  Many high dimensional integrals can be reduced to the problem of finding the
relative measures of two sets. Often one set will be exponentially larger than
the other, making it difficult to compare the sizes. A standard method of
dealing with this problem is to interpolate between the sets with a sequence of
nested sets where neighboring sets have relative measures bounded above by a
constant. Choosing such a well balanced sequence can be very difficult in
practice. Here a new approach that automatically creates such sets is
presented. These well balanced sets allow for faster approximation algorithms
for integrals and sums, and better tempering and annealing Markov chains for
generating random samples. Applications such as finding the partition function
of the Ising model and normalizing constants for posterior distributions in
Bayesian methods are discussed.
