Acta Numerica: High dimensional integration - the Quasi-Monte Carlo way

By Josef Dick, Frances Y. Kuo, Ian H. Sloan. 2014

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Control functionals for Monte Carlo integration (2014, background): “tegration in high dimensions is fundamentally challenging. QMC studies the impact of such regularity assumptions on high-dimensional integration (see e.g. the discussion of weighted Sobolev spaces in Dick et al., 2013). Integration and functional approximation are two sides to the same coin, so that”
Collider physics at the precision frontier (2020, background): “For certain classes of 1-periodic functions and generating vectors a linear error scaling ∼ 1/n can be achieved [898].”
Monte Carlo with determinantal point processes (2016, methodology): “In particular, fast constructions of nets in QMC [Dick et al., 2013] could yield fast sampling algorithms for DPPs.”
Constrained Bayesian Optimization with Noisy Experiments (2017, methodology): “Theoretically, QMC methods achieve a convergence rate of O ((log N ) d /N ), and typically achieve much faster convergence in practice (Dick et al. 2013).”
Stein Variational Gradient Descent as Gradient Flow (2017, unknown): “ons and Trends R in Machine Learning, 1(1–2):1–305, 2008. [3] Y. Chen, M. Welling, and A. Smola. Super-samples from kernel herding. In Conference on Uncertainty in Artiﬁcial Intelligence (UAI), 2010. [4] J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: the quasi-monte carlo way. Ac”
Hyperbolic Cross Approximation (2016, background): “For more details and further references see also the recent survey by Dick, Kuo and Sloan [82].”
Support points (2016, background): “Such a projective property is enjoyed by most QMC point sets in the literature [11], and new methodology is needed to incorporate this within the support points framework.”
Approximate Log-Hilbert-Schmidt Distances between Covariance Operators for Image Classification (2016, methodology): “An alternative approach, proposed by [34], employs Quasi-Monte Carlo integration [9], in which the ω j ’s are deterministic points arising from a low-discrepancy sequence in [0 , 1] n .”
Demystifying Orthogonal Monte Carlo and Beyond (2020, methodology): “The theory of MC is rich with various techniques improving the accuracy of base MC estimators such as: antithetic couplings and importance sampling [6], variance reduction via carefully designed control variate terms [29, 33] and finally: the vast field of the so-called Quasi Monte Carlo (QMC) metho”