Entropy, Randomization, Derandomization, and Discrepancy

The star discrepancy is a measure of how uniformly distributed a finite point set is in the d-dimensional unit cube. It is related to high-dimensional numerical integration of certain function classes as expressed by the Koksma-Hlawka inequality. A sharp version of this inequality states that the worst-case error of approximating the integral of functions from the unit ball of some Sobolev space by an equal-weight cubature is exactly the star discrepancy of the set of sample points. In many applications, as, e.g., in physics, quantum chemistry or finance, it is essential to approximate high-dimensional integrals. Thus with regard to the Koksma- Hlawka inequality the following three questions are very important: (i) What are good bounds with explicitly given dependence on the dimension d for the smallest possible discrepancy of any n-point set for moderate n? (ii) How can we construct point sets efficiently that satisfy such bounds? (iii) How can we calculate the discrepancy of given point sets efficiently? We want to discuss these questions and survey and explain some approaches to tackle them relying on metric entropy, randomization, and derandomization

Randomized Algorithms for High-Dimensional Integration and Approximation

We prove upper and lower error bounds for error of the randomized Smolyak algorithm and provide a thorough case study of applying the randomized Smolyak algorithm with the building blocks being quadratures based on scrambled nets for integration of functions coming from Haar-wavelets spaces. Moreover, we discuss different notions of negative dependence of randomized point sets which find applications in discrepancy theory and randomized quasi-Monte Carlo integration