# application of oscillatory high-dimensional functions

Has anybody stumbled upon any kind of application of high-frequency high-dimensional problems ($d\geq 4$)?

My interest comes from the following: there is quite a decent amount of papers where people consider efficient representation of discretized integral operators in $d$ dimensions that allows to avoid a curse of dimensionality (tensor decomposition methods); namely, $d$-dimensional Laplace operator seems to be of an importance for quantum mechanics community. Is there any interest in working with $d$-dimensional Helmholtz equation, for example?

I heard as well that statistical applications require efficient representation of high-dimensional functions, but still, I have never seen anybody dealing with low-rank representations of multidimensional oscillatory functions unless it was a question of mainly theoretical interest.

• Just to clarify, are you asking for an alternative to FMM for high-frequency Helmholtz whose coefficient scales better than $\log(1/\epsilon)^{d-1}$? If so, I am not aware of any such algorithm. – Jack Poulson May 25 '12 at 15:36
• @JackPoulson Well, I would rather say that I am looking if somebody needs an analogue of fmm (in a slightly modified form ) for the problems whose dimension is bigger than 3. P.S. Since you've mentioned it, have you ever seen a proof that an interaction rank of two well-separable boxes (with a usual admissibility condition used in Greengard's classical Laplace FMM paper) scales as $O(\kappa)$ in 3 dimensions? – Mary May 25 '12 at 20:10
• @JackPoulson Thanks. It in fact assumes that the interaction rank is $O(k^2)$ rather than $O(k)$, but I am not sure if there is a version of fmm exploiting this. – Mary May 30 '12 at 19:57