Robust numerical integration and pairwise independent random variables (Q5957965)

This paper deals with random sampling methods for numerical integration of complicated functions, and proposes the development of several robust numerical integration methods. The most used sampling method utilizes independently identical distributed (i.i.d.) random variables as sampling points. By the law of large numbers, there is a robust numerical integration method.NEWLINENEWLINENEWLINEA first result of the paper is that for any random sampling method there always exists an integrand for which the error becomes almost as large as (or larger than) the i.i.d.-sampling. This implies that any sampling method cannot be significantly more efficient than the i.i.d.-sampling. But i.i.d.-sampling is not stable in practice because the large number of samples requires too much randomness, which may entail a fatal error by an amplified statistical bias of pseudo-random numbers. In order to reduce the randomness of robust numerical integration, the author proposes the use of the pairwise random samples instead of i.i.d. samples, also because there exists in computer science a well-known method to generate pairwise independent random variables with least possible randomness. Another strongly recommended robust integration technique is the so-called discrete random Weyl sampling, whose quick generation of pairwise independent samples implies simpler and practical implementations.