Discrepancy, Integration and Tractability

Abstract

The discrepancy function of a point distribution measures the deviation from the uniform distribution. Different versions of the discrepancy function capture this deviation with respect to different geometric objects. Via Koksma-Hlawka inequalities the norm of the discrepancy function in a function space is intimately connected to the worst case integration error of the quasi-Monte Carlo integration rule determined by the point set for functions from the unit ball of a related function space. So the a priori very geometric concept of the discrepancy function is a crucial tool for studying numerical integration. In this survey article we want to discuss aspects of this interplay between discrepancy, integration and tractability questions. The main focus is on the exposition of some more recent results as well as on identifying open problems whose solution might advance our knowledge about this interplay of discrepancy, integration and randomization. Via the Koksma-Hlawka connection, the construction of point sets with small discrepancy automatically yields good quasi-Monte Carlo rules. Here we discuss how the explicit point sets constructed by Chen and Skriganov as low discrepancy sets in L p for 1 < p < ∞ provide also good quasi-Monte Carlo rules in Besov spaces of dominating mixed smoothness. Lower bounds for norms of the discrepancy function show the limits of this approach using function values and deterministic algorithms for the computation of integrals. Randomized methods may perform better, especially if the dimension of the problem is high. In this context we treat recent results on the power of importance sampling. The study of average discrepancies is of interest to gain insight into the behavior of typical point sets with respect to discrepancy and integration errors. Very general notions of the discrepancy function are related to empirical processes, average discrepancies then are expectations of certain norms of such empirical processes. We explain this connection and discuss some recent results on the limit behavior of average discrepancies as the number of points goes to infinity. © Springer-Verlag Berlin Heidelberg 2013.