J. Beck and W. Chen. Irregularities of distribution. Cambridge University Press, 1987.

....to use sampling patterns with high frequency spectrum. These patterns drive aliasing noise to higher frequencies, where it is less visible, and can be reduced with supersampling. One promising approach is the application of the theory of discrepancy or irregularities of distribution, introduced by [BC], and applied to computer graphics first by [S] and [N92] The discrepancy theory focuses on the problem of approximating one measure (typically a continuous one) with another (typically a discrete one) It s main application is in Quasi Monte Carlo numerical integration, where we use a point set ....

....Results by [Ko] and [TW] which show that the error in evaluating an integral (under bounded variation conditions) is proportional to the discrepancy of the sampling point set, further justify its use. A lot of work has to be done to produce point sets with low discrepancy in this model (see [BC]) and in fact almost optimal sequences (Hammersley points, Hal] are known. Algorithms that compute the exact or approximate maximum numerical discrepancy of point sets are useful however to compare point sets, to find patterns with very low discrepancy and to produce point sets when other ....

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J. Beck and W.W.L. Chen, Irregularities of distribution. Cambridge University Press (1987).

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J. Beck and W. W. L. Chen, Irregularities of Distribution, Cambridge University Press, Cambridge, England, 1987.

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Beck J., Chen W. L. (1987): Irregularities of Distribution. Cambridge Univ. Press.

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Beck, J., Chen, W., "Irregularities of distribution", Cambridge University Press, 1987.

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