Quasi-Monte Carlo Methods

Abstract

There are many scientific as well as real-world applications where we run into the problem of computing a definite integral. In calculus courses you are taught that a definite integral \(\int _{a}^{b}f(u)du\) is evaluated by the fundamental theorem of integral calculus which says that $$\displaystyle{ \int _{a}^{b}f(u)du = F(b) - F(a), }$$ (4.1) where the function F is an antiderivative of the integrand f. What you are often not told is that there are many cases where F cannot be expressed in finite terms by means of elementary functions, and in such situations the formula (4.1) is useless for computational purposes. Examples are \(\int _{0}^{1}e^{-u^{2} }du\) and \(\int _{0}^{1}(\sin u)(u + 1)^{-1}du\). We then have to settle for numerical approximations of \(\int _{a}^{b}f(u)du\). The process of approximately computing definite integrals with a sufficient degree of precision is called numerical integration.