Integration and Approximation Based on Scramble Sampling in Arbitrary Dimensions

This paper considers integration in the worst case setting and approximation in the average case setting based on the scramble sampling scheme proposed by A. B. Owen (1995, Lecture Notes in Statistics, Vol. 106, pp. 299?317, Springer-Verlag, New York.) The tractability and strong tractability exponents are found for function spaces with reproducing/covariance kernels that are scramble-invariant. Integration and approximation for a space with a non-scramble-invariant kernel are no harder than the corresponding problems with the associated scramble-invariant kernel. This enables us to derive tractability results for weighted Sobolev spaces.

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