On numerical methods for functions depending on a very large number of variables

The question under discussion is why do optimal algorithms on classes of functions sometimes become useless in practice. As an example let's consider the class of functions which satisfy a general Lipschitz condition. The methods of integral evaluation over a unit cube of $d$ dimensions, where $d$ is significantly large, are discussed. It is assumed that the integrand is square integrable. A crude Monte Carlo estimation can be used. In that case the probable error of estimation is proportional $1/\sqrt{N}$, where $N$ is the number of values of the integrand. If we use a quasi-Monte Carlo method instead of Monte Carlo one, then the error does not depend on the dimension $d$, numerous examples show that it depends on the average dimension $\hat{d}$ of the integrand. For small $\hat{d}$ the order of the error is close to $1/N$.