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Proceedings of the Bulgarian Academy of Sciences

Quasi-Monte Carlo Integration in the Mixed Korobov–Sobolev Type Space $$H_{Mix, s, \alpha, \gamma, w}$$

Vassil Grozdanov, Atidzhe Hodzhova, Elmi Shabani

Published May 29, 2026• 0 citations
Publisher

Abstract

In this article, the authors construct a special reproducing kernel Hilbert space of mixed Korobov–Sobolev type. To address the main problems, an appropriate hybrid function system is employed. The arithmetic associated with this function system is used to define the notion of the mean square worst-case error of the integration in the considered space. An explicit formula for this error expressed in terms of the functions of the employed system is obtained.

DOI
10.7546/crabs.2026.05.01

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Quasi-Monte Carlo Integration in the Mixed Korobov–Sobolev Type Space HMix,s,α,γ,wH_{Mix, s, \alpha, \gamma, w}HMix,s,α,γ,w​
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Recently, quasi-Monte Carlo methods have been successfully used for approximating multiple integrals in hundreds of dimensions in mathematical finance, and were significantly more efficient than Monte Carlo methods. To understand the apparent success of quasi-Monte Carlo methods for multiple integration, one popular approach is to study worst-case error bounds in weighted function spaces in which the importance of the variables is moderated by some sequences of weights. Ideally, a family of quasi-Monte Carlo methods in some weighted function space should be strongly tractable. Strong tractability means that the minimal number of quadrature points n needed to reduce the initial error by a factor of ε is bounded by a polynomial in ε⁻¹ independently of the dimension d. Several recent publications show the existence of lattice rules that satisfy the strong tractability error bounds in weighted Korobov spaces of periodic integrands and weighted Sobolev spaces of non-periodic integrands. However, those results were non-constructive and thus give no clues as to how to actually construct these lattice rules. In this thesis, we focus on the construction of quasi-Monte Carlo methods that are strongly tractable. We develop and justify algorithms for the construction of lattice rules that achieve strong tractability error bounds in weighted Korobov and Sobolev spaces. The parameters characterizing these lattice rules are found ‘component-by-component’: the (d + 1)-th components are obtained by successive 1-dimensional searches, with the previous d components kept unchanged. The cost of these algorithms vary from O(nd²) to O(n³d²) operations. With currently available technology, they allow construction of rules easily with values of n up to several million and dimensions d up to several hundred.
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