Fast CBC construction of randomly shifted lattice rules achieving O(n-1+δ) convergence for unbounded integrands over R5 in weighted spaces with POD weights

Abstract This paper provides the theoretical foundation for the component-by-component (CBC) construction of randomly shifted lattice rules that are tailored to integrals over R s arising from practical applications. For an integral of the form ∫ R s f ( y ) ∏ j = 1 s ϕ ( y j ) d y with a univariate probability density ϕ , our general strategy is to first map the integral into the unit cube [ 0 , 1 ] s using the inverse of the cumulative distribution function of ϕ , and then apply quasi-Monte Carlo (QMC) methods. However, the transformed integrand in the unit cube rarely falls within the standard QMC setting of Sobolev spaces of functions with mixed first derivatives. Therefore, a non-standard function space setting for integrands over R s , previously considered by Kuo, Sloan, Wasilkowski and Waterhouse (2010), is required for the analysis. Motivated by the needs of three applications, the present paper extends the theory of the aforementioned paper in several non-trivial directions, including a new error analysis for the CBC construction of lattice rules with general non-product weights, the introduction of an unanchored variant for the setting, the use of coordinate-dependent weight functions in the norm, and the strategy for fast CBC construction with POD (“product and order dependent”) weights.

[1]  Frances Y. Kuo,et al.  The smoothing effect of the ANOVA decomposition , 2010, J. Complex..

[2]  Mario Rometsch,et al.  Quasi-Monte Carlo Methods in Finance , 2009 .

[3]  M. Wand,et al.  Quasi-Monte Carlo for Highly Structured Generalised Response Models , 2008 .

[4]  Frances Y. Kuo,et al.  On the Choice of Weights in a Function Space for Quasi-Monte Carlo Methods for a Class of Generalised Response Models in Statistics , 2013 .

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

[6]  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..

[7]  Masatake Mori,et al.  Double Exponential Formulas for Numerical Integration , 1973 .

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

[9]  Frances Y. Kuo,et al.  High-dimensional integration: The quasi-Monte Carlo way*† , 2013, Acta Numerica.

[10]  Frances Y. Kuo,et al.  CORRECTION TO “QUASI-MONTE CARLO METHODS FOR HIGH-DIMENSIONAL INTEGRATION: THE STANDARD (WEIGHTED HILBERT SPACE) SETTING AND BEYOND” , 2012, The ANZIAM Journal.

[11]  Henryk Wozniakowski,et al.  Finite-order weights imply tractability of multivariate integration , 2004, J. Complex..

[12]  R. L. Naff,et al.  High‐resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media: 2. Transport results , 1998 .

[13]  G. Dagan.  Solute transport in heterogeneous porous formations , 1984, Journal of Fluid Mechanics.

[14]  Henryk Wozniakowski,et al.  On decompositions of multivariate functions , 2009, Math. Comput..

[15]  Grzegorz W. Wasilkowski,et al.  Tractability of Approximation and Integration for Weighted Tensor Product Problems over Unbounded Domains , 2002 .

[16]  J. Dick.  THE DECAY OF THE WALSH COEFFICIENTS OF SMOOTH FUNCTIONS , 2009, Bulletin of the Australian Mathematical Society.

[17]  D. Hunter.  Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension , 2000 .

[18]  Josef Dick,et al.  QMC Rules of Arbitrary High Order: Reproducing Kernel Hilbert Space Approach , 2009 .

[19]  Grzegorz W. Wasilkowski,et al.  Randomly shifted lattice rules with the optimal rate of convergence for unbounded integrands , 2010, J. Complex..

[20]  Grzegorz W. Wasilkowski,et al.  Complexity of Weighted Approximation over R , 2000 .

[21]  Frances Y. Kuo,et al.  Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications , 2011, J. Comput. Phys..

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

[23]  Frances Y. Kuo,et al.  The smoothing effect of integration in Rd and the ANOVA decomposition , 2013, Math. Comput..

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

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

[26]  Dirk Nuyens,et al.  Efficient calculation of the worst-case error and (fast) component-by-component construction of higher order polynomial lattice rules , 2011, Numerical Algorithms.

[27]  Josef Dick,et al.  Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order , 2008, SIAM J. Numer. Anal..

[28]  M. Wand,et al.  General design Bayesian generalized linear mixed models , 2006, math/0606491.

[29]  James A. Nichols,et al.  Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients , 2014, Numerische Mathematik.

[30]  You-Kuan Zhang,et al.  Numerical simulations of non-ergodic solute transport in three-dimensional heterogeneous porous media , 2004 .

[31]  Josef Dick.  On the convergence rate of the component-by-component construction of good lattice rules , 2004, J. Complex..

[32]  P. Glasserman,et al.  A Comparison of Some Monte Carlo and Quasi Monte Carlo Techniques for Option Pricing , 1998 .

[33]  Henryk Wozniakowski,et al.  When Are Quasi-Monte Carlo Algorithms Efficient for High Dimensional Integrals? , 1998, J. Complex..

[34]  N. Aronszajn.  Theory of Reproducing Kernels. , 1950 .

[35]  Josef Dick,et al.  Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces , 2005, J. Complex..

[36]  Benjamin J. Waterhouse,et al.  Quasi-Monte Carlo for finance applications , 2008 .

[37]  I. H. SLOAN,et al.  Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces , 2002, SIAM J. Numer. Anal..

[38]  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.

[39]  R. L. Naff,et al.  High‐resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media: 1. Methodology and flow results , 1998 .

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

[41]  Frances Y. Kuo,et al.  Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces , 2003, J. Complex..

[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 , 2011 .

[43]  M. Mori.  Discovery of the Double Exponential Transformation and Its Developments , 2005 .

[44]  David H. Bailey.  Tanh-Sinh High-Precision Quadrature , 2006 .

[45]  J. Borwein,et al.  Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Effective Error Bounds in Euler-Maclaurin-Based Quadrature Schemes Permalink , 2005 .

[46]  G. Rodríguez-Yam,et al.  ESTIMATION FOR STATE-SPACE MODELS BASED ON A LIKELIHOOD APPROXIMATION , 2005 .