IMPORTANCE Sampling Basic
Use of Bounding
Use of Bounding Boxes in Complicated Geometries --- II
is discontinuous at
. This violates assumptions in analytic methods which require
be approximated by a linear (trapezoidal), quadratic (Simpson) or higher order polynomial.
This assumption is violated by a function
which is discontinuous. Even if
happens to be zero at
, it would rarely have required zero derivatives (of higher order for Gauss methods).
On the other hand, Monte Carlo method only requires finite standard deviation. If has NO requirement on continuity of values; derivatives need not even exist.
Thus, only Monte Carlo methods can use this ``bounding box'' approach to complicated integration domains. In practice, the latter are very common.
Northeast Parallel Architectures Center
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