Multidimensional Monte Carlo Integration Based on Factorized Approximation Functions

Most of the variance-reducing techniques for Monte Carlo integrations are based on approximations of integrands. Useful approximations are, however, very difficult to find when many variables are involved. This paper investigates the use of approximations of the factorized form such as = h₁ (x₁, x₂) h₂ (x₂, x₃) h₍ - ₁ (x₍ - ₁, xₙ) in a suitably chosen approximation scheme. Such approximations are very simple in form enabling automatic calculations of them. An efficient method for iteratively calculating such functions is given. It is shown that they serve as control variates or as importance sampling weight functions conveniently in even large dimensions. The results of numerical experiments performed on several six-dimensional integrals show that our method is much more powerful than the stratified sampling method as long as the variations of the integrands are not too large.