Algorithm 958: Lattice Builder: A General Software Tool for Constructing Rank-1 Lattice Rules

We introduce a new software tool and library named Lattice Builder, written in C++, that implements a variety of construction algorithms for good rank-1 lattice rules. It supports exhaustive and random searches, as well as component-by-component (CBC) and random CBC constructions, for any number of points, and for various measures of (non)uniformity of the points. The measures currently implemented are all shiftinvariant and represent the worst-case integration error for certain classes of integrands. They include for example the weighted Pα square discrepancy, the Rα criterion, and figures of merit based on the spectral test, with projection-dependent weights. Each of these measures can be computed as a finite sum. For the Pα and Rα criteria, efficient specializations of the CBC algorithm are provided for projection-dependent, order-dependent and product weights. For numbers of points that are integer powers of a prime base, the construction of embedded rank-1 lattice rules is supported through any of the above algorithms, and also through a fast CBC algorithm, with a variety of possibilities for the normalization of the merit values of individual embedded levels and for their combination into a single merit value. The library is extensible, thanks to the decomposition of the algorithms into decoupled components, which makes it easy to implement new types of weights, new search domains, new figures of merit, etc.

[1]  Pierre L'Ecuyer,et al.  An Implementation of the Lattice and Spectral Tests for Multiple Recursive Linear Random Number Generators , 1997, INFORMS J. Comput..

[2]  Fred J. Hickernell,et al.  The existence of good extensible rank-1 lattices , 2003, J. Complex..

[3]  Dirk Nuyens,et al.  Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces , 2006, Math. Comput..

[4]  P. L’Ecuyer,et al.  On Figures of Merit for Randomly-Shifted Lattice Rules , 2012 .

[5]  Xiaoqun Wang,et al.  Constructing Robust Good Lattice Rules for Computational Finance , 2007, SIAM J. Sci. Comput..

[6]  Sandeep Koranne,et al.  Boost C++ Libraries , 2011 .

[7]  Pierre L'Ecuyer,et al.  Quasi-Monte Carlo methods with applications in finance , 2008, Finance Stochastics.

[8]  I. Sloan.  Lattice Methods for Multiple Integration , 1994 .

[9]  Leszek Plaskota,et al.  Monte Carlo and Quasi-Monte Carlo methods 2010 , 2012 .

[10]  Andrei Alexandrescu,et al.  Modern C++ design: generic programming and design patterns applied , 2001 .

[11]  A. Owen,et al.  Estimating Mean Dimensionality of Analysis of Variance Decompositions , 2006 .

[12]  Pierre L'Ecuyer,et al.  Tables of linear congruential generators of different sizes and good lattice structure , 1999, Math. Comput..

[13]  I. Sloan,et al.  QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND , 2011, The ANZIAM Journal.

[14]  P. L’Ecuyer,et al.  On the distribution of integration error by randomly-shifted lattice rules , 2010 .

[15]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[16]  Pierre L'Ecuyer,et al.  Variance bounds and existence results for randomly shifted lattice rules , 2012, J. Comput. Appl. Math..

[17]  Art B. Owen,et al.  Latin supercube sampling for very high-dimensional simulations , 1998, TOMC.

[18]  Harald Niederreiter,et al.  New methods for pseudorandom numbers and pseudorandom vector generation , 1992, WSC '92.

[19]  Henryk Wozniakowski,et al.  Good Lattice Rules in Weighted Korobov Spaces with General Weights , 2006, Numerische Mathematik.

[20]  Fred J. Hickernell,et al.  A generalized discrepancy and quadrature error bound , 1998, Math. Comput..

[21]  Pierre L'Ecuyer,et al.  On the Behavior of the Weighted Star Discrepancy Bounds for Shifted Lattice Rules , 2009 .

[22]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[23]  I. Sobola,et al.  Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates , 2001 .

[24]  Dirk Nuyens.  The construction of good lattice rules and polynomial lattice rules , 2014, Uniform Distribution and Quasi-Monte Carlo Methods.

[25]  Dirk Nuyens,et al.  Fast Component-by-Component Construction, a Reprise for Different Kernels , 2006 .

[26]  C. Lemieux.  Monte Carlo and Quasi-Monte Carlo Sampling , 2009 .

[27]  R. Caflisch.  Monte Carlo and quasi-Monte Carlo methods , 1998, Acta Numerica.

[28]  Pierre L'Ecuyer,et al.  Tables of maximally equidistributed combined LFSR generators , 1999, Math. Comput..

[29]  Fred J. Hickernell,et al.  Extensible Lattice Sequences for Quasi-Monte Carlo Quadrature , 2000, SIAM J. Sci. Comput..

[30]  Frances Y. Kuo,et al.  Component-by-Component Construction of Good Lattice Rules with a Composite Number of Points , 2002, J. Complex..

[31]  Ian H. Sloan,et al.  Efficient Weighted Lattice Rules with Applications to Finance , 2006, SIAM J. Sci. Comput..

[32]  Grzegorz W. Wasilkowski,et al.  Randomly shifted lattice rules for unbounded integrands , 2006, J. Complex..

[33]  Dirk Nuyens,et al.  Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points , 2006, J. Complex..

[34]  Dominique Maisonneuve.  Recherche et Utilisation des “Bons Treillis.” Programmation et Résultats Numériques , 1972 .

[35]  R. Cranley,et al.  Randomization of Number Theoretic Methods for Multiple Integration , 1976 .

[36]  Frances Y. Kuo,et al.  Constructing Embedded Lattice Rules for Multivariate Integration , 2006, SIAM J. Sci. Comput..

[37]  Henryk Wozniakowski,et al.  Lattice rule algorithms for multivariate approximation in the average case setting , 2008, J. Complex..

[38]  Henryk Wozniakowski,et al.  Liberating the weights , 2004, J. Complex..

[39]  Harald Niederreiter,et al.  Random number generation and Quasi-Monte Carlo methods , 1992, CBMS-NSF regional conference series in applied mathematics.

[40]  Josef Dick,et al.  The construction of good extensible rank-1 lattices , 2008, Math. Comput..

[41]  P. L’Ecuyer,et al.  Variance Reduction via Lattice Rules , 1999 .

[42]  Frances Y. Kuo,et al.  Multi-level quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients , 2012, 1208.6349.

[43]  F. Pillichshammer,et al.  Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration , 2010 .

[44]  F. J. Hickernell.  Lattice rules: how well do they measure up? in random and quasi-random point sets , 1998 .