Multidimensional Variation for Quasi-Monte Carlo

Abstract

This paper collects together some properties of multidimensional definitions of the total variation of a real valued function. The subject has been studied for a long time. Many of the results presented here date back at least to the early 1900s. The main reason for revisiting this topic is that there has been much recent work in theory and applications of Quasi-Monte Carlo (QMC) sampling. For an account of quasi-Monte Carlo integration see Fang and Wang (1994) and Niederreiter (1992). QMC is especially competitive for multidimensional integrands with bounded variation in the sense of Hardy and Krause (BVHK). For such integrands, over d dimensional domains, one sees QMC errors that are O(n−1(log n)) when using n function evaluations. When d = 1, competing methods are usually preferred to QMC. For even modestly large d, Monte Carlo and quasi-Monte Carlo sampling become the methods of choice. When the integrand is in BVHK, then QMC has superior asymptotic behavior, compared to Monte Carlo sampling. Therefore we may like to know when a specific function is in BVHK. Recent introductory text books on real analysis typically cover the notion of total variation for functions of a single real variable. Few of them say much about multidimensional variation. The not very recent book, Hobson (1927, Chapter 5), does include some discussion of variation beyond the one dimensional case. Discussions of multidimensional variation usually require ungainly expressions that grow in complexity with the dimension d. For this reason, many authors work out details for d = 2 and report that the same results hold for all d. Yet some results that hold for d = 2 do not hold for d > 2. For example an indicator function in two dimensions must either have positive variation in Vitali’s sense, or must have at least one input variable on which it does not truly depend. The same is not true for d ≥ 3. Similarly, if f(x) and g(x) are linear functions on the d dimensional cube, then min(f(x), g(x)) is BVHK when d = 2 but is not necessarily so when d > 2.