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.
Read more about this topic: Quasi-Monte Carlo Method
Famous quotes containing the words error, bounds and/or carlo:
“Truth is one, but error proliferates. Man tracks it down and cuts it up into little pieces hoping to turn it into grains of truth. But the ultimate atom will always essentially be an error, a miscalculation.”
—René Daumal (19081944)
“Nature seems at each mans birth to have marked out the bounds of his virtues and vices, and to have determined how good or how wicked that man shall be capable of being.”
—François, Duc De La Rochefoucauld (16131680)
“If there is anything so romantic as that castle-palace-fortress of Monaco I have not seen it. If there is anything more delicious than the lovely terraces and villas of Monte Carlo I do not wish to see them. There is nothing beyond the semi-tropical vegetation, the projecting promontories into the Mediterranean, the all-embracing sweep of the ocean, the olive groves, and the enchanting climate! One gets tired of the word beautiful.”
—M. E. W. Sherwood (18261903)