Journal
SIAM JOURNAL ON NUMERICAL ANALYSIS
Volume 53, Issue 5, Pages 2488-2503Publisher
SIAM PUBLICATIONS
DOI: 10.1137/15M1007963
Keywords
randomized quasi-Monte Carlo; numerical integration; discontinuous function; mean squared error; convex set
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Funding
- National Natural Science Foundation of China [71471100]
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This paper studies the convergence rate of randomized quasi-Monte Carlo (RQMC) for discontinuous functions, which are often of infinite variation in the sense of Hardy and Krause. It was previously known that the root mean square error (RMSE) of RQMC is only o(n(-1)/(2)) for discontinuous functions. For certain discontinuous functions in d dimensions, we prove that the RMSE of RQMC is O(n(-1/2-1/(4d-2)+epsilon)) for any epsilon > 0 and arbitrary n. If some discontinuity boundaries are parallel to some coordinate axes, the rate can be improved to O(n(-1/2-1/(4du-2)+epsilon)), where d(u) denotes the so-called irregular dimension, that is, the number of axes which are not parallel to the discontinuity boundaries. Moreover, this paper shows that the RMSE is O(n(-1/2-1/(2d))) for certain indicator functions.
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