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241
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 183 (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 ...
A generalized discrepancy and quadrature error bound
 Math. Comp
, 1998
"... Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which dep ..."
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Cited by 140 (13 self)
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Abstract. An error bound for multidimensional quadrature is derived that includes the KoksmaHlawka inequality as a special case. This error bound takes the form of a product of two terms. One term, which depends only on the integrand, is defined as a generalized variation. The other term, which depends only on the quadrature rule, is defined as a generalized discrepancy. The generalized discrepancy is a figure of merit for quadrature rules and includes as special cases the L pstar discrepancy and Pα that arises in the study of lattice rules.
On the Relationship Between Classical Grid Search and Probabilistic Roadmaps
"... We present, implement, and analyze a spectrum of closelyrelated planners, designed to gain insight into the relationship between classical grid search and probabilistic roadmaps (PRMs). Building on quasiMonte Carlo sampling literature, we have developed deterministic variants of the PRM that use ..."
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Cited by 136 (10 self)
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We present, implement, and analyze a spectrum of closelyrelated planners, designed to gain insight into the relationship between classical grid search and probabilistic roadmaps (PRMs). Building on quasiMonte Carlo sampling literature, we have developed deterministic variants of the PRM that use lowdiscrepancy and lowdispersion samples, including lattices. Classical grid search is extended using subsampling for collision detection and also the optimaldispersion Sukharev grid, which can be considered as a kind of latticebased roadmap to complete the spectrum. Our experimental results show that the deterministic variants of the PRM offer performance advantages in comparison to the original PRM and the recent Lazy PRM. This even includes searching using a grid with subsampled collision checking. Our theoretical analysis shows that all of our deterministic PRM variants are resolution complete and achieve the best possible asymptotic convergence rate, which is shown superior to that obtained by random sampling. Thus, in surprising contrast to recent trends, there is both experimental and theoretical evidence that some forms of grid search are superior to the original PRM.
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
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Cited by 119 (6 self)
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Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Latin Supercube Sampling for Very High Dimensional Simulations
, 1997
"... This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables ..."
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Cited by 84 (8 self)
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This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...
Methods for the Computation of Multivariate tProbabilities
 Computing Sciences and Statistics
, 2000
"... This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems ..."
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Cited by 83 (11 self)
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This paper compares methods for the numerical computation of multivariate tprobabilities for hyperrectangular integration regions. Methods based on acceptancerejection, sphericalradial transformations and separationofvariables transformations are considered. Tests using randomly chosen problems show that the most efficient numerical methods use a transformation developed by Genz (1992) for multivariate normal probabilities. These methods allow moderately accurate multivariate tprobabilities to be quickly computed for problems with as many as twenty variables. Methods for the noncentral multivariate tdistribution are also described. Key Words: multivariate tdistribution, noncentral distribution, numerical integration, statistical computation. 1 Introduction A common problem in many statistics applications is the numerical computation of the multivariate t (MVT) distribution function (see Tong, 1990) defined by T(a; b; \Sigma; ) = \Gamma( +m 2 ) \Gamma( 2 ) p j\Sigma...
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 82 (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.
QuasiRandomized Path Planning
 In Proc. IEEE Int’l Conf. on Robotics and Automation
, 2001
"... We propose the use of quasirandom sampling techniques for path planning in highdimensional conguration spaces. Following similar trends from related numerical computation elds, we show several advantages oered by these techniques in comparison to random sampling. Our ideas are evaluated in the con ..."
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Cited by 78 (8 self)
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We propose the use of quasirandom sampling techniques for path planning in highdimensional conguration spaces. Following similar trends from related numerical computation elds, we show several advantages oered by these techniques in comparison to random sampling. Our ideas are evaluated in the context of the probabilistic roadmap (PRM) framework. Two quasirandom variants of PRMbased planners are proposed: 1) a classical PRM with quasirandom sampling, and 2) a quasirandom LazyPRM. Both have been implemented, and are shown through experiments to oer some performance advantages in comparison to their randomized counterparts. 1 Introduction Over two decades of path planning research have led to two primary trends. In the 1980s, deterministic approaches provided both elegant, complete algorithms for solving the problem, and also useful approximate or incomplete algorithms. The curse of dimensionality due to highdimensional conguration spaces motivated researchers from the 199...
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.