On a projection-corrected component-by-component construction

The component-by-component construction is the standard method of finding good lattice rules or polynomial lattice rules for numerical integration. Several authors have reported that in numerical experiments the generating vector sometimes has repeated components. We study a variation of the classical component-by-component algorithm for the construction of lattice or polynomial lattice point sets where the components are forced to differ from each other. This avoids the problem of having projections where all quadrature points lie on the main diagonal. Since the previous results on the worst-case error do not apply to this modified algorithm, we prove such an error bound here. We also discuss further restrictions on the choice of components in the component-by-component algorithm

Hot new directions for quasi-Monte Carlo research in step with applications

This article provides an overview of some interfaces between the theory of quasi-Monte Carlo (QMC) methods and applications. We summarize three QMC theoretical settings: first order QMC methods in the unit cube $[0,1]^s$ and in $\mathbb{R}^s$, and higher order QMC methods in the unit cube. One important feature is that their error bounds can be independent of the dimension $s$ under appropriate conditions on the function spaces. Another important feature is that good parameters for these QMC methods can be obtained by fast efficient algorithms even when $s$ is large. We outline three different applications and explain how they can tap into the different QMC theory. We also discuss three cost saving strategies that can be combined with QMC in these applications. Many of these recent QMC theory and methods are developed not in isolation, but in close connection with applications

Multilevel Quasi-Monte Carlo Methods for Lognormal Diffusion Problems

In this paper we present a rigorous cost and error analysis of a multilevel estimator based on randomly shifted Quasi-Monte Carlo (QMC) lattice rules for lognormal diffusion problems. These problems are motivated by uncertainty quantification problems in subsurface flow. We extend the convergence analysis in [Graham et al., Numer. Math. 2014] to multilevel Quasi-Monte Carlo finite element discretizations and give a constructive proof of the dimension-independent convergence of the QMC rules. More precisely, we provide suitable parameters for the construction of such rules that yield the required variance reduction for the multilevel scheme to achieve an $\varepsilon$-error with a cost of $\mathcal{O}(\varepsilon^{-\theta})$ with $\theta < 2$, and in practice even $\theta \approx 1$, for sufficiently fast decaying covariance kernels of the underlying Gaussian random field inputs. This confirms that the computational gains due to the application of multilevel sampling methods and the gains due to the application of QMC methods, both demonstrated in earlier works for the same model problem, are complementary. A series of numerical experiments confirms these gains. The results show that in practice the multilevel QMC method consistently outperforms both the multilevel MC method and the single-level variants even for non-smooth problems.Comment: 32 page

Quasi-Monte Carlo methods for high-dimensional integration: the standard (weighted Hilbert space) setting and beyond

This 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. doi:10.1017/S144618111200007

Quasi-Monte Carlo for finance applications

Monte Carlo methods are used extensively in computational finance to estimate the price of financial derivative options. We review the use of quasi-Monte Carlo methods to obtain the same accuracy at a much lower computational cost, and focus on three key ingredients: the generation of Sobol' and lattice points, reduction of effective dimension using the principal component analysis approach at full potential, and randomization by shifting or digital shifting to give an unbiased estimator with a confidence interval. Our aim is to provide a starting point for finance practitioners new to quasi-Monte Carlo methods. References P. Acworth, M. Broadie, and P. Glasserman, A comparison of some Monte Carlo and quasi-Monte Carlo techniques for option pricing, in: Monte Carlo and quasi-Monte Carlo methods 1996 (P. Hellekalek, G. Larcher, H. Niederreiter, and P. Zinterhof, eds.), Springer Verlag, Berlin, 1--18 (1998). J. Baldeaux, qmc for finance beyond Black-Scholes, submitted to ANZIAM J. Proc. CTAC 2008. R. E. Caflisch, W. Morokoff, and A. B. Owen, Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension, J. Comp. Finance 1, 27--46 (1997). http://www.thejournalofcomputationalfinance.com/public/showPage.html?page=919 R. Cools, F. Y. Kuo, and D. Nuyens, Constructing embedded lattice rules for multivariate integration, SIAM J. Sci. Comput. 28, 2162--2188 (2006). doi:10.1137/06065074X J. Dick, F. Pillichshammer, and B. J. Waterhouse, The construction of good extensible rank-$1$ lattices, Math. Comp. 77, 2345--2373 (2008). doi:10.1090/S0025-5718-08-02009-7 P. L'Ecuyer, Quasi-Monte Carlo methods in finance, in: Proceedings of the 2004 Winter Simulation Conference (R. G. Ingalls, M.D. Rossetti, J. S. Smith,and B. A. Peters, eds.), IEEE Computer Society Press, Los Alamitos, 1645--1655 (2004). doi:10.1109/WSC.2004.1371512 M. B. Giles and B. J. Waterhouse, Multilevel quasi-Monte Carlo path simulation, in preparation. P. Glasserman, Monte Carlo methods in financial engineering, Springer--Verlag, New York, 2004. S. Joe and F. Y. Kuo, Constructing Sobol' sequences with better two-dimensional projections, SIAM J. Sci. Comput. 30, 2635--2654 (2008). doi:10.1137/070709359 J. Keiner and B. J. Waterhouse, Fast implementation of the pca method for finance problems with unequal time steps, in preparation. F. Y. Kuo and I. H. Sloan, Lifting the curse of dimensionality, Notices Amer. Math. Soc. 52, 1320--1329 (2005). http://www.ams.org/notices/200511/index.html H. Niederreiter, Random number generation and quasi-Monte Carlo methods, SIAM, Philadelphia, 1992. D. Nuyens and B. J. Waterhouse, Adaptive quasi-Monte Carlo in finance, in preparation. K. Scheicher, Complexity and effective dimension of discrete Levy areas, J. Complexity 23, 152--168 (2007). doi:10.1016/j.jco.2006.12.006 I. H. Sloan, X. Wang, and H. Wozniakowski, Finite-order weights imply tractability of multivariate integration, J. Complexity 20, 46--74 (2004). doi:10.1016/j.jco.2003.11.003 X. Wang and I. H. Sloan, Quasi-Monte Carlo methods in financial enginnering: an equivalence principle and dimension reduction, in preparation