What Is the Complexity of Volume Calculation?

We study the worst case complexity of computing ε-approximations of volumes of d-dimensional regions g([0, 1]d), by sampling the function g. Here, g is an s times continuously differentiable injection from [0, 1]d to Rd , where we assume that s ≥ 1. Since the problem can be solved exactly when d = 1, we concentrate our attention on the case d ≥ 2. This problem is a special case of the surface integration problem studied in [12]. Let c be the cost of one function evaluation. The results of [12] might suggest that the εcomplexity of volume calculation should be proportional to c(1/ε)d/s when s ≥ 2. However, using integration by parts to reduce the dimension, we show that if s ≥ 2, then the complexity is proportional to c(1/ε)(d−1)/s. Next, we consider the case s = 1, which is the minimal smoothness for which our volume problem is well-defined. We show that when s = 1, an ε-approximation ∗This research was supported in part by the National Science Foundation under Grant CCR9987858. †This research was supported in part by the National Science Foundation under Grant CCR-9731858.