# [petsc-users] EPSSolve finds eigensolutions in deflation space

Giacomo Mulas giacomo.mulas84 at gmail.com
Sun May 8 06:20:49 CDT 2016

```I found a problem in the default iterative eigensystem solver of SLEPC,
namely the Krilov-Schur one: it does not respect the deflation space that it
is given.  This makes it very inefficient even in a case in which it should
be extremely effective.

The problem I want to solve: I have a symmetric operator (an approximated
vibrational Hamiltonian, but that's unimportant) and I want to selectively
find a subset of eigenvectors so that some predefined vectors (which I call
"target" vectors) are very nearly contained in the eigenvector subspace.

I therefore have both a well defined selection criterion (find eigenstates
which have maximum projection on the subspace of "target" vectors) and a
well defined condition for stopping (say, >99% of each of the "target"
vectors is contained in the subspace of the eigenstates found).

The first one can be explicitly cast in a function and given to SLEPC via
EPSSetArbitrarySelection and EPSSetWhichEigenpairs, so that the solver is
guided to the solutions I am most interested in.  I also make use of
EPSSetInitialSpace to have the solver start from a guess which is "rich" in
the "target" vectors (just the "target" vector which has the smallest
projection on the eigenstate space so far).

As to stopping when I found just enough eigenstates, unfortunately it is
thus far impossible to give a custom function to slepc so that it will keep
finding eigenvectors until all are found or the user-defined function tells
it it's enough.  Therefore, my second best solution was to repeatedly run
EPSSolve, aiming at a small number of eigensolutions in each call, putting
previously found solutions in the deflation space and passing it to the
solver via EPSSetDeflationSpace.  Unfortunately, this did not simply work
out of the box.

The first problem is that after the first few, easiest to find
eigensolutions are found, EPSSolve frequently does not find any solutions,
if run with a very conservative EPSSetDimensions (e.g.  like "find me just
one eigensolution").  I thus wrote my code so that the number of requested
solutions in EPSSolve are increased if no new solutions are found by an
EPSSolve call, and decreased if too many were found in just one call.
Inefficient as it may be, due to many useless EPSSolve calls, it works.

However, when trying to use this on real cases, another problem arose, the
big one: frequently, EPSSolve would find and return eigenvectors that were
in the deflation space!  Some eigensolutions would therefore appear more
than once in the set of solutions, of course breaking everything down the
road of my physics calculation before I figured out why.
When I found out, I set up a check in my code that explicitly computes the
projection of newly found eigenstates on previously found ones, and discards
them if they are duplicates.  But this is ineffective for several reasons:
first, such a check is a waste of time and should not be necessary,
otherwise what is the point in setting up a deflation space?  second, and
most important, I have to run EPSSolve looking for lots of solutions to only
find one or two really new ones, because all the first ones are duplicates.
Even if the initial guess is rich in the solution I actually want, and poor
in the duplicate ones. As a result, even if I would just need to find a
(relatively) small subset of the eigenvectors, lapack turns out to be always
much faster, which should not be the case.

I am attaching a test code that shows this behaviour, in the hope that it
will help the SLEPC developers in finding why the deflation space is not
respected and, if possible, correct it.  I know that being able to reproduce
a problem is the first, sometimes actually the most important, step to
attempt to solve it, so there it goes.  Of course I also have a tiny hope
that they might actually include support for a user-defined check for
"enough solutions" in EPSSolve...  There is in fact something not too
different already implemented, for EPSSolve can find all solutions with
eigenvalues in a given interval, without knowing in advance how many they
will be, without the user having to loop over many tentative EPSSolve calls
and finding lots of unneeded additional solutions in the process.

To run the test code, just compile the source, and run it in the same
directory of the two data files.  This is a small test case, so that it runs
fast enough, but I have much larger ones and can produce them more or less
as large as wanted on request.  Upon running the code, you will see that
lots of tentative EPSSolve calls are made, many times finding no solutions
or only solutions already in the deflation space (which contains previously
found eigenvectors).
In this code I privileged clarity over efficiency, no attempt was made to
overoptimise anything.  I put lots of comments in it, so that hopefully it
is self-explanatory.  But of course I am more than willing to explain
whatever is not clear in it.  Just ignore the part that reads the input
files, that's irrelevant.

Of course, I would also very much appreciate it if I could get some clever
suggestions, from SLEPC developers or anyone else on the list, as to how to
solve my problem more efficiently.  And of course I am perfectly willing
to thank them and/or to include them as authors in the scientific publication
describing my molecular spectroscopy code, if some substantial contribution
is offered.

Best regards
Giacomo Mulas

--
_________________________________________________________________

Giacomo Mulas <gmulas at oa-cagliari.inaf.it>
_________________________________________________________________

INAF - Osservatorio Astronomico di Cagliari
via della scienza 5 - 09047 Selargius (CA)

tel.   +39 070 71180244
mob. : +39 329  6603810
_________________________________________________________________

"When the storms are raging around you, stay right where you are"
(Freddy Mercury)
_________________________________________________________________
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