On efficient weighted integration via a change of variables
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Numerische Mathematik (2020) 146:545–570 https://doi.org/10.1007/s00211-020-01147-7
On efficient weighted integration via a change of variables P. Kritzer1 · F. Pillichshammer2 · L. Plaskota3 · G. W. Wasilkowski4 Received: 24 December 2018 / Revised: 24 August 2020 / Accepted: 24 August 2020 / Published online: 24 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, we study the approximation of d-dimensional -weighted integrals over unbounded domains Rd+ or Rd 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 ∈ [1, +∞]. The main results give sufficient conditions on the change of variables ν 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 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. Mathematics Subject Classification 65D30
1 Introduction We consider in this paper the approximation of d-variate d -weighted integrals of the form Id, ( f ) =
Dd
f (x) d (x) dx, where d (x) =
d
(x j ),
(1)
j=1
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. Extended author information available on the last page of the article
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over an unbounded domain D d , where D = R+ or D = R and for a given probability density function : D → R+ . In the univariate case, such integrals are often approximated by Gaussian quadratures that enjoy exponential rates of convergence, see, e.g., [1]. There are also generalized Gaussian rules, see, e.g., [4] and the papers cited there, that achieve exponential rates for integrands with singularities at infinity. We stress that those results are about the asymptotic behavior of the integration error and require analytic integrands. In the current paper, we consider integrands of regularity one only, and we analyze the worst case error with respect to a class of integrands. Indeed, we follow the Information-Based Complexity approach (see, e.g., [10]) providing worst case results for all integrands f from the Sobolev s
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