Introduction and Examples

Abstract

An essential part of many scientific problems is the computation of integral

$$I = \int_D {g\left( x \right)} {\kern 1pt} dx$$

((1))

, where D is often a region in a high-dimensional space and g(x) is the target function of interest. If we can draw independent and identically distributed (i.i.d.) random samples x⁽¹⁾ ... , x^(m) uniformly from D (by a computer), an approximation to I can be obtained as

$$ {\hat I_m} = \frac{1}{m}\left\{ {g\left( {{x^{\left( 1 \right)}}} \right) + \cdots + g\left( {{x^{\left( m \right)}}} \right)} \right\}. $$