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14
When are QuasiMonte Carlo Algorithms Efficient for High Dimensional Integrals?
 J. Complexity
, 1997
"... Recently quasiMonte Carlo algorithms have been successfully used for multivariate integration of high dimension d, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasiMonte Carlo algorithms does not explain this phenomenon. ..."
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Cited by 188 (23 self)
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Recently quasiMonte Carlo algorithms have been successfully used for multivariate integration of high dimension d, and were significantly more efficient than Monte Carlo algorithms. The existing theory of the worst case error bounds of quasiMonte Carlo algorithms does not explain this phenomenon. This paper presents a partial answer to why quasiMonte Carlo algorithms can work well for arbitrarily large d. It is done by identifying classes of functions for which the effect of the dimension d is negligible. These are weighted classes in which the behavior in the successive dimensions is moderated by a sequence of weights. We prove that the minimal worst case error of quasiMonte Carlo algorithms does not depend on the dimension d iff the sum of the weights is finite. We also prove that under this assumption the minimal number of function values in the worst case setting needed to reduce the initial error by " is bounded by C " \Gammap , where the exponent p 2 [1; 2], and C depends ...
Numerical Integration using Sparse Grids
 NUMER. ALGORITHMS
, 1998
"... We present and review algorithms for the numerical integration of multivariate functions defined over ddimensional cubes using several variants of the sparse grid method first introduced by Smolyak [51]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor ..."
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Cited by 91 (16 self)
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We present and review algorithms for the numerical integration of multivariate functions defined over ddimensional cubes using several variants of the sparse grid method first introduced by Smolyak [51]. In this approach, multivariate quadrature formulas are constructed using combinations of tensor products of suited onedimensional formulas. The computing cost is almost independent of the dimension of the problem if the function under consideration has bounded mixed derivatives. We suggest the usage of extended Gauss (Patterson) quadrature formulas as the onedimensional basis of the construction and show their superiority in comparison to previously used sparse grid approaches based on the trapezoidal, ClenshawCurtis and Gauss rules in several numerical experiments and applications.
The Curse of Dimension and a Universal Method For Numerical Integration
 in Multivariate Approximation and Splines
, 1998
"... . Many high dimensional problems are difficult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension, however, can be compensated by a high degree of smoothness. We study numeri ..."
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Cited by 29 (3 self)
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. Many high dimensional problems are difficult to solve for any numerical method. This curse of dimension means that the computational cost must increase exponentially with the dimension of the problem. A high dimension, however, can be compensated by a high degree of smoothness. We study numerical integration and prove that such a compensation is possible by a recently invented method. The method is shown to be universal, i.e., simultaneously optimal up to logarithmic factors, on two different smoothness scales. The first scale is defined by isotropic smoothness conditions, while the second scale involves anisotropic smoothness and is related to partially separable functions. 1. Introduction Several applications require the computation of high dimensional integrals. They are present, for example, in statistical mechanics, see [28] for an introduction. Another important example is the fast valuation of financial derivatives, see [16]. Some applications even require approxima...
A Multigrid Algorithm For Higher Order Finite Elements On Sparse Grids
 ETNA
, 1997
"... . For most types of problems in numerical mathematics, efficient discretization techniques are of crucial importance. This holds for tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as accurate as possible as well as for the numerical sol ..."
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Cited by 6 (1 self)
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. For most types of problems in numerical mathematics, efficient discretization techniques are of crucial importance. This holds for tasks like how to define sets of points to approximate, interpolate, or integrate certain classes of functions as accurate as possible as well as for the numerical solution of differential equations. Introduced by Zenger in 1990 and based on hierarchical tensor product approximation spaces, sparse grids have turned out to be a very efficient approach in order to improve the ratio of invested storage and computing time to the achieved accuracy for many problems in the areas mentioned above. Concerning the sparse grid finite element discretization of elliptic partial differential equations, recently, the class of problems that can be tackled has been enlarged significantly. First, the tensor product approach led to the formulation of unidirectional algorithms which are essentially independent of the number d of dimensions. Second, techniques for the treatm...
Asymptotically Optimal Weighted Numerical Integration
, 1997
"... We study numerical integration of Höldertype functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma whi ..."
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We study numerical integration of Höldertype functions with respect to weights on the real line. Our study extends previous work by F. Curbera, [2] and relies on a connection between this problem and the approximation of distribution functions by empirical ones. The analysis is based on a lemma which is important within the theory of optimal designs for approximating stochastic processes. As an application we reproduce a variant of the well known result for weighted integration of Brownian paths, see e.g., [8].
On the L 2 discrepancy for anchored boxes
"... The L 2 discrepancy for anchored axisparallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of th ..."
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Cited by 5 (1 self)
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The L 2 discrepancy for anchored axisparallel boxes has been used in several recent computational studies, mostly related to numerical integration, as a measure of the quality of uniform distribution of a given point set. We point out that if the number of points is not large enough in terms of the dimension (e.g., fewer than 10 4 points in dimension 30) then nearly the lowest possible L 2 discrepancy is attained by a pathological point set, and hence the L 2 discrepancy may not be very relevant for relatively small sets. Recently, Hickernell obtained a formula for the expected L 2 discrepancy of certain randomized lowdiscrepancy set constructions introduced by Owen. We note that his formula remains valid also for several modifications of these constructions which admit a very simple and efficient implementation. We also report results of computational experiments with various constructions of lowdiscrepancy sets. Finally, we present a fairly precise formula for the performanc...
Finding optimal volume subintervals with k points and computing the star discrepancy are NPhard problems
 JOURNAL OF COMPLEXITY
, 2008
"... The wellknown star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics. We study here the complexity of calculating the star discrepancy of po ..."
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The wellknown star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics. We study here the complexity of calculating the star discrepancy of point sets in the ddimensional unit cube and show that this is an NPhard problem. To establish this complexity result, we first prove NPhardness of the following related problems in computational geometry: Given n points in the ddimensional unit cube, find a subinterval of minimum or maximum volume that contains k of the n points. Our results for the complexity of the subinterval problems settle a conjecture of
Fibonacci sets and symmetrization in discrepancy theory
"... We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn} ∞ n=0 be the sequence of Fibonacci numbers. The bnpoint Fibonacci set F n ⊂ [0, 1] 2 is defined as Fn: = {(µ/bn, {µbn−1/bn})} bn µ=1, where {x} is the fractional part ..."
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We study the Fibonacci sets from the point of view of their quality with respect to discrepancy and numerical integration. Let {bn} ∞ n=0 be the sequence of Fibonacci numbers. The bnpoint Fibonacci set F n ⊂ [0, 1] 2 is defined as Fn: = {(µ/bn, {µbn−1/bn})} bn µ=1, where {x} is the fractional part of a number x ∈ R. It is known that cubature formulas based on Fibonacci set Fn give optimal rate of error of numerical integration for certain classes of functions with mixed smoothness. We give a Fourier analytic proof of the fact that the symmetrized Fibonacci set F ′ n = Fn ∪ {(p1, 1 − p2) : (p1, p2) ∈ Fn} has asymptotically minimal L2 discrepancy. This approach also yields an exact formula for this quantity, which allows us to evaluate the constant in the discrepancy estimates. Numerical computations indicate that these sets have the smallest currently known L2 discrepancy among twodimensional point sets. We also introduce quartered Lp discrepancy which is a modification of the Lp discrepancy symmetrized with respect to the center of the unit square. We prove that the Fibonacci set Fn has minimal in the sense of order quartered Lp discrepancy for all p ∈ (1, ∞). This in turn implies that certain twofold symmetrizations of the Fibonacci set Fn are optimal with respect to the standard Lp discrepancy.
A new randomized algorithm to approximate the star discrepancy based on threshold accepting
 SIAM J. Numer. Anal
"... ar ..."
Computing Discrepancies Related to Spaces of Smooth Periodic Functions
"... A notion of discrepancy is introduced, which represents the integration error on spaces of rsmooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2  discrepancy as well as rsmooth versions of it introduced recently by Paskov [Pas93]. Based ..."
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A notion of discrepancy is introduced, which represents the integration error on spaces of rsmooth periodic functions. It generalizes the diaphony and constitutes a periodic counterpart to the classical L2  discrepancy as well as rsmooth versions of it introduced recently by Paskov [Pas93]. Based on previous work [FH96], we develop an efficient algorithm for computing periodic discrepancies for quadrature formulas possessing certain tensor product structures, in particular, for Smolyak quadrature rules (also called sparse grid methods). Furthermore, fast algorithms of computing periodic discrepancies for lattice rules can easily be derived from wellknown properties of lattices. On this basis we carry out numerical comparisons of discrepancies between Smolyak and lattice rules. 1 Introduction Discrepancies are a quantitative measure of the precision of multivariate quadratures. Their computation, however, often is a very complex task. Therefore algorithms are of interest which red...