Walsh Spaces Containing Smooth Functions and Quasi-Monte Carlo Rules of Arbitrary High Order

By Josef Dick. 2008

147 citations recorded by the source corpus.

We define a Walsh space which contains all functions whose partial mixed derivatives up to order $\delta \ge 1$ exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on digital $(t,\alpha,s)$-sequences achieve the optimal rate of convergence of the worst-case error for numerical integration. This rate of convergence is also optimal for the subspace of smooth functions. Explicit constructions of digital $(t,\alpha,s)$-sequences are given, hence providing explicit quasi-Monte Carlo rules which achieve the optimal rate of convergence of the integration error for arbitrarily smooth functions.

Subfields: Algebra and Number Theory, Computer Vision and Pattern Recognition, Numerical Analysis

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Citations in the corpus

Convergence guarantees for kernel-based quadrature rules in misspecified settings (2016, background): “We also note that there are certain quadratures which are adaptive to less smooth integrands [8, 9, 10].”
Monte Carlo and Quasi-Monte Carlo Sampling (2009, background): “Finally, another line of research that has been pursued recently is to construct digital nets and sequences that work well for integrands belonging to certain classes of smooth functions [77, 78].”
Discrepancy Theory and Quasi-Monte Carlo Integration (2014, methodology): “The following explicit construction method of suitable generating matrices was introduced in [17, 18].”
Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands (2010, unknown): “In this section, we prove a bound on d,l(Pbm), where the point set is a digital (t,m, s)-net as constructed in [4].”
Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration (2010, background): “A general theory of higher order digital nets and sequences has been developed in [36] for periodic functions, and for the general case in [37].”