New development in eigenvector calculation?

[Earl Fairall]

Hi everyone,

I apologize this is not directly Deal.ii related, but I wanted to share it in case it contains something enlightening for deal.ii developers.

A friend of mine in the physics community shared this article:

https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/?fbclid=IwAR0BRt0HKUFwsOhpzb2gORhqnu159faVDPgt-_3mHtkJGCHZXiKKWct32so

I was hesitant to get excited about potential clickbait, but then Fermi National Accelerator Laboratory shared this article

https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/?fbclid=IwAR2Tu0yoKQKN9Y-69d718PebxDzwBKwBlTaxWEecwveTGeYpfCK6Pixn8Rw

My knowledge of eigenvector and eigenvalue calculation is very rusty, but I understand they are claiming to have simplified eigenvector calculation under certain conditions. Does anyone know what this means in a practical sense? Is this clickbait? Does anyone understand this article well enough to say if it has computational implications outside quantum mechanics?

Have a great weekend everyone,

Earl

[Wolfgang Bangerth]

Earll,

> I apologize this is not directly Deal.ii related, but I wanted to share it in
> case it contains something enlightening for deal.ii developers.
>
> A friend of mine in the physics community shared this article:

> [...]

>
> My knowledge of eigenvector and eigenvalue calculation is very rusty, but I
> understand they are claiming to have simplified eigenvector calculation under
> certain conditions. Does anyone know what this means in a practical sense? Is
> this clickbait? Does anyone understand this article well enough to say if it
> has computational implications outside quantum mechanics?

I'm no eigenvalue expert either, but it's an interesting formula. In general,
if Terence Tao speaks, it's worth listening :-)

As for practical impact, I think this comment is useful:
https://terrytao.wordpress.com/2019/08/13/eigenvectors-from-eigenvalues/#comment-529043

But at least as far as the community on this mailing list is concerned, it's
worth keeping in mind two issues:

* The matrices we have are sparse; that makes a lot of things more
complicated, but also some much easier.

* The matrices for which we consider eigenvalues are generally
finite-dimensional approximations of infinite-dimensional operators. That
means that in general only the matrix eigenvalues at either the low or the
high end of the spectrum are good approximations of the operator eigenvalues,
and those are the ones we're generally interested in. In other words, we
almost never compute all eigenvalues of a matrix because most of them are
meaningless. So we only care about a few eigenvalues and corresponding
eigenvectors (and have efficient Krylov space methods to compute them) -- but
then a formula that requires us to know *all* eigenvalues of a matrix (plus
all eigenvalues of all of the M matrices) is not going to be of great help to
our community.

Best
W.

--
------------------------------------------------------------------------
Wolfgang Bangerth email: bang...@colostate.edu
www: http://www.math.colostate.edu/~bangerth/

[Earl Fairall]

Thanks for the replay Wolfgang.

And no disrespect to Terry Tao. I had never heard of him before these articles. I guess I am pretty disconnected from the math community!

You successfully answered all my questions; however, you now have me curious why the physics community would want to know all (or a larger portion of) the eigenvalues.  If they're using these methods to somehow characterize the wide frequency range of light or radiation, I can imagine that would get nasty very quickly!

Kind regards,

Earl

[Wolfgang Bangerth]

On 11/20/19 9:53 PM, Earl Fairall wrote:
>
> You successfully answered all my questions; however, you now have me curious
> why the physics community would want to know all (or a larger portion of) the
> eigenvalues.  If they're using these methods to somehow characterize the wide
> frequency range of light or radiation, I can imagine that would get nasty very
> quickly!

In the context of the calculations that these people cared about, their
matrices were 3x3 or 8x8 or something similarly small. Their matrices are not
finite-dimensional approximations of infinite-dimensional operators, and so
the eigenvalues have physical reality. On the other hand, in the finite
element context, our matrices are approximations of infinite-dimensional
operators, and so only those matrix eigenvalues that approximate operator
eigenvalues well have physical relevance -- these are generally the first few
(dozens, hundreds) smallest or largest eigenvalues of a matrix.