Abstract
In this paper, we study the approximation of d-dimensional \(\varrho \)-weighted integrals over unbounded domains \({\mathbb {R}}_+^d\) or \({\mathbb {R}}^d\) using a special change of variables, so that quasi-Monte Carlo (QMC) or sparse grid rules can be applied to the transformed integrands over the unit cube. We consider a class of integrands with bounded \(L_p\) norm of mixed partial derivatives of first order, where \(p\in [1,+\infty ].\) The main results give sufficient conditions on the change of variables \(\nu \) which guarantee that the transformed integrand belongs to the standard Sobolev space of functions over the unit cube with mixed smoothness of order one. These conditions depend on \(\varrho \) and p. The proposed change of variables is in general different than the standard change based on the inverse of the cumulative distribution function. We stress that the standard change of variables leads to integrands over a cube; however, those integrands have singularities which make the application of QMC and sparse grids ineffective. Our conclusions are supported by numerical experiments.
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Notes
We use lattice rules with generating vectors taken from the website of Frances Y. Kuo. The generating vectors used in this example were generated for equal product weights \(\gamma _j=1\), referred to as “lattice-28001” at https://web.maths.unsw.edu.au/~fkuo/lattice/index.html.
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Acknowledgements
P. Kritzer, L. Plaskota, and G.W. Wasilkowski would like to thank the MATRIX institute in Creswick, VIC, Australia, and its staff for supporting their stay during the program “On the Frontiers of High-Dimensional Computation” in June 2018. Furthermore, the authors thank the RICAM Special Semester Program 2018, during which parts of the paper were written.
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P. Kritzer is supported by the Austrian Science Fund (FWF): Project F5506-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”. F. Pillichshammer is supported by the Austrian Science Fund (FWF): Project F5509-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Application”. L. Plaskota is supported by the National Science Centre, Poland, under Grant 2017/25/B/ST1/00945.
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Kritzer, P., Pillichshammer, F., Plaskota, L. et al. On efficient weighted integration via a change of variables. Numer. Math. 146, 545–570 (2020). https://doi.org/10.1007/s00211-020-01147-7
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DOI: https://doi.org/10.1007/s00211-020-01147-7