What is the complexity of volume calculation?

  • Authors:

  • A. G. Werschulz;H. Woź/niakowski

  • Affiliations:

  • Department of Computer and Information Sciences, Fordham University, New York, New York 10023/ and Department of Computer Science, Columbia University, New York, New York;Department of Computer Science, Columbia University, New York, New York/ and Institute of Applied Mathematics, University of Warsaw, Warsaw, Poland

  • Venue:
  • Journal of Complexity
  • Year:
  • 2002

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Abstract

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 we studied earlier (J. Complexity 17, 442-446). Let c be the cost of one function evaluation. The earlier results (cited above) might suggest that the ε-complexity 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 can be computed with cost proportional to at most c(1/ε)(d-1)/d/2. Since a lower bound proportional to c(1/ε)d-1 holds when s = 1, it follows that the complexity in the minimal smoothness case is proportional to c(1/ε) when d = 2, and that there is a gap between the lower and upper bounds when d ≥ 3.