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528
Nonconventional ergodic averages and nilmanifolds
"... Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cube ..."
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Cited by 156 (16 self)
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Abstract. We study the L2convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions appearing in Furstenberg’s proof of Szemerédi’s Theorem. The second average is taken along cubes whose sizes tend to +∞. For each average, we show that it is sufficient to prove the convergence for special systems, the characteristic factors. We build these factors in a general way, independent of the type of the average. To each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold. From the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density. 1.
Monte Carlo and QuasiMonte Carlo methods
 Acta Numerica
, 1998
"... Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N ~ 1 ^ 2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including conve ..."
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Cited by 104 (3 self)
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Monte Carlo is one of the most versatile and widely used numerical methods. Its convergence rate, O(N ~ 1 ^ 2), is independent of dimension, which shows Monte Carlo to be very robust but also slow. This article presents an introduction to Monte Carlo methods for integration problems, including convergence theory, sampling methods and variance reduction techniques. Accelerated convergence for Monte Carlo quadrature is attained using quasirandom (also called lowdiscrepancy) sequences, which are a deterministic alternative to random or pseudorandom sequences. The points in a quasirandom sequence are correlated to provide greater uniformity. The resulting quadrature method, called quasiMonte Carlo, has a convergence rate of approximately O((log N^N ' 1). For quasiMonte Carlo, both theoretical error estimates and practical limitations are presented. Although the emphasis in this article is on integration, Monte Carlo simulation of rarefied gas dynamics is also discussed. In the limit of small mean free path (that is, the fluid dynamic limit), Monte Carlo loses its effectiveness because the collisional distance is much less
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 92 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
QuasiMonte Carlo Integration
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1995
"... The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved con ..."
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Cited by 72 (6 self)
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The standard Monte Carlo approach to evaluating multidimensional integrals using (pseudo)random integration nodes is frequently used when quadrature methods are too difficult or expensive to implement. As an alternative to the random methods, it has been suggested that lower error and improved convergence may be obtained by replacing the pseudorandom sequences with more uniformly distributed sequences known as quasirandom. In this paper the Halton, Sobol' and Faure quasirandom sequences are compared in computational experiments designed to determine the effects on convergence of certain properties of the integrand, including variance, variation, smoothness and dimension. The results show that variation, which plays an important role in the theoretical upper bound given by the KoksmaHlawka inequality, does not affect convergence; while variance, the determining factor in random Monte Carlo, is shown to provide a rough upper bound, but does not accurately predict performance. In ge...
Variance Reduction via Lattice Rules
 Management Science
"... All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. ..."
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Cited by 64 (13 self)
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All intext references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately.
Asymptotic estimates for best and stepwise approximation of convex bodies III
, 1997
"... . We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the difference of the mean width of the convex body and the approximating polytopes. The following results ..."
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Cited by 61 (5 self)
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. We consider approximations of a smooth convex body by inscribed and circumscribed convex polytopes as the number of vertices, resp. facets tends to infinity. The measure of deviation used is the difference of the mean width of the convex body and the approximating polytopes. The following results are obtained. (i) An asymptotic formula for best approximation. (ii) Upper and lower estimates for stepby step approximation in terms of the socalled dispersion. (iii) For a sequence of best approximating inscribed polytopes the sequence of vertex sets is uniformly distributed in the boundary of the convex body where the density is specified explicitly. 1. Introduction and statement of results 1.1 Let C denote the space of convex bodies in Euclidean dspace IE d , i.e. of all compact convex subsets of IE d with nonempty interior. For notions not explained below we refer to [20]. Given C 2 C and k = 0; : : : ; d, let W k (C) be the kth quermassintegral of C. W 0 = V is the volume, dW 1...
On the Random Character of Fundamental Constant Expansions
 EXPERIMENTAL MATHEMATICS
, 2001
"... We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 no ..."
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Cited by 59 (15 self)
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We propose a theory to explain random behavior for the digits
in the expansions of fundamental mathematical constants. At
the core of our approach is a general hypothesis concerning the distribution of the iterates generated by dynamical maps. On this main hypothesis, one obtains proofs of base2 normality—namely bit randomness in a specific technical sense— for a collection of celebrated constants, including , log 2, (3), and others. Also on the hypothesis, the number (5) is either rational or normal to base 2. We indicate a research connection between our dynamical model and the theory of pseudorandom number
generators.
Sixty years of Bernoulli convolutions
 In: Fractal Geometry and Stochastics, II (Greifswald/Koserow, 1998) Progress in Probability
, 2000
"... ..."
Quasirandom methods for estimating integrals using relatively small samples
 SIAM Review
, 1994
"... Abstract. Much of the recent work dealing with quasirandom methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finitedimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to ..."
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Cited by 53 (1 self)
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Abstract. Much of the recent work dealing with quasirandom methods has been aimed at establishing the best possible asymptotic rates of convergence to zero of the error resulting when a finitedimensional integral is replaced by a finite sum of integrand values. In contrast with this perspective to concentrate on asymptotic convergence rates, this paper emphasizes quasirandom methods that are effective for all sample sizes. Throughout the paper, the problem of estimating finitedimensional integrals is used to illustrate the major ideas, although much of what is done applies equally to the problem of solving certain Fredholm integral equations. Some new techniques, based on errorreducing transformations of the integrand, are described that have been shown to be useful both in estimating highdimensional integrals and in solving integral equations. These techniques illustrate the utility of carrying over to the quasiMonte Carlo method certain devices that have proven to be very valuable in statistical (pseudorandom) Monte Carlo applications. Key words, quasiMonte Carlo, asymptotic rate of convergence, numerical integration
Invariant Distributions and Time Averages for Horocycle Flows
 Duke J. of Math
"... There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu = f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regula ..."
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Cited by 47 (9 self)
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There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu = f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the