3.5 Monte Carlo integration

The terms comprising a Neumann series contain high-dimensional integrals which can be efficiently calculated using stochastic sampling by Monte Carlo techniques.

Consider an integral

    ∫ b         ∫
I ≡    dxϕ (x) =   dxp (x )ψ(x) ,                                                  (3.23)
     a
where p is a probability density function such that abdxp(x) = 1 and ψ(x) = ϕ(x)
p(x). The integral I corresponds to the mean value of ψ                        (x).

The choice of the distribution p determines various qualities of the Monte Carlo algorithm [106], namely the computational efficiency, the convergence rate and the associated trade-off with reliability (variance in the result). Often physical considerations are used to choose the distribution p.

Consider the random variables X and Ψ[X ]: A sequence of N numbers {xi} is generated according to p and is used to sample Ψ                  [X ], thereby approximating the mean value by an expected value:

                  N
I ≈ E [Ψ [X ]] =-1 ∑  ψ (x) .                                                    (3.24)
               N  i=1     i

This establishes the link between the Neumann series and the Monte Carlo algorithm, which will be discussed in Section 3.7.