Journal ArticleDOI
High-dimensional integration: The quasi-Monte Carlo way
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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.read more
Citations
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Multilevel Monte Carlo methods
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.
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Control functionals for Monte Carlo integration
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Full top quark mass dependence in Higgs boson pair production at NLO
Sophia Borowka,Nicolas Greiner,Gudrun Heinrich,S. P. Jones,Matthias Kerner,Johannes Schlenk,T. Zirke +6 more
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Journal ArticleDOI
Sequential quasi Monte Carlo
Mathieu Gerber,Nicolas Chopin +1 more
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.
Journal ArticleDOI
Quasi-Monte Carlo finite element methods for elliptic PDEs with lognormal random coefficients
Ivan G. Graham,Frances Y. Kuo,James A. Nichols,Robert Scheichl,Christoph Schwab,Ian H. Sloan +5 more
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.
References
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Journal ArticleDOI
On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals
TL;DR: In this paper, the efficiency of such an integration formula can be measured by considering how it fares when the indicator function of the hyperbrick is defined by an arbitrary point in the unit hypercube.
Journal ArticleDOI
Multilevel Monte Carlo Path Simulation
TL;DR: It is shown that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using Monte Carlo path simulations.
Book
The Geometry of Random Fields
TL;DR: In this article, the authors present a survey of random fields and excursion sets and their spectral properties, including sample function regularity, sample function erraticism, and the Markov property for Gaussian fields.