Approximation Error Bounds of Quasi-Monte Carlo
The approximation error of the quasi-Monte Carlo method is bounded by a term proportional to the discrepancy of the set x1, ..., xN. Specifically, the Koksma-Hlawka inequality states that the error
is bounded by
- ,
where V(f) is the Hardy-Krause variation of the function f (see Morokoff and Caflisch (1995) for the detailed definitions). DN is the discrepancy of the set (x1,...,xN) and is defined as
- ,
where Q is a rectangular solid in s with sides parallel to the coordinate axes. The inequality can be used to show that the error of the approximation by the quasi-Monte Carlo method is, whereas the Monte Carlo method has a probabilistic error of . Though we can only state the upper bound of the approximation error, the convergence rate of quasi-Monte Carlo method in practice is usually much faster than its theoretical bound. Hence, in general, the accuracy of the quasi-Monte Carlo method increases faster than that of the Monte Carlo method.
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