Monte Carlo Methods
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چکیده
so that the error in I goes down as 1/ √ M and is smaller if the variance σ 2 f of f is smaller. For a one dimensional integration the Monte Carlo method is not compelling. However consider a d dimensional integral evaluated withM points. For a uniform mesh each dimension of the integral getsM1/d points, so that the separation is h = M−1/d . The error in the integration over one h cube is of order hd+2, since we are approximating the surface by a linear interpolation (a plane) with an O(h2) error. The total error in the integral is Mhd+2 = M−2/d . The error in the Monte Carlo method remains M−1/2, so that this method wins for d > 4. We can reduce the error in I by reducing the effective σf . This is done by concentrating the sampling where f (x) is large, using a weight function w(x) (i.e. w(x) > 0, ∫ 1 0 w(x) = 1) I = ∫ 1
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