Local antithetic sampling with scrambled nets. (English) Zbl 1157.65006

Many problems in science and engineering require multidimensional quadratures. The Monte Carlo integration can be improved by the use of variance reduction methods.
The author extends the classical variance reduction method of antithetic sampling and combine it with randomized quasi-Monte Carlo methods. Several ways of combining and randomized digital nets are considered. The main result is that for any \(\epsilon > 0\) such method reduces the rate of the root mean square error (RMSE) of the integration of smooth integrands to \({\mathcal O}\left( n^{-{3 \over 2} - {1 \over d} + \epsilon}\right),\) where \(d\) is the dimension.
In section 2 the information on scrambled nets is reminded. The concepts of \((q,m,d)\)-nets, \((\lambda,q,m,d)\)-nets and \((q,d)\)-sequences in base \(b \geq 2\) are discussed. Techniques of random scrambles as a random linear scrambling, an affine matrix scramble with a different structure of the matrices are presented. The ANOVA decomposition of functions is reminded.
In Section 3 \(d\)-dimensional folding operations are used to introduce local antithetic properties into digital nets. Three specific methods – reflection nets, box folded nets and monomial nets – are introduced for inducing local antithetic properties in some \((q,m,2)\)-nets.
In Section 4 these three methods are illustrated to concrete two-dimensional integrand.
In Section 5 the variance for scrambled digital nets quadrature of smooth functions is considered. Theorem 2 shows some upper bounds of the gain coefficients for a \((0,m,d)\)-net, a \((\lambda,0,m,d)\)-net and a \((\lambda,q,m,d)\)-net in base \(b.\) Theorem 3 gives an order \({\mathcal O}\left( n^{-{3 \over 2}} (\log n)^{d-1 \over 2}\right)\) of the RMSE of the integration of smooth functions using a randomized relaxed \((\lambda,q,m,d)\)-net in base \(b.\)
In Section 6 the effect of the reflection scheme on a scrambled net is investigated. The concept of the box fold scheme is introduced. Theorem 4 states that for doubly smooth functions under the box folding on a randomized relaxed \((\lambda,0,m,d)\)-net in base \(b\) the RMSE of the integration has an order \({\mathcal O}\left( n^{-{3 \over 2} - {1 \over d}} (\log n)^{d-1 \over 2}\right).\)
In Section 7 the box folding scheme is presented as a hybrid of a monomial cubature rule with scrambled net sampling.


65C05 Monte Carlo methods
11K45 Pseudo-random numbers; Monte Carlo methods
41A55 Approximate quadratures
41A63 Multidimensional problems


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