Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation
Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasi-Monte Carlo set of tools. A theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper....
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| Veröffentlicht in: | SIAM journal on scientific computing Jg. 24; H. 5; S. 1768 - 1789 |
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| Sprache: | Englisch |
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Philadelphia, PA
Society for Industrial and Applied Mathematics
01.01.2003
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| Abstract | Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasi-Monte Carlo set of tools. A theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, are special cases of this method. |
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| AbstractList | Lattice rules are among the best methods to estimate integrals in a large number of dimensions. They are part of the quasi-Monte Carlo set of tools. A theoretical framework for a class of lattice rules defined in a space of polynomials with coefficients in a finite field is developed in this paper. A randomized version is studied, implementations and criteria for selecting the parameters are discussed, and examples of its use as a variance reduction tool in stochastic simulation are provided. Certain types of digital net constructions, as well as point sets constructed by taking all vectors of successive output values produced by a Tausworthe random number generator, are special cases of this method. |
| Author | Lemieux, Christiane L'Ecuyer, Pierre |
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| Keywords | Polynomial Finite field Monte Carlo method Integration Numerical analysis Integral Stochastic method Random number generation Implementation Variance |
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| References | Caflisch R. E. (R3) 1997; 1 Beauchamp K. (R2) 1984 R41 R40 R21 R23 R22 R24 R27 R29 Hellekalek Peter (R15) 1997; 80 R28 Cassels J. (R4) 1997 Korobov N. M. (R19) 1959; 124 R1 Davis Philip (R8) 1984 R5 R7 R9 Sloan I. (R34) 1994 R30 R32 Couture Raymond (R6) 1993; 60 R12 R36 R13 Knuth Donald (R18) 1981 R35 R16 |
| References_xml | – volume: 60 start-page: 0 year: 1993 ident: R6 publication-title: Math. Comp. – volume-title: An introduction to the geometry of numbers year: 1997 ident: R4 – ident: R27 doi: 10.1016/S0378-4754(00)00254-8 – ident: R22 doi: 10.1090/S0025-5718-1994-1234426-9 – ident: R28 doi: 10.1137/S0036144597317959 – ident: R36 doi: 10.1137/1036002 – ident: R32 doi: 10.1145/272991.273010 – ident: R30 doi: 10.1137/1.9781611970081 – ident: R23 doi: 10.1090/S0025-5718-96-00696-5 – ident: R35 doi: 10.1016/0041-5553(67)90144-9 – volume-title: Methods of numerical integration year: 1984 ident: R8 – volume-title: Lattice methods for multiple integration year: 1994 ident: R34 – ident: R13 doi: 10.1007/978-94-011-3288-6 – ident: R7 doi: 10.1137/0713071 – ident: R29 doi: 10.1007/BF01294651 – ident: R40 doi: 10.1145/116890.116892 – volume: 124 start-page: 1207 issn: 0002-3264 year: 1959 ident: R19 publication-title: Dokl. Akad. Nauk SSSR – ident: R24 doi: 10.1287/mnsc.46.9.1214.12231 – ident: R21 doi: 10.1007/BF01298952 – ident: R41 doi: 10.1145/321765.321778 – volume-title: Applications of Walsh and related functions year: 1984 ident: R2 – volume-title: The art of computer programming. Vol. 2 year: 1981 ident: R18 – volume: 80 start-page: 187 year: 1997 ident: R15 publication-title: Acta Arith. doi: 10.4064/aa-80-2-187-196 – ident: R12 doi: 10.1145/358150.358159 – ident: R16 doi: 10.1090/S0025-5718-98-00894-1 – ident: R5 doi: 10.1090/S0025-5718-99-01112-6 – ident: R9 doi: 10.1214/aos/1176345462 – volume: 1 start-page: 27 year: 1997 ident: R3 publication-title: J. Comput. Finance doi: 10.21314/JCF.1997.005 – ident: R1 doi: 10.1287/opre.44.2.327 |
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| SubjectTerms | Algebra Combinatorics. Ordered structures Estimates Exact sciences and technology Mathematical analysis Mathematics Measure and integration Monte Carlo simulation Number theory Numerical analysis Numerical analysis. Scientific computation Numerical methods in probability and statistics Order, lattices, ordered algebraic structures Random variables Scholarships & fellowships Sciences and techniques of general use |
| Title | Randomized Polynomial Lattice Rules for Multivariate Integration and Simulation |
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