2
$\begingroup$

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

Improve this question
$\endgroup$
4
  • $\begingroup$ 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
1
$\begingroup$

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.

Improve this answer
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.