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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.

Thank you in advance

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  • $\begingroup$ 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. $\endgroup$ – Jack Poulson May 25 '12 at 15:36
  • $\begingroup$ @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? $\endgroup$ – Mary May 25 '12 at 20:10
  • $\begingroup$ I don't have access to the article from where I am right now, but I would assume that you could find such a proof either in this paper or one of its references. $\endgroup$ – Jack Poulson May 28 '12 at 15:48
  • $\begingroup$ @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. $\endgroup$ – Mary May 30 '12 at 19:57
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Semiclassical methods in quantum mechanics based on the path integral (Herman--Kluk formula) lead to highly oscillatory integrals. See, e.g., http://arxiv.org/pdf/0908.0847v1.pdf , which looks at stationary path approximations to such integrals. For related papers, look into http://scholar.google.com for citations of reference [19] in this paper.

I never saw low rank approximations to such integrals, and do not know whether these would be useful.

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