A Comparison of Monte Carlo, Lattice Rules and Other Low-Discrepancy Point Sets

2000 ◽  
pp. 326-340 ◽  
Author(s):  
Christiane Lemieux ◽  
Pierre L’Ecuyer
Keyword(s):  
Monte Carlo ◽  
Point Sets ◽  
Lattice Rules

scholarly journals Derandomised lattice rules for high dimensional integration

2019 ◽  
Vol 60 ◽  
pp. C247-C260
Author(s):  
Y. Kazashi ◽  
F. Y. Kuo ◽  
I. H. Sloan
Keyword(s):  
Monte Carlo ◽  
Sobolev Spaces ◽  
Monte Carlo Integration ◽  
High Dimensional ◽  
Lattice Rules ◽  
Worst Case ◽  
Multiple Integration ◽  
Quasi Monte Carlo ◽  
Lattice Methods ◽  
Numer Math

We seek shifted lattice rules that are good for high dimensional integration over the unit cube in the setting of an unanchored weighted Sobolev space of functions with square-integrable mixed first derivatives. Many existing studies rely on random shifting of the lattice, whereas here we work with lattice rules with a deterministic shift. Specifically, we consider 'half-shifted' rules in which each component of the shift is an odd multiple of \(1/(2N)\) where \(N\) is the number of points in the lattice. By applying the principle that there is always at least one choice as good as the average, we show that for a given generating vector there exists a half-shifted rule whose squared worst-case error differs from the shift-averaged squared worst-case error by a term of only order \({1/N^2}\). We carry out numerical experiments where the generating vector is chosen component-by-component (CBC), as for randomly shifted lattices, and where the shift is chosen by a new `CBC for shift' algorithm. The numerical results are encouraging. References J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-Monte Carlo way. Acta Numer., 22:133–288, 2013. doi:10.1017/S0962492913000044. J. Dick, D. Nuyens, and F. Pillichshammer. Lattice rules for nonperiodic smooth integrands. Numer. Math., 126(2):259–291, 2014. doi:10.1007/s00211-013-0566-0. T. Goda, K. Suzuki, and T. Yoshiki. Lattice rules in non-periodic subspaces of sobolev spaces. Numer. Math., 141(2):399–427, 2019. doi:10.1007/s00211-018-1003-1. F. Y. Kuo. Lattice rule generating vectors. URL http://web.maths.unsw.edu.au/ fkuo/lattice/index.html. D. Nuyens and R. Cools. Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput., 75:903–920, 2006. doi:10.1090/S0025-5718-06-01785-6. I. H. Sloan and S. Joe. Lattice methods for multiple integration. Oxford Science Publications. Clarendon Press and Oxford University Press, 1994. URL https://global.oup.com/academic/product/lattice-methods-for-multiple-integration-9780198534723. I. H. Sloan and H. Wozniakowski. When are quasi-Monte Carlo algorithms efficient for high dimensional integrals? J. Complex., 14(1):1–33, 1998. doi:10.1006/jcom.1997.0463. I. H. Sloan, F. Y. Kuo, and S. Joe. On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces. Math. Comput., 71:1609–1641, 2002. doi:10.1090/S0025-5718-02-01420-5.

scholarly journals A computable figure of merit for quasi-Monte Carlo point sets

2013 ◽  
Vol 83 (287) ◽  
pp. 1233-1250 ◽  
Author(s):  
Makoto Matsumoto ◽  
Mutsuo Saito ◽  
Kyle Matoba
Keyword(s):  
Monte Carlo ◽  
Figure Of Merit ◽  
Point Sets ◽  
Quasi Monte Carlo

scholarly journals QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND

2011 ◽  
Vol 53 (1) ◽  
pp. 1-37 ◽  
Author(s):  
F. Y. KUO ◽  
CH. SCHWAB ◽  
I. H. SLOAN
Keyword(s):  
Monte Carlo ◽  
Hilbert Spaces ◽  
Unit Cube ◽  
High Dimensional ◽  
Equal Weight ◽  
Lattice Rules ◽  
Quasi Monte Carlo ◽  
Strong Focus ◽  
The Family ◽  
Alternative Settings

AbstractThis paper is a contemporary review of quasi-Monte Carlo (QMC) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0,1]s. It first introduces the by-now standard setting of weighted Hilbert spaces of functions with square-integrable mixed first derivatives, and then indicates alternative settings, such as non-Hilbert spaces, that can sometimes be more suitable. Original contributions include the extension of the fast component-by-component (CBC) construction of lattice rules that achieve the optimal convergence order (a rate of almost 1/N, where N is the number of points, independently of dimension) to so-called “product and order dependent” (POD) weights, as seen in some recent applications. Although the paper has a strong focus on lattice rules, the function space settings are applicable to all QMC methods. Furthermore, the error analysis and construction of lattice rules can be adapted to polynomial lattice rules from the family of digital nets.

Circle-Point Containment, Monte Carlo Method for Shape Matching Based on Feature Points Using the Technique of Sparse Uniform Grids

2018 ◽  
Vol 18 (1) ◽  
Author(s):  
Xiangzhi Wei ◽  
Jie Zhao ◽  
Siqi Qiu
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Method ◽  
Minimum Distance ◽  
Shape Matching ◽  
Feature Points ◽  
Point Sets ◽  
Two Phase ◽  
Critical Feature ◽  
Uniform Grids ◽  
Practical Algorithm

Shape matching using their critical feature points is useful in mechanical processes such as precision measure of manufactured parts and automatic assembly of parts. In this paper, we present a practical algorithm for measuring the similarity of two point sets A and B: Given an allowable tolerance ε, our target is to determine the feasibility of placing A with respect to B such that the maximum of the minimum distance from each point of A to its corresponding matched point in B is no larger than ε. For sparse and small point sets, an improved algorithm is achieved based on a sparse grid, which is used as an auxiliary structure for building the correspondence relationship between A and B. For large point sets, allowing a trade-off between efficiency and accuracy, we approximate the problem as computing the directed Hausdorff distance from A to B, and provide a two-phase nested Monte Carlo method for solving the problem. Experimental results are presented to validate the proposed algorithms.

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