[Trilinos-Users] PyTrilinos + Anasazi - Generalized eigenproblem
Marko Budisic
mbudisic at engineering.ucsb.edu
Sat Dec 20 18:16:21 MST 2008
Dear Chris,
thank you for your prompt response. Let me state at the beginning that I
will usually be working with N on the order of 1e3 up to possibly 1e5.
So I thought Anasazi would be a good pick. I was using a 5-dim problem
because I could easily (and quickly) generate something that resembled
my intended application and check whether I was using the code properly.
Now I understand that some of the problems were rooted in the small size
of the operator. I'm including the rest of the responses where
appropriate below.
Chris Baker wrote:
> Dear Marko,
>
> 1) It seems that the setM() operation is ignored by the code, i.e.
> setting setM(D) still computes just the regular eigenproblem, instead of
> the generalized one.
>
>
> This should not be the case. I'll look into it some more. To be clear,
> you are saying that you get the exact same eigenvectors and eigenvalues
> whether or not you include the call to setM() ? My recollection of the
> code is that it should have the intended effect, of scaling the
> eigenvalues (the resulting eigenvectors will probably not be unit
> length; I don't recall).
Correct. Even after passing a non-identity matrix to setM(), I get an
eigenvector and eigenvalue of the non-generalized problem (due to
problems below I couldn't compute any other eigenvectors, I will try
generating something larger and let you know). Since I was using the LM
and LR settings to check this, I get the evector corresponding to evalue
4.3... The eigenvector is of unit length.
>
>
>
> 2) Whenever I try to set Nev > 1, e.g. Nev = 2, the solver complains
> about orthogonality constraints producing the following error:
>
> "Anasazi::BlockKrylovSchurSolMgr: Potentially impossible orthogonality
> requests. Reduce basis size."
>
> I am guessing there are some issues with my setting the blocksize. I am
> passing a randomized storage multivector to setInitVec(). I have tried
> setting the size of multivector to both 1 and to Nev (the number of
> eigenvalues requested) and in both cases I get the above complaint.
>
> Could anyone shed some light on how one would go about figuring the
> right blocksize for this problem and in general?
>
>
> The operation of this solver is that blocks of vectors (of size "Block
> Size") are added to an orthonormal basis up to a certain basis size
> ("Block Size" times "Number of Blocks"). At that point, the basis is
> restarted. However, the restart must be large enough to capture at least
> the number of desired eigenvalues ("NEV"). Lastly, fundamental linear
> algebra dictates that the size of the basis cannot exceed the length of
> the vector space (for this problem, 5).
>
> The default number of blocks for this solver manager is 3. This is the
> minimum that you can use for this solver. Typically, one uses a
> significantly larger basis. In your case, the block size and basis size
> depend on how many eigenvectors you want to compute, but my advice is
> the following: if you will only be solving 5x5 eigenvalue problems, you
> don't want to use Anasazi. In particular, a current limitation of the
> BlockKrylovSchur eigensolver is that the basis can only be run up to
> size n-1, where n is the size of the linear operator (in your case,
> n=5). This makes it impossible (even using auxilliary vectors) to
> compute all eigenvalues of a sparse operator.
So if I had a larger operator size (say 1e3 dimensions), to compute the
first say 10 evalues/evectors using the default options, it should be
sufficient to pass 10 to setNev() and pass a multivector of size 1 to
setInitVec()?
>
>
> 3) I am setting only "Block Size", "Which" and "Verbosity" options for
> the parameter list passed to BlockKrylovSchurSolMgr. Should other
> options relating to blocksize be set too? Should I set even the "Block
> Size" manually?
>
>
> You are welcome to manually set the block size, but for this problem,
> you can't. Block size larger than one will build a basis larger than 6,
> which is not possible for a 5-dimensional vector space.
>
>
> 4) Should AuxVec also be set prior to solving the problem/is there a
> benefit for doing so? If yes, how would one go about choosing the size
> of this multivector and should it be initialized in a certain way?
>
>
> Auxilliary vectors define a space which the solver should avoid. This is
> useful for restarting the eigensolve if some number of eigenvector are
> already known. I apologize; this is not documented very well.
I have just remembered that I tried passing a randomized MultiVector of
size 1 to setAuxVec() just to see what happens for the above problem.
The code seemed to freeze and after 10 seconds or so I killed the
process. I understand that this might again be due to the small size of
the problem I used but perhaps the code could made to fail with some
error in future, to facilitate debugging.
>
>
> In the end, I am including my python code for this problem. At the very
> end, numpy.linalg is used to verify the results obtained by the Anasazi.
>
>
> If you are going to continue solving eigenvalue problems on the order of
> tens or hundreds of variables, or if you are going to compute a large
> proportion of the eigenvalues, I suggest you use dense eigensolvers like
> those provided by numpy.
>
> Please respond with more questions, or if we can provide any other advice.
>
> Best regards,
> Chris Baker
>
Thank you for the detailed explanation of how the code works. Perhaps
some of it could be added to the documentation in upcoming versions of
Anasazi, I believe I understand how the code works much better now.
Best,
Marko Budisic
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