  This paper deals with the equation $-\Delta u+\mu u=f$ on high-dimensional
spaces $\mathbb{R}^m$, where the right-hand side $f(x)=F(Tx)$ is composed of a
separable function $F$ with an integrable Fourier transform on a space of a
dimension $n>m$ and a linear mapping given by a matrix $T$ of full rank and
$\mu\geq 0$ is a constant. For example, the right-hand side can explicitly
depend on differences $x_i-x_j$ of components of $x$. We show that the solution
of this equation can be expanded into sums of functions of the same structure
and develop in this framework an equally simple and fast iterative method for
its computation. The method is based on the observation that in almost all
cases and for large problem classes the expression $\|T^ty\|^2$ deviates on the
unit sphere $\|y\|=1$ the less from its mean value the higher the dimension $m$
is, a concentration of measure effect. The higher the dimension $m$, the faster
the iteration converges.
