Iterations, Tensors, and beyond

This talk deals with the equation $-\mathrm{\Delta }u+\mu u=f$ on high-dimensional spaces ${\mathbb{R}}^m$, where $\mu$ is a positive constant. If the right-hand side $f$ is a rapidly converging series of separable functions, the solution $u$ can be represented in the same way. These constructions are based on approximations of the function $1/r$ by sums of exponential functions. I will present results of similar kind for more general right-hand sides $f(x)=F(Tx)$ that are composed of a separable function on a space of a dimension $n$ greater than $m$ and a linear mapping given by a matrix $T$ of full rank. These results are based on the observation that in the high-dimensional case, for $\omega$ in most of the ${\mathbb{R}}^n$, the euclidean norm of the vector $T^t\omega$ in the lower dimensional space ${\mathbb{R}}^m$ behaves like the euclidean norm of $\omega$. This effect has much to do with the random projection theorem, which plays an important role in the data sciences, and can be seen as a concentration of measure phenomenon.