Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration

By Josef Dick, Friedrich Pillichshammer. 2010

736 citations recorded by the source corpus.

Subfields: Applied Mathematics, Numerical Analysis, Statistical and Nonlinear Physics

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “…(Rubinstein and Kroese, 2011), (c) antithetic variables (Green and Han, 1992) and more generally (randomized) quasi- Monte-Carlo (RQMC) sampling (Dick and Pillichshammer, 2010), (d) Rao–Blackwellization (Robert and Casella, 2004; Douc and Robert, 2011; Ghosh and Clyde, 2011; Olsson and Ryden,…”
Interpreting Black Box Predictions using Fisher Kernels (2018, methodology): “Other methods include kernel herding [3] and quasi-Monte carlo [7], both of which use wi = 1/n but use specific schemes to draw xi.”
Discrepancy Theory and Quasi-Monte Carlo Integration (2014, background): “Further results on the tractability of weighted discrepancy can be found in [33], Hinrichs, P.”
Unifying Orthogonal Monte Carlo Methods (2019, methodology): “As mentioned in Section 1, there are many scenarios where estimation based on OMC yields great statistical beneﬁts over i.i.d. Monte Carlo.”
High-dimensional peaks-over-threshold inference (2018, positive): “The existence of generating vectors achieving a nearly optimal convergence rate, with integration error independent of the dimension, has been proved and methods for their construction exist (Dick and Pillichshammer, 2010).”
Least squares polynomial chaos expansion: A review of sampling strategies (2017, methodology): “…convergence rate of standard MC technique, especially for high-dimensional integrations in ﬁnance applications [120], researchers have proposed a deterministic version of the MC sampling method based on the low-discrepancy sequences [26] known as quasi-Monte Carlo (QMC) technique [89, 14, 8, 79].”
Frank-Wolfe Bayesian Quadrature: Probabilistic Integration with Theoretical Guarantees (2015, background): “Alternatives such as Quasi-Monte Carlo (QMC) can exploit knowledge on the smoothness of the integrand to obtain optimal convergence rates [7].”
Bayesian Probabilistic Numerical Methods (2017, background): “…of the information complexity literature (Traub et al., 1988; Novak and Woźniakowski, 2010), while other fields such as quasi-Monte Carlo (QMC, Dick and Pillichshammer, 2010) attempt to develop asymptotically optimal information operators for specific numerical tasks, such as the choice of…”
Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands (2010, neutral): “A comprehensive introduction to digital nets can be found in [6, 12].”
Support points (2016, background): “Indeed, the reverse-engineering of a GOF test forms the basis for state-of-the-art integration techniques called Quasi-Monte Carlo (QMC) methods (see [12] and [11] for a modern overview).”
Monte Carlo with determinantal point processes (2016, methodology): “Quasi-Monte Carlo methods (QMC; [Dick and Pilichshammer, 2010, Dick et al., 2013]) are deterministic constructions that focus on the uniform case, µ(dx) = dx in (1.2).”