High-dimensional integration: The quasi-Monte Carlo way
TL;DR: A survey of recent developments in lattice methods, digital nets, and related themes can be found in this paper, where the authors present a contemporary review of QMC (quasi-Monte Carlo) methods, that is, equalweight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s, w heres may be large, or even infinite.
Abstract: This paper is a contemporary review of QMC (‘quasi-Monte Carlo’) methods, that is, equal-weight rules for the approximate evaluation of high-dimensional integrals over the unit cube [0, 1] s ,w heres may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes. Among those recent developments are methods of construction of both lattices and digital nets, to yield QMC rules that have a prescribed rate of convergence for sufficiently smooth functions, and ideally also guaranteed slow growth (or no growth) of the worst-case error as s increases. A crucial role is played by parameters called ‘weights’, since a careful use of the weight parameters is needed to ensure that the worst-case errors in an appropriately weighted function space are bounded, or grow only slowly, as the dimension s increases. Important tools for the analysis are weighted function spaces, reproducing kernel Hilbert spaces, and discrepancy, all of which are discussed with an appropriate level of detail.
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TL;DR: A review of the progress in multilevel Monte Carlo path simulation can be found in this article, where the authors highlight the simplicity, flexibility and generality of the multi-level Monte Carlo approach.
Abstract: The author’s presentation of multilevel Monte Carlo path simulation at the MCQMC 2006 conference stimulated a lot of research into multilevel Monte Carlo methods. This paper reviews the progress since then, emphasising the simplicity, flexibility and generality of the multilevel Monte Carlo approach. It also offers a few original ideas and suggests areas for future research.
590 citations
TL;DR: In this article, a non-parametric extension of control variates is presented, which leverages gradient information on the sampling density to achieve substantial variance reduction, and it is not required that sampling density be normalized.
Abstract: Summary
A non-parametric extension of control variates is presented These leverage gradient information on the sampling density to achieve substantial variance reduction It is not required that the sampling density be normalized The novel contribution of this work is based on two important insights: a trade-off between random sampling and deterministic approximation and a new gradient-based function space derived from Stein's identity Unlike classical control variates, our estimators improve rates of convergence, often requiring orders of magnitude fewer simulations to achieve a fixed level of precision Theoretical and empirical results are presented, the latter focusing on integration problems arising in hierarchical models and models based on non-linear ordinary differential equations
232 citations
TL;DR: In this article, the effects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider were studied.
Abstract: We study the effects of the exact top quark mass-dependent two-loop corrections to Higgs boson pair production by gluon fusion at the LHC and at a 100 TeV hadron collider. We perform a detailed comparison of the full next-to-leading order result to various approximations at the level of differential distributions and also analyse non-standard Higgs self-coupling scenarios. We find that the different next-to-leading order approximations differ from the full result by up to 50 percent in relevant differential distributions. This clearly stresses the importance of the full NLO result.
216 citations
TL;DR: Quasi Monte Carlo (QMC) as mentioned in this paper is an alternative to Monte Carlo, where random points are replaced with low-discrepancy sequences, and the advantage is that QMC estimates usually converge faster than their Monte Carlo counterparts.
Abstract: So far, the algorithms we have discussed rely on Monte Carlo, that is, on averages of random variables. QMC (quasi-Monte Carlo) is an alternative to Monte Carlo where random points are replaced with low-discrepancy sequences. The advantage is that QMC estimates usually converge faster than their Monte Carlo counterparts.
149 citations
TL;DR: A rigorous error analysis for methods constructed from standard continuous and piecewise linear finite element approximation in physical space, truncated Karhunen–Loève expansion for computing realizations of a and lattice-based quasi-Monte Carlo quadrature rules for computing integrals over parameter space which define the expected values.
Abstract: In this paper we analyze the numerical approximation of diffusion problems over polyhedral domains in $$\mathbb {R}^d$$Rd ($$d = 1, 2,3$$d=1,2,3), with diffusion coefficient $$a({\varvec{x}},\omega )$$a(x,?) given as a lognormal random field, i.e., $$a({\varvec{x}},\omega ) = \exp (Z({\varvec{x}},\omega ))$$a(x,?)=exp(Z(x,?)) where $${\varvec{x}}$$x is the spatial variable and $$Z({\varvec{x}}, \cdot )$$Z(x,·) is a Gaussian random field. The analysis presents particular challenges since the corresponding bilinear form is not uniformly bounded away from $$0$$0 or $$\infty $$? over all possible realizations of $$a$$a. Focusing on the problem of computing the expected value of linear functionals of the solution of the diffusion problem, we give a rigorous error analysis for methods constructed from (1) standard continuous and piecewise linear finite element approximation in physical space; (2) truncated Karhunen---Loeve expansion for computing realizations of $$a$$a (leading to a possibly high-dimensional parametrized deterministic diffusion problem); and (3) lattice-based quasi-Monte Carlo (QMC) quadrature rules for computing integrals over parameter space which define the expected values. The paper contains novel error analysis which accounts for the effect of all three types of approximation. The QMC analysis is based on a recent result on randomly shifted lattice rules for high-dimensional integrals over the unbounded domain of Euclidean space, which shows that (under suitable conditions) the quadrature error decays with $$\mathcal {O}(n^{-1+\delta })$$O(n-1+?) with respect to the number of quadrature points $$n$$n, where $$\delta >0$$?>0 is arbitrarily small and where the implied constant in the asymptotic error bound is independent of the dimension of the domain of integration.
140 citations
References
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Book•
01 Mar 1990
TL;DR: In this paper, a theory and practice for the estimation of functions from noisy data on functionals is developed, where convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework.
Abstract: This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a number of problems within this framework. Methods for including side conditions and other prior information in solving ill posed inverse problems are provided. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals.
6,120 citations
TL;DR: In this paper, a short historical introduction is given to indicate the different manners in which these kernels have been used by various investigators and discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter.
Abstract: : The present paper may be considered as a sequel to our previous paper in the Proceedings of the Cambridge Philosophical Society, Theorie generale de noyaux reproduisants-Premiere partie (vol 39 (1944)) which was written in 1942-1943 In the introduction to this paper we outlined the plan of papers which were to follow In the meantime, however, the general theory has been developed in many directions, and our original plans have had to be changed Due to wartime conditions we were not able, at the time of writing the first paper, to take into account all the earlier investigations which, although sometimes of quite a different character, were, nevertheless, related to our subject Our investigation is concerned with kernels of a special type which have been used under different names and in different ways in many domains of mathematical research We shall therefore begin our present paper with a short historical introduction in which we shall attempt to indicate the different manners in which these kernels have been used by various investigators, and to clarify the terminology We shall also discuss the more important trends of the application of these kernels without attempting, however, a complete bibliography of the subject matter (KAR) P 2
5,760 citations
Book•
20 Dec 1990TL;DR: In this article, a representation of stochastic processes and response statistics are represented by finite element method and response representation, respectively, and numerical examples are provided for each of them.
Abstract: Representation of stochastic processes stochastic finite element method - response representation stochastic finite element method - response statistics numerical examples.
5,495 citations
TL;DR: This chapter discusses Convergence: Weak, Almost Uniform, and in Probability, which focuses on the part of Convergence of the Donsker Property which is concerned with Uniformity and Metrization.
Abstract: 1.1. Introduction.- 1.2. Outer Integrals and Measurable Majorants.- 1.3. Weak Convergence.- 1.4. Product Spaces.- 1.5. Spaces of Bounded Functions.- 1.6. Spaces of Locally Bounded Functions.- 1.7. The Ball Sigma-Field and Measurability of Suprema.- 1.8. Hilbert Spaces.- 1.9. Convergence: Almost Surely and in Probability.- 1.10. Convergence: Weak, Almost Uniform, and in Probability.- 1.11. Refinements.- 1.12. Uniformity and Metrization.- 2.1. Introduction.- 2.2. Maximal Inequalities and Covering Numbers.- 2.3. Symmetrization and Measurability.- 2.4. Glivenko-Cantelli Theorems.- 2.5. Donsker Theorems.- 2.6. Uniform Entropy Numbers.- 2.7. Bracketing Numbers.- 2.8. Uniformity in the Underlying Distribution.- 2.9. Multiplier Central Limit Theorems.- 2.10. Permanence of the Donsker Property.- 2.11. The Central Limit Theorem for Processes.- 2.12. Partial-Sum Processes.- 2.13. Other Donsker Classes.- 2.14. Tail Bounds.- 3.1. Introduction.- 3.2. M-Estimators.- 3.3. Z-Estimators.- 3.4. Rates of Convergence.- 3.5. Random Sample Size, Poissonization and Kac Processes.- 3.6. The Bootstrap.- 3.7. The Two-Sample Problem.- 3.8. Independence Empirical Processes.- 3.9. The Delta-Method.- 3.10. Contiguity.- 3.11. Convolution and Minimax Theorems.- A. Appendix.- A.1. Inequalities.- A.2. Gaussian Processes.- A.2.1. Inequalities and Gaussian Comparison.- A.2.2. Exponential Bounds.- A.2.3. Majorizing Measures.- A.2.4. Further Results.- A.3. Rademacher Processes.- A.4. Isoperimetric Inequalities for Product Measures.- A.5. Some Limit Theorems.- A.6. More Inequalities.- A.6.1. Binomial Random Variables.- A.6.2. Multinomial Random Vectors.- A.6.3. Rademacher Sums.- Notes.- References.- Author Index.- List of Symbols.
4,600 citations
Book•
01 Jan 1992TL;DR: This chapter discusses Monte Carlo methods and Quasi-Monte Carlo methods for optimization, which are used for numerical integration, and their applications in random numbers and pseudorandom numbers.
Abstract: Preface 1. Monte Carlo methods and Quasi-Monte Carlo methods 2. Quasi-Monte Carlo methods for numerical integration 3. Low-discrepancy point sets and sequences 4. Nets and (t,s)-sequences 5. Lattice rules for numerical integration 6. Quasi- Monte Carlo methods for optimization 7. Random numbers and pseudorandom numbers 8. Nonlinear congruential pseudorandom numbers 9. Shift-Register pseudorandom numbers 10. Pseudorandom vector generation Appendix A. Finite fields and linear recurring sequences Appendix B. Continued fractions Bibliography Index.
3,815 citations