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43
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 91 (11 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.
Monte Carlo Techniques for Direct Lighting Calculations
 ACM Transactions on Graphics
, 1996
"... In a distribution ray tracer, the crucial part of the direct lighting calculation is the sampling strategy for shadow ray testing. Monte Carlo integration with importance sampling is used to carry out this calculation. Importance sampling involves the design of integrandspecific probability density ..."
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Cited by 86 (8 self)
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In a distribution ray tracer, the crucial part of the direct lighting calculation is the sampling strategy for shadow ray testing. Monte Carlo integration with importance sampling is used to carry out this calculation. Importance sampling involves the design of integrandspecific probability density functions which are used to generate sample points for the numerical quadrature. Probability density functions are presented that aid in the direct lighting calculation from luminaires of various simple shapes. A method for defining a probability density function over a set of luminaires is presented that allows the direct lighting calculation to be carried out with one sample, regardless of the number of luminaires. CR Categories and Subject Descriptors: G.1.4 [Mathematical Computing]: Quadrature and Numerical Differentiation; I.3.0 [Computer Graphics]: General; I.3.7 [Computer Graphics]: ThreeDimensional Graphics and Realism. Additional Key Words and Phrases: direct lighting, importanc...
Using Randomization to Break the Curse of Dimensionality
 Econometrica
, 1997
"... Abstract: This paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon Markovian decision problems (MDPs). We prove that these algorithms succeed in breaking the “curse of dimensionality ” fo ..."
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Cited by 85 (0 self)
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Abstract: This paper introduces random versions of successive approximations and multigrid algorithms for computing approximate solutions to a class of finite and infinite horizon Markovian decision problems (MDPs). We prove that these algorithms succeed in breaking the “curse of dimensionality ” for a subclass of MDPs known as discrete decision processes (DDPs). 1
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 73 (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...
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 67 (5 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
Explicit Cost Bounds of Algorithms for Multivariate Tensor Product Problems
 J. Complexity
, 1994
"... We study multivariate tensor product problems in the worst case and average case settings. They are defined on functions of d variables. For arbitrary d, we provide explicit upper bounds on the costs of algorithms which compute an "approximation to the solution. The cost bounds are of the form (c( ..."
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Cited by 44 (10 self)
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We study multivariate tensor product problems in the worst case and average case settings. They are defined on functions of d variables. For arbitrary d, we provide explicit upper bounds on the costs of algorithms which compute an "approximation to the solution. The cost bounds are of the form (c(d) + 2) fi 1 ` fi 2 + fi 3 ln 1=" d \Gamma 1 ' fi 4 (d\Gamma1) ` 1 " ' fi 5 : Here c(d) is the cost of one function evaluation (or one linear functional evaluation), and fi i 's do not depend on d; they are determined by the properties of the problem for d = 1. For certain tensor product problems, these cost bounds do not exceed c(d) K " \Gammap for some numbers K and p, both independent of d. We apply these general estimates to certain integration and approximation problems in the worst and average case settings. We also obtain an upper bound, which is independent of d, for the number, n("; d), of points for which discrepancy (with unequal weights) is at most ", n("; d) 7:26 ...
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 42 (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...
Computing the Maximum Bichromatic Discrepancy, with applications to Computer Graphics and Machine Learning
 in Computer Graphics and Machine Learning. Journal of Computer and Systems Sciences
, 1996
"... Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, ..."
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Cited by 39 (8 self)
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Computing the maximum bichromatic discrepancy is an interesting theoretical problem with important applications in computational learning theory, computational geometry and computer graphics. In this paper we give algorithms to compute the maximum bichromatic discrepancy for simple geometric ranges, including rectangles and halfspaces. In addition, we give extensions to other discrepancy problems. 1. Introduction The main theme of this paper is to present efficient algorithms that solve the problem of computing the maximum bichromatic discrepancy for axis oriented rectangles. This problem arises naturally in different areas of computer science, such as computational 1 The research work of these authors was supported by NSF Grant CCR9301254 and the Geometry Center. learning theory, computational geometry and computer graphics ([Ma], [DG]), and has applications in all these areas. In computational learning theory, the problem of agnostic PAClearning with simple geometric hypothese...
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 35 (1 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
Methods for Approximating Integrals in Statistics with Special Emphasis on Bayesian Integration Problems
 Statistical Science
"... This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain method ..."
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Cited by 32 (4 self)
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This paper is a survey of the major techniques and approaches available for the numerical approximation of integrals in statistics. We classify these into five broad categories; namely, asymptotic methods, importance sampling, adaptive importance sampling, multiple quadrature and Markov chain methods. Each method is discussed giving an outline of the basic supporting theory and particular features of the technique. Conclusions are drawn concerning the relative merits of the methods based on the discussion and their application to three examples. The following broad recommendations are made. Asymptotic methods should only be considered in contexts where the integrand has a dominant peak with approximate ellipsoidal symmetry. Importance sampling, and preferably adaptive importance sampling, based on a multivariate Student should be used instead of asymptotics methods in such a context. Multiple quadrature, and in particular subregion adaptive integration, are the algorithms of choice for...