Summary
This article considers quasi-Monte Carlo sampling for integrands having isolated point singularities. It is usual for such singular functions to be approached via importance sampling. Indeed one might expect that very uniform sampling, such as QMC uses, should be unhelpful in such problems, and the Koksma-Hlawka inequality seems to indicate as much. Perhaps surprisingly, we find that the expected errors in randomized QMC converge to zero at a faster rate than holds for Monte Carlo sampling, under growth conditions for which 2 + ε moments of the integrand are finite. The growth conditions do place constraints on certain partial derivatives of the integrand, but unlike importance sampling, they do not require knowledge of the locations of the singularities.
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Owen, A.B. (2006). Quasi-Monte Carlo for Integrands with Point Singularities at Unknown Locations. In: Niederreiter, H., Talay, D. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-31186-6_24
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DOI: https://doi.org/10.1007/3-540-31186-6_24
Publisher Name: Springer, Berlin, Heidelberg
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